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Review Acer Aspire One Cloudbook 14 AO1-431-C6QM Review Case amp Konektivitas Desain, material, dan warna dari Cloudbook 14 identik dengan saudaranya yang berukuran 11,6 inci. Terlepas dari bingkai layar hitamnya, Acer menggunakan plastik tipis berwarna abu-abu. Tutup dan bagian bawah bertekstur. Sisi atas unit dasar kasar. Kualitas membangunnya tanpa cela. Namun, stabilitas harus ditingkatkan - unit dasar twists terlalu mudah. Dengan demikian, bagian depan kaki kiri laptop diangkat saat tekanan diaplikasikan tepat disamping keyboard. Selain itu, Cloudbook 14 dan Cloudbook 11 berbagi konektivitas yang sama. Salah satu dari dua port USB mendukung standar USB 3.0. Port duduk di kedua sisi ke arah belakang. Akibatnya, ruang di samping telapak tangan tetap bebas dari kabel. Pembaca kartu memori bekerja perlahan. Ini mentransfer 250 file gambar JPG (sekitar 5 MB masing-masing) hanya 18,6 MB. Kami selalu menguji pembaca kartu memori dengan kartu referensi kami (Toshiba Exceria Pro SDXC 64 GB UHS-II). Display Response Times Waktu respon display menunjukkan seberapa cepat layar bisa berubah dari satu warna ke warna berikutnya. Waktu respon yang lambat bisa menyebabkan afterimages dan bisa menyebabkan benda bergerak tampak buram (ghosting). Gamer dari judul 3D yang serba cepat harus memberi perhatian khusus pada waktu respon yang cepat. Harr Waktu Respon Hitam ke Putih 21,6 ms. Naik dan turun gabungan 8599 kenaikan 16,4 ms 8600 5,2 ms jatuh Layar menunjukkan tingkat respons yang baik dalam pengujian kami, namun mungkin terlalu lambat bagi gamer yang kompetitif. Sebagai perbandingan, semua perangkat yang diuji berkisar dari 0,8 (minimum) sampai 240 (maksimal) ms. Raquo 14 dari semua perangkat lebih baik. Ini berarti bahwa waktu respon terukur lebih baik daripada rata-rata semua perangkat yang diuji (27,4 ms). Harr Waktu Respon 50 Grey sampai 80 Grey 49,6 ms. Naik dan turun gabungan 8599 23,6 ms naik Perangkat Penyimpanan Cloudbook dilengkapi dengan modul penyimpanan eMMC dengan kapasitas 32 GB. Hanya di bawah 15 GB dari total kapasitas bebas out-of-the-box. Tingkat transfer berada pada tingkat normal untuk jenis memori ini. Hal ini tidak berguna untuk memperluas kapasitas penyimpanan dengan kartu SD. Kartu SD keluar dari pembaca kartu sekitar 5 mm dan tidak terkunci di pembaca kartu. Namun demikian, pengguna akan memiliki cukup ruang penyimpanan yang tersedia: Pembeli komputer mendapatkan 1 ruang penyimpanan TB dari layanan MicrosoftS Cloud OneDrive (gratis selama satu tahun). Graphics card Graphics dilakukan oleh Intels HD Graphics (Braswell) GPU. Ini mendukung DirectX 12 dan bekerja sampai 600 MHz. Inti grafis Braswell secara signifikan lebih cepat daripada pendahulunya Bay Trail. Akibatnya, kinerjanya jauh lebih baik dalam benchmark 3DMark - meskipun pada tingkat rendah. Meskipun Cloudbook hanya membawa prosesor yang lemah, namun bisa digunakan untuk pemutaran video. Ini berkat decoder yang terintegrasi ke dalam GPU, yang mengambil beban dari prosesor saat memutar video. Ini mendukung semua format biasa termasuk H.265 (pengganti format H.264, yang tersebar luas hari ini), yang memungkinkan file berukuran lebih kecil dengan kualitas gambar yang sama. Beban CPU di bawah 20 saat kami memutar video uji kami (4k, H.265, 60 fps). 3DMark 11 Performance 3DMark Ice Storm Standard Score 3DMark Cloud Gate Skor Standar 3DMark Fire Strike Score 3DMark 11 - 1280x720 Kinerja Lenovo E31-70 80KX015RGE HD Graphics 5500, 5005U, Seagate ST500LM000 Solid State Hybrid Drive 978 Poin sim100 126 Acer Aspire ES1-331-C5KL HD Graphics (Braswell), N3150, Seagate Momentus Tipis ST500LT012-1DG142 462 Poin sim47 7 Acer Aspire ES1-420-377F Radeon HD 8240, E1-2500, Western Digital Scorpio Blue WD5000LPVT 453 Poin sim46 5 Grafik HD Acer Extensa 2519-C7DC ( Braswell), N3050, Seagate Momentus Tipis ST500LT012-1DG142 442 Poin sim45 2 433 Poin sim44 HP Stream 13-c102ng HD Graphics (Braswell), N3050, 32 GB eMMC Flash 310 Poin Sim32 -28 1920x1080 Skor Strike Api Acer Aspire ES1-420- 377F Radeon HD 8240, E1-2500, Digital Digital Scorpio Blue WD5000LPVT 259 Poin sim100 7 241 Poin sim93 Acer Extensa 2519-C7DC HD Graphics (Braswell), N3050, Seagate Momentus Tipis ST500LT012-1DG142 236 Poin sim91 -2 Acer Aspire ES1-331 -C5KL HD Grafik (Braswell), N3150, Seagate Momentus Tipis ST500LT012-1DG142 219 Poin sim85 -9 1280x720 Skor Langit Diver Acer Aspire ES1-331-C5KL HD Graphics (Braswell), N3150, Seagate Momentus Tipis ST500LT012-1DG142 1058 Poin sim100 5 Acer Extensa 2519 -C7DC HD Graphics (Braswell), N3050, Seagate Momentus Tipis ST500LT012-1DG142 1032 Poin sim98 3 1003 Poin sim95 1280x720 Nilai Standar Awan Ganda Acer Aspire ES1-331-C5KL HD Graphics (Braswell), N3150, Seagate Momentus Tipis ST500LT012-1DG142 1879 Poin sim100 22 1542 Poin sim82 Acer Extensa 2519-C7DC HD Graphics (Braswell), N3050, Seagate Momentus Tipis ST500LT012-1DG142 1529 Poin sim81 -1 Grafik HD 13-c102ng HD HD (Braswell), N3050, 32 GB eMMC Flash 1404 Poin sim75 -9 1280x720 Skor Standar Badai Es Acer Aspire ES1-331-C5KL HD Graphics (Braswell), N3150, Seagate Momentus Tipis ST500LT012-1DG142 21443 Poin sim100 18 Grafis Acer Extensa 2519-C7DC HD (Braswell), N3050, Seagate Momentus Tipis ST500LT012- 1DG142 18186 Poin si M85 0 18186 Poin sim85 Acer Aspire ES1-420-377F Radeon HD 8240, E1-2500, Western Digital Scorpio Blue WD5000LPVT 15693 Poin sim73 -14 HP Stream 13-c102ng HD Graphics (Braswell), N3050, 32 GB eMMC Flash 15693 Poin sim73 -14 Gaming Performance Games yang cocok untuk Cloudbook dapat ditemukan di bagian game kasual di Windows Store. Perangkat kerasnya terlalu lemah untuk game modern seperti Hitman. Bahkan game seperti FIFA 16 dengan hanya persyaratan hardware moderat, jangan sampai frame rate yang mulus. Selain itu, Cloudbook tidak menawarkan kapasitas penyimpanan yang cukup untuk kebanyakan game. Cloudbook 14 inci memberikan kinerja yang cukup untuk aplikasi kantor dan internet sederhana. Berkat decoder yang terintegrasi di inti grafis, komputer juga bisa memutar video. Berbeda dengan pesaing Chromebook-nya, Cloudbook mengizinkan penginstalan menggunakan perangkat lunak desktop biasa. Namun, kapasitas penyimpanannya terbatas. Sekitar 15 GB bebas out-of-the-box. Cloudbook bekerja tanpa suara dan hampir tidak terasa hangat. Layar tidak akan memenangkan hadiah apapun, tapi tidak apa-apa mengingat titik harga. Ini memberikan kontras yang dapat diterima dan permukaan matte. Namun, kecerahan harus lebih tinggi. Cloudbook 14 dimaksudkan untuk menjadi mesin ketik dan mesin penjelajah seluler. Komputer sepenuhnya memenuhi profil aplikasi ini. Masa pakai baterai cukup untuk satu hari di universitas sekolah. Hal ini dilengkapi dengan keyboard yang layak. Selain itu, lisensi satu tahun untuk MS Office 365 Personal disertakan. Dengan harga sekitar 280 Euro (310), bundelnya terjangkau. Selamat datang di situs Alberts. Tempat untuk mendapatkan informasi yang sangat sederhana namun gratis. Its juga sedikit situs hobi, tapi saya mencoba untuk menempatkan barang-barang yang layak di sini. Anda bisa menemukan beberapa dokumen baru di sini, tapi saya juga takut pada beberapa orang tua. (Maaf). Tapi semua entry level nya. Hal yang saya tulis sendiri, benar-benar gratis (yaitu: jika Anda menyukainya, gunakan seluruhnya, atau copypaste hanya bagian, atau apapun yang Anda inginkan). Alberts Series: Barang dasar, tapi serius. I. Catatan Matematika pendek dan sederhana: (R: siap, D: dalam pembangunan) II. Catatan yang agak lebih besar (tapi sederhana) dari Fisika: (R: siap, D: dalam pengembangan) III. Catatan TI pendek dan sederhana: (R: ready, D: in development) Dokumen lain: -Fun stuff: -Lainnya: -DutchNederlands: EPR dan Steering in Quantum Mechanics (QM). Versi: 16 Februari, 2017. Status: Hampir selesai. Beberapa gagasan tentang Quantum Entanglement dan non-lokalitas ditemukan kembali dalam 15 atau 20 tahun terakhir (atau lebih), sebagian didasarkan pada gagasan Schrodinger pada kemudi EPR, yang dinyatakan pada tahun 1935. Memang ada perbedaan halus antara, apa yang kita Menggambarkan sebagai belitan, Bell non-lokalitas, dan kemudi. Saya suka mengatakan sesuatu pada topik itu, karena hal-hal yang benar-benar bermanfaat. Jadi, jika Anda agak tidak terbiasa dengan topik seperti itu, catatan ini mungkin menarik. Lima bab pertama akan menjelaskan beberapa efek keterikatan yang diketahui dengan baik, yang secara tradisional mengarah pada paradigma EPR yang terkait. Jadi, kelima bab pertama ini agak tua-skool, kurasa. Sementara setelah banyak usaha baru-baru ini, dan, habiskan untuk menemukan esensi kemudi. Belitan, dan nonlokalitas. Sekarang tampaknya pandangan yang berkembang di tahun-tahun sebelum tahun 90an, mungkin memerlukan beberapa revisi, terutama karena semua penelitian sejak tahun 2000an. Namun, lima bab pertama akan mempresentasikan ide skool lama (pra 90an) lebih dulu, karena cara ini mungkin masih merupakan praktik terbaik untuk menyajikan materi semacam itu. Bab 1 adalah tentang paradigma EPR yang terkenal, dan fenomena itu adalah tema penting untuk seluruh teks ini. Di bab awal ini, keterikatan, dan bahkan mungkin kemudi, akan disajikan dalam kasus yang aneh. Kemudian, bab 2 akan menghabiskan beberapa kata di Quantum Teleportation (QT), yang merupakan efek nyata. Di sini juga, keterkaitan dan kemungkinan non-lokalitas ditunjukkan. Bab 3 dan 4 dapat dipandang sebagai pengantar nano untuk QM, menunjukkan konsep dan notasi paling dasar yang digunakan di QM. Ini bisa membantu jika Anda suka membaca artikel profesional mengenai berbagai topik. Bab 5 akan menjadi intro pendek untuk ketidaksetaraan Bell, yang sebenarnya adalah kriteria stokastik untuk memutuskan apakah hasil pengukuran disebabkan oleh efek non-lokal QM, disebabkan oleh mekanisme lokal, seperti variabel Tersembunyi, yang sebenarnya mewakili Pandangan realistis lokal tentang kenyataan. Di Bab 6, saya akan mencoba menjelaskan beberapa spesifikasi kemudi EPR, namun sifatnya sangat ringan, dan hanya bisa menarik perhatian jika Anda benar-benar tidak mengenal subjek ini. Tak pelak lagi, masalah yang berkaitan dengan interpretasi QM harus ditangani: Di â€‹â€‹Bab 7, saya akan mencoba mengatakan sesuatu yang berguna mengenai masalah pengukuran, dan peran pengamat. Di Bab 8, saya ingin menyentuh beberapa gagasan radikal mengenai interpretasi QM, yaitu beberapa teori paralel baru seperti MIW (dan juga teori MWI Everetts yang lebih tua). Bab 10, adalah intro yang sangat sederhana pada beberapa gagasan baru, bahwa SpaceTime sendiri dapat dibangun dari keterikatan, dan juga keterkaitan itu mungkin memainkan peran besar dalam model gravitasi baru. Di sini, satu tema sentral adalah paradigma EREPR. Saya pribadi berpikir bahwa ini adalah jalur yang menjanjikan dalam fisika. Apa pun yang Anda pikirkan. Tentu cukup spectecular. Karena keterbatasan saya sendiri, keseluruhan teksnya cukup mendasar, dan hampir secara eksklusif menggunakan (seperti yang kita katakan dalam bahasa Belanda) semacam Jip en Janneke bahasa. Maaf untuk itu Jadi, Anda diperingatkan Tetapi jika Anda ingin mencobanya pula. Maka mari kita lihat apa ini semua tentang. Isi sekilas pandang: Bab 1. Pengantar paradigma EPR. Bab 2. Beberapa kata pada artikel Teleportasi Kuantum yang asli. Bab 3. Pengantar nano untuk QM: beberapa konsep dan notasi penting di QM. Bab 4. Into nano untuk QM: keadaan murni dan negara campuran. Bab 5. Ketidaksamaan teorema Bell, atau Bells. Bab 6. Beberapa kata tentang Steering, Entanglement, dan Bell non-locality. Bab 7. Beberapa kata pada masalah pengukuran. Bab 8. Beberapa Penafsiran Dunia Banyak di QM. Bab 9. Beberapa kata pada entropi. Bab 10. Model dan ide EREPR yang lebih baru. Lampiran Pendek: peralatan A1: Radar Kuantum. Bab 1. Pengantar paradigma EPR. 1.1 Pendahuluan dan Beberapa informasi latar belakang. Kita tahu bahwa itu tidak diperbolehkan untuk informasi, untuk melakukan perjalanan lebih cepat dari c. Namun, ada beberapa situasi di Quantum Mechanics (QM), di mana tampaknya peraturan ini rusak. Saya segera tergesa-gesa untuk mengatakan, bahwa hampir semua fisikawan percaya bahwa peraturan tersebut masih berlaku, namun ada sesuatu yang lain sedang bekerja. Apa sebenarnya yang lain, belum sepenuhnya jelas, walaupun ada beberapa gagasan yang didanai dengan baik, memang ada. QM menggunakan beberapa rasa untuk menggambarkan entitas secara matematis (misal: partikel), properti (misal: posisi, putaran), dan peristiwa. Salah satu rasa yang sering digunakan adalah Dirac (vektor) notasi. Di bab 3, untuk orang-orang yang tidak terlalu tertarik dengan QM, saya akan memperkenalkan beberapa hal menarik dari QM, dan fungsi gelombangnya, dan sesuatu pada notasi Dirac. Untuk saat ini, bagaimanapun, bahkan jika Anda hanya memiliki gagasan dasar tentang vektor, maka kita akan menempuh perjalanan jauh dalam bab ini. Misalkan kita memiliki partikel tertentu yang dapat diamati (propery). Misalkan yang ini bisa diamati memang bisa diukur dengan beberapa alat ukur. Untuk tujuan teks semacam ini, putaran partikel sering digunakan sebagai karakteristik yang dapat diamati. Putaran ini, menyerupai momentum magnet sudut, dan bisa naik (atau sering ditulis sebagai atau 1), atau turun (atau ditulis sebagai - atau 0), bila diukur sepanjang arah tertentu (seperti misalnya sumbu z di R 3. Sebagian besar fisikawan menyepakati fakta bahwa kerangka kerja QM secara intrinsik probabilistik. Artinya, jika putaran partikel tidak terukur, putarannya merupakan kombinasi linier dari atas ke bawah pada waktu yang bersamaan. Sebenarnya, ini menyerupai Vektor dalam ruang 2 dimensi (sebenarnya bola Bloch 3D), jadi secara umum, keadaan seperti itu dapat ditulis sebagai: 966 a1 b0 (persamaan 1) (Catatan: argumen dapat ditemukan untuk berbicara tentang bola Bloch 3D, tapi kami tidak Pikiran tentang ini, pada saat ini.) Dimana 1 dan 0 mewakili vektor basis superposisi semacam itu. Ini memang luar biasa dengan sendirinya Tapi beginilah cara kerjanya dalam kerangka QM. Ada kemungkinan menemukan 9661 dan sebuah Probabilitas tertentu untuk menemukan 9660 saat pengukuran dilakukan. Untuk probabilitas tersebut, harus dilakukan ho Jika 2 b 2 1, karena total probabilitasnya harus bertambah menjadi satu. By the way, sistem yang dijelaskan dalam persamaan 1, sering disebut qubit, sebagai bit Quantum Mechanical di Quantum Computing. Demikian pula, qutrit dapat ditulis sebagai kombinasi linear 0 dan 1 dan 2, yang merupakan tiga basis ortogonal. Semuanya benar-benar terlihat seperti di kalkulus vektor. Qutrit dapat ditunjukkan oleh: 966 a0 b1 c2 (persamaan 2) Namun, dalam kebanyakan diskusi, qubit seperti pada persamaan 1, memainkan peran sentral. Sekarang, misalkan kita memiliki dua sistem yang tidak berinteraksi dua qubit966 1 dan 966 2 (berdekatan). Kemudian keadaan gabungan, atau keadaan produk mereka (yaitu: bila tidak terjerat), dapat dinyatakan dengan: 936 966 1 8855 966 â€‹â€‹2 a 00 00 a 01 01 a 10 10 a 11 11 (persamaan 3) Negara tersebut adalah Juga disebut terpisah, karena negara gabungan adalah produk dari masing-masing negara bagian. Jika Anda memiliki status produk seperti itu, kemungkinan untuk menentukan (atau memisahkan) setiap sistem individual dari gabungan persamaan. Catatan: Bila kita mengatakan persamaan 1, 966 a1 b0, itu berarti kombinasi linear dari atas dan ke bawah pada waktu bersamaan. Dengan demikian secara simultan, maka pernyataan ini merupakan interpretasi umum dalam QM. Kebanyakan orang tidak mempertanyakan penafsiran ini. Namun, beberapa orang masih melakukannya. Dalam beberapa kasus, persamaan seperti persamaan 3, tidak bekerja untuk sistem gabungan partikel. Dalam kasus seperti itu, partikel saling terkait, dan sedemikian rupa, bahwa pengukuran pada satu, mempengaruhi keadaan yang kedua. Pernyataan yang terakhir sangat luar biasa, dan inilah yang sekarang orang sebut kemudi, sebagai bagian khusus dari keterikatan istilah yang lebih umum. Misalkan kita memulai dengan sistem kuantum dengan spin 0. Sekarang anggap lebih jauh bahwa ia meluruh dalam dua partikel. Karena putaran total adalah 0, pasti benar bahwa jumlah putaran partikel baru juga nol. Tapi, kita tidak bisa mengatakan bahwa seseorang pasti berputar, dan yang lainnya pasti berputar ke bawah. Namun, bisa kita katakan bahwa kombinasi mereka membawa spin nol. Ini mungkin tampak aneh, tapi cara yang baik untuk menunjukkan pernyataan sebelumnya, adalah dengan persamaan berikut: 936 187302. (01 10) (persamaan 4) Perhatikan bahwa ini adalah superposisi dua negara, yaitu 01 dan 10. Dalam pembicaraan QM , Kita mengatakan bahwa kita memiliki probabilitas 189 untuk mengukur 01 untuk kedua partikel, dan juga probabilitas 189 untuk mengukur 10 untuk kedua partikel. Artinya, setelah pengukuran. Perhatikan bahwa sebuah ekspresi seperti 10 sebenarnya mengatakan bahwa satu partikel ditemukan ada, sementara yang lain ditemukan turun. Tapi superposisi berarti, bahwa kedua partikel bisa berada dalam keadaan apapun, pada saat bersamaan. Sebelum pengukuran, kita sama sekali tidak tahu. Kita hanya tahu bahwa 936 adalah 936. 1.2. Menggambarkan kasus aneh paradoks EPR. Perspektif yang lebih historis disajikan pada bagian 1.3. Tapi saya suka langsung terjun ke apa yang dikenal sebagai paradoks EPR. Kasus berikut menggunakan dua orang, yaitu Alice dan Bob, masing-masing di lokasi terpencil yang terpisah. Masing-masing memiliki satu partikel anggota, dari sistem terjerat (seperti persamaan 4), di laboratorium mereka. Deskripsi kasus di bawah ini, bukan tanpa kritik: - Salah satu pertanyaan penting adalah, jika seseorang mempertimbangkan kasus ini dengan menggunakan interpretasi negara bagian murni, atau interpretasi negara campuran. Untuk lebih memahami hal itu, seharusnya kita sudah tahu telah mempelajari bab 4. Untuk saat ini, bagaimanapun, kita anggap remeh bahwa kasus ini berurusan dengan satu keadaan murni, seperti yang dijelaskan oleh persamaan 4. Namun, dalam bentuk eksperimen yang sebenarnya. Kita akan memiliki banyak eksperimen, dan dengan demikian mengenalkan sebuah ansambel. Ini juga berarti bahwa sampai tingkat tertentu, statistik dan korelasi juga ikut bermain. -Selanjutnya, saat pemisahan ruang-waktu meningkat, beberapa juga mungkin mengungkapkan keraguan tentang bagaimana sebenarnya (atau efektif) sebuah dekripsi sebagai persamaan 3 tetap merupakan kenyataan. Namun, argumen ini agak lemah. Keterikatan benar-benar cukup mapan, dan dikonfirmasi, bahkan dalam jarak yang jauh seperti dalam eksperimen Teleportasi Kuantum. Hanya disipasi keadaan terjerat, karena interaksi dengan lingkungan (decoherence), bisa melemahkan atau menghancurkan keterikatan (bahkan mungkin sangat tiba-tiba). -Juga, orang mungkin berpendapat bahwa baik Alice atau Bob hanya akan mengukur atas atau bawah partikel anggotanya, dan tidak ada senar yang terpasang. Penglihatan turun ke Bumi menyatakan bahwa, tidak ada apa yang Bob lakukan, atau apa yang Alice lakukan (atau tindakan), akan mengubah sesuatu pada ukuran pribadi mereka. Orang mungkin menduga, bahwa hanya jika Alice memberi tahu Bob, atau sebaliknya, orang mungkin menemukan korelasi. Sudut pandang semacam itu mungkin mempersulit bagaimana menafsirkan hasilnya. Namun, kemudi temuan Alice, pada partikel anggota Bobs, diyakini benar, karena hasil eksperimen mendukung pandangan ini. Apapun yang benar Atau apa yang perlu kita perhatikan, saya ingin menyajikan kasus ini dalam bentuk aslinya. Namun, itu akan menjadi penyederhanaan gagasan asli, dan kemudian pembuatan eksperimental. Untungnya, cukup diterima untuk mempresentasikan kasus ini dengan cara ini. Lihat kembali persamaan 4. Kedua negara bagian, 01 dan 10, tampaknya memiliki probabilitas yang sama untuk menjadi kenyataan atau ditemukan, setelah pengukuran dilakukan. Jika pengukuran dilakukan, keadaan berkurang menjadi 01 atau 10, tergantung jarak apapun. Ini adalah jantung dari masalah appark. Catatan: Seluruh sistem tampaknya murni, dalam posisi superposisi, sementara setiap istilah tampaknya tercampur. Perbedaan akan disinggung di Bab 4 (pada keadaan murni dan campuran). Misalkan kita memiliki sistem etangled lagi, yang dapat digambarkan dengan persamaan 4. Sebelum kita melakukan pengukuran, misalkan kita memiliki cara untuk memisahkan dua partikel. Katakanlah jarak memisahkan partikel, menjadi sangat besar. Alice ada di lokasi 1, tempat partikel bergerak, dan Bob berada di lokasi 2, di mana partikel 2 baru saja tiba. Sekarang, Alice melakukan pengukuran untuk menemukan spin partikel 1. Hal yang menakjubkan adalah, bahwa jika dia mengukur sepanjang sumbu tertentu, maka Bob harus menemukan sumbu yang sama. Jangan terlalu memikirkan hal ini. Kami memulai dengan mengatakan bahwa (dalam bahasa QM), kedua negara bagian, 10 dan 01, memiliki probabilitas yang sama untuk menjadi kenyataan. Total keadaan, selalu merupakan superposisi 01 dan 10 dimana masing-masing memiliki probabilitas yang sama. Bagaimana partikel 2 tahu, partikel itu ditemukan oleh Alice, berada di dalam keadaan 1, sehingga partikel 2 sekarang tahu bahwa itu pasti 0 Anda mungkin mengatakan bahwa partikel 1 dengan cepat menginformasikan partikel 2 keadaan. Tapi ini menjadi sangat aneh jika jarak antara kedua partikel begitu besar, sehingga hanya sinyal (semacam) yang lebih cepat dari kecepatan cahaya, yang terlibat. Itu tentu saja tidak masuk akal. (1). Persamaan 4 bukan merupakan keadaan campuran socalled. Ini adalah superposisi, dan keadaan murni. Probabilitas dihitung dengan keadaan campuran, sedikit berbeda dibanding keadaan murni murni. Lihat bab 3 dan 4 untuk perbandingan antara negara campuran dan murni. (2). Paradoks EPR pertama kali disadari oleh Einstein, Podolski dan Rosen (1935). Informasi lebih lanjut dapat ditemukan di banyak bagian di bawah ini. Paradoks yang tampak jelas adalah bahwa pengukuran pada salah satu partikel tampaknya meruntuhkan keadaan 936 187302. (01 10), sehingga keseluruhan sistem yang terjerat, menjadi 01 atau 10. Namun, superposisi (persamaan 4) telah berlaku, semua Waktu. Mengapa keruntuhan itu, yang selalu menentukan keadaan partikel kedua. Efeknya, jika Anda mengamati satu partikel di sepanjang sumbu pengukuran, maka yang satunya selalu ditemukan sebaliknya. Hal ini tampaknya terjadi seketika. Yang mana kami tidak memiliki penjelasan klasik. Efeknya telah dikonfirmasi secara eksperimental, oleh Stuart Freedman (et al) di awal 70an, dan yang cukup terkenal adalah eksperimen Aspek pada awal tahun 80an. Namun, karena eksperimen itu bersifat statistik, mereka tidak sepenuhnya bebas. Nanti lebih lanjut tentang ini. Omong-omong, eksperimen bebas celah pertama dilakukan pada tahun 2015 (Delft), hampir meyakinkan untuk mengonfirmasi efek aneh seperti yang dijelaskan di atas. Dalam setting ini, benar-benar terlihat seolah Alice mengarahkan apa yang bisa Bob temukan. - Satu penjelasan (sementara) dengan konsensus tertentu di kalangan fisikawan: Tidak baik jika mempertahankan paradigma apparant, terbuka sepenuhnya, pada saat ini. Yang benar bahwa banyak rincian harus dikerjakan lebih jauh, karena semua deskripsi di atas, disajikan secara sangat sederhana dan tidak lengkap. Jika jarak antara Alice dan Bob cukup besar, maka jika Anda berasumsi bahwa partikel terukur pertama, menginformasikan partikel kedua yang harus diambilnya, maka sinyal semacam itu harus berjalan lebih cepat daripada kecepatan cahaya. Ini sangat tidak bisa diterima, bagi sebagian besar fisikawan. Pada tahun tiga puluhan dari abad sebelumnya, beberapa model diajukan (Einstein: lihat bab 2), yang teori variabel Tersembunyi (Lokal) adalah yang paling menonjol. Intinya begini: Saat ini pasangan yang terjerat (seperti pada bagian 1.2) dibuat, sebuah kontrak tersembunyi ada yang sepenuhnya menentukan perilaku mereka dalam kejadian yang tampaknya bukan lokal. Itu hanya karena kurangnya pengetahuan tentang variabel tersembunyi tersebut, yang membuat kita memikirkan tindakan seram dari kejauhan. Di satu sisi, hipotesis ini adalah kembalinya realisme lokal. - Sebuah pemahaman modern (tanpa konsensus penuh di kalangan fisikawan): Pemahaman modern terletak pada superposisi keadaan terjerat seperti yang diungkapkan oleh persamaan 4. Alice dapat mengukur qubitnya, dan dia menemukan keduanya naik atau turun, masing-masing memiliki probabilitas 50. Dia tidak tahu apa-apa tentang pengukuran Bob, jika dia benar-benar melakukan sesuatu terhadapnya. Satu interpretasi modern kemudian mengatakan, bahwa Alice tidak tahu pasti apa yang ditemukan oleh Bobs, atau akan. Kecuali jika Bob melakukan pengukurannya pada partikel anggotanya (sepanjang arah yang sama). Sekarang, keajaiban itu benar-benar ada dalam kata-kata kecuali jika Bob melakukan pengukurannya. Yang juga menyiratkan bahwa Alice dan Bob (di lain waktu) membandingkan hasilnya. Keajaiban ini berada pada keterikatan, atau non-lokalitas, di mana kedua istilah itu agak mirip, saat mempertimbangkan keadaan murni. Kini, para periset masih dihadapkan pada kerja keras yang luar biasa dari keterikatan, non-lokalitas, dan kemudi, yang saya harap catatan sederhana ini dapat menjelaskannya. Teka-teki lain, namun terkait: Saya juga harus menyebutkan, bahwa banyak periset mengatakan, bahwa untuk mengetahui bahwa kedua partikel benar-benar membentuk keadaan murni (seperti persamaan 4), Anda memerlukan akses ke kedua partikel tersebut. Jika Anda hanya bisa mengamati satu partikel, Anda akan secara efektif melihat entitas yang terlacak, yang, jika terjadi sistem seperti persamaan 4, akan mengurangi keadaan campuran. Jika seseorang mencoba memisahkan persamaan untuk satu partikel, dari pasangan yang terjerat, beberapa hasil aneh muncul ke permukaan. Dengan menggunakan operasi matematika yang baik, yaitu, melacak satu partikel dari persamaan 4, mengembalikan matriks kerapatan yang dikurangi (lihat bab 3 dan 4): 961 189 (0 60 0 1 60 1) (persamaan 5) Meskipun kita tidak membahas bagaimana Untuk melakukan jejak parsial, persamaan 5 di atas adalah keadaan campuran, yang berarti ansambel statistik (lihat pendamping 4). Katakan bahwa kita melakukan sebagian jejak untuk partikel Bobs, maka matriks kepadatan hanya memberi tahu kita bahwa ada kemungkinan untuk menemukan 1, dan 50 kesempatan untuk menemukan 0. Fakta ini cukup menakjubkan, karena perhitungan probabilitasnya nampaknya menjadi efek lokal. Pernyataan semacam ini bisa membingungkan, tapi juga karena kita masih melewatkan beberapa teori penting pada saat ini. Di bab selanjutnya, saya akan mengatur semuanya dengan lurus. Pandangan lain: Saya juga harus menjelaskan pada saat ini, bahwa beberapa fisikawan tidak melihat non-lokalitas sebagai mekanisme yang bekerja di sini. Beberapa masih berpegang pada variabel Tersembunyi yang terpusat (lihat 1.3 dan Bab 6), dan beberapa bahkan menganggap teori yang cukup eksotis seperti interpretasi Dunia paralel. Pertama, kita membutuhkan lebih banyak informasi tentang latar belakang sejarah, dan tindakan seram dari kejauhan, seperti yang terjadi pada usia 30an, dan bahkan sampai tahun 90an dari abad sebelumnya. Jadi, mungkin ada efek yang aneh, dan ada konsensus di antara banyak fisikawan, keterikatan, dan non-lokalitas (dan kemudi), menjadi dasar efek seperti dijelaskan di atas, dan yang agak tidak biasa seperti Dunia yang kita kenal dari fisika klasik. Dan, sebagai semacam musim panas critism: selalu ada tapi. Harus ditekankan, bahwa beberapa fisikawan mengatakan bahwa EPR mungkin tampak tidak lokal. Tapi karena lebih cepat dari cahaya (FTL), tidak mungkin, kita tidak benar-benar memiliki non-lokalitas, tapi ada hal lain yang terjadi. Misalnya, variabel tersembunyi, atau hal lain yang belum kita ketahui. Dengan demikian perhatikan bahwa ada fisikawan yang cukup letih terhadap non-lokalitas. Tampaknya fisikawan tersebut, mengukur non-lokalitas terhadap kemungkinan FTL. Karena menyarankan FTL cukup berdosa dalam fisika, non-lokalitas pastilah salah. Namun, mohon perhatikan fakta bahwa sebagian besar fisikawan memandang non-lokal sebagai penjelasan terbaik. Masih belum ada jawaban pasti disini Jadi mari kita lanjutkan dengan latar belakang yang lebih historis, dan kemudian fokus pada wawasan yang lebih modern. 1.3 EPR dan alternatif yang mungkin. Beberapa ilmuwan terkenal berkontribusi pada QM, kira-kira pada periode 1890-1940. Tentu, juga di kemudian hari, penyempurnaan dan penemuan yang tak terhitung jumlahnya terjadi. Namun, fundamental dasar dasar QM diletakkan pada periode yang disebutkan sebelumnya. Eistein juga berkontribusi besar-besaran. Kesan saya adalah, bahwa positivisme aslinya terhadap teori tersebut, perlahan-lahan berkurang sampai perpanjangan tertentu, dan terutama di bidang interpretasi QM, dan yang lebih penting, pertanyaan mengenai teori perpanjangan QM ini benar-benar mewakili kenyataan. Bersama dengan beberapa perguruan tinggi, pada tahun 1935, dia menerbitkan artikel EPR yang terkenal: Dapatkah Deskripsi Kuantum Mekanik Realitas Fisik Dipertimbangkan Lengkap (1935). Ada banyak tempat di mana Anda bisa menemukan artikel klasik ini, misalnya: Bahkan dalam judul ini Anda sudah bisa melihat beberapa tema penting yang diduduki Einstein: Physical Reality and Complete. Ada beberapa keadaan dan deskripsi QM teoritis, yang bermasalah dengan Einstein. Di sini, saya ingin mendeskripsikan (dalam beberapa kata), empat tema berikut: (1): Sebagai contoh keraguan Einstein, dapat melayani penentuan posisi dan momentum, yang merupakan sifat duniawi di dunia Klasik. Namun, dengan pengamatan Quantum Mechanical non-commuting, tidak mungkin mengukur (atau mengamati) keduanya secara bersamaan dengan ketepatan yang tidak terbatas. Hal ini terutama cukup cepat untuk disimpulkan, dengan menggunakan notasi gelombang fungsi partikel. Hal ini juga diungkapkan oleh salah satu prinsip ketidakpastian Heisenbergs: 916 p 916 v 189 8463 Hubungan ini sebenarnya mengatakan, bahwa jika Anda mampu mengukur kecepatan (v) sangat tepat, maka momentum (p) akan (secara otomatis) sangat tidak tepat . Dan sebaliknya. Hasil QM semacam ini, membuat Einstein (cukup benar) untuk dipertanyakan, bagaimana dengan kenyataan bahwa kita dapat menghubungkan hasil QM ini. (2 :) Kalau begitu kita juga punya masalah realisme lokal. Dalam pandangan klasik, realisme lokal hanya alami. Misalnya, jika dua bola bilyar bertabrakan, maka itu merupakan tindakan yang menyebabkan momentum untuk dipertukarkan antar partikel tersebut. Sebagai contoh lain: partikel bermuatan di Lapangan listrik, perhatikan efek lokal dari bidang itu, dan ini dapat mempengaruhi kecepatannya. Seperti yang telah kita lihat di bagian 1.2, pengukuran satu partikel dari sepasang terjerat, yang tampaknya langsung (seketika) berpengaruh pada pengukuran partikel lainnya, bahkan jika jaraknya begitu besar sehingga kecepatan cahaya tidak dapat menyampaikan informasi dari Partikel pertama ke yang kedua, pada waktunya. Ini adalah contoh non-lokalitas. Banyak orang lain juga memiliki keberatan yang kuat terhadap non-lokalitas. Beberapa hipotesis konservatif (dalam beberapa hal) muncul, terutama teori variabel Tersembunyi. Singkatnya, ini berarti: Saat ini pasangan yang terjerat (seperti pada bagian 1.2) dibuat, sebuah kontrak tersembunyi ada yang sepenuhnya menentukan perilaku mereka dalam kejadian yang tampaknya bukan lokal. Itu hanya karena kurangnya pengetahuan tentang variabel tersembunyi tersebut, yang membuat kita memikirkan tindakan seram dari kejauhan. Di satu sisi, hipotesis ini adalah kembalinya realisme lokal. Saya harus mengatakan bahwa beberapa alternatif untuk variabel Tersembunyi juga ada. Pada tahun 1964, fisikawan John Stewart Bell mengajukan ketidaksetaraan Bell-nya, yang merupakan derivasi matematis yang, pada prinsipnya, akan memungkinkan jika teori realistis lokal dapat menghasilkan hasil yang sama seperti QM. Teorema Bell direvisi pada saat berikutnya, membuatnya menjadi argumen kuat untuk sebuah tes konklusif. Meski teori Bells tidak kontroversial di kalangan fisikawan, masih ada beberapa yang keberatan. Ketidaksetaraan Bell yang direvisi memang harus diuji dalam berbagai percobaan, yang mendukung QM. Tes ini tampaknya membuat teori lokal tidak sesuai, seperti variabel Lokal Tersembunyi, dan mempromosikan fitur non-lokal QM. (3): Penulis EPR juga memiliki beberapa keraguan serius tentang bagaimana menangani sistem yang terjerat, seperti yang dijelaskan di atas dalam 1.2. Sebagai contoh, jika Alice ingin mengubah himpunan vektor dasar, bagaimana pengaruhnya terhadap sistem Bobs Sebenarnya, terutama keraguan ini merupakan dasar dari pemindaian Einstein pada representasi QM tentang kenyataan. (4): Tema ini lagi tentang sistem yang terjerat. Kali ini, para penulis EPR menganggap keterkaitan terutama dengan memperhatikan posisi dan momentum. Menurut QM, kedua pengamatan tersebut tidak dapat diamati secara bersamaan. Penulis kemudian memberikan argumen mengapa QM gagal memberikan deskripsi realitas yang lengkap. Given the fact that QM was fairly new at that time, it seems to be a quite understandable viewpoint, although various physicists strongly disagreed with those arguments. As of the 90s, it seems to me that more and more people started to doubt the argumentation of the EPR authors, partly due to newer insights or theoretical developments. But, as already mentioned, also in the period of the 30s, some physicists fundamentally disagreed with Einsteins views (like for example Bohr). Before we go to EPR steering and some other great proposals, lets take a look at a nice example which has hit the spotlights the last few decades, namely Quantum Teleportation. I really do not have a particular reason for this example. But it exhibits strong characteritics of non-locality, and something which many folks call the EPR channel. And amazingly, we will see that we need to use classical bits and a classical channel too Chapter 2: a few words on the first article on Quantum Teleportation (QT) (1993). The following classical article, published in 1993: Teleporting an Unknown Quantum State via Dual Classical and EinsteinPodolskyRosen Channels (1993) by Charles H. Bennett, Gilles Brassard, Claude Crpeau, Richard Jozsa, Asher Peres, and William K. Wootters. really started to set the QT train in motion. Quantum Teleportation is not about the teleportation of matter, like for example a particle. Its about teleporting the information which we can associate with that particle, like the state of its spin. For example, the state of the system described by equation 1 above. A collarly of Quantum Information Theory says, that unknown Quantum Information cannot be cloned. This means that if you would succeed in teleporting Quantum Information to another location, the original information is lost . This is also often referred to as the no-cloning theorem. It might seem rather bizar, since in the classical world, many examples exists where you can simply copy unknown information to another location (e.g. copying the content of a computer register, to another computer). In QM, its actually not so bizar, because if you look at equation 1 again, you see an example of an unknow state. Its also often called a qubit as the QM representative of a classical bit. Unmeasured, it is a superposition of the basis states 0 and 1, using coefficients a and b. Indeed, unmeasured, we do not know this state. If you would like to copy it, you must interact with it, meaning that in fact you are observing it (or measuring it) . which means that it flips into one of its basis states. So, it would fail. Hence, the no-cloning theorem of unknown information. Note that if you would try to (stronly) interact with a qubit, it collapses (or flips) from the superpostion into one of the basis states. Instead of the small talk above, you can also formally work with an Operator on the qubit, which tries to copy it, and then it gets proven that it cant be done. One of the latest records in achieved distances, over which Quantum Teleportation succeeded, is about 150 km. What is it, and how does an experimental might look like Again, we have Alice and Bob. Alice is in Lab1, and Bob is in Lab2, which is about 100km away from Alice. Suppose Alice is able to create an entangled 2 particle system, with respect to the spin. So, the state might be written as 936 187302 ( 01 10 ), just like equation 4 above. Its very important to realize, that we need this equation (equation 4) to describe both particles , just as if they are melted into one entity. As a side remark, I like to mention that actually four of such (Bell) states would be possible, namely: 936 1 187302 ( 00 11 ) 936 2 187302 ( 00 - 11 ) 936 3 187302 ( 01 10 ) 936 4 187302 ( 01 - 10 ) In the experiment below, we can use any of those, to describe an entangled pair in our experiment. Now, lets return to the experimental setup of Alice and Bob. Lets call the particle which Alice claims, particle 2, and which Bob claims particle 3. Why not 1 and 2 Well, in a minute, a third particle will be introduced. I like to call that particle 1. This new particle (particle 1), is the particle which state will be teleported to Bobs location. At this moment, only the entangled particles 2 and 3, are both at Alices location. Next, we move particle 3 to Bobs location. The particles 2 and 3, remain entangled, so they stay strongly correlated. After a short while, particle 3 arrived at Bobs Lab. Next, a new particle (particle 1), a qubit, is introduced at Alices location. In the picture below, you see the actions above, be represented by the subfigures 1, 2, and 3. The particles 2 and 3, are ofcourse still entangled. This situation, or non-local property, is often also expressed (or labeled) as an EPR channel between the particles. This is presumably not to be understood as a real channel between the particles, like in the sense of a channel in the classical world. In chapter 2, we try to see what physicists are suggesting today, of which physical principles may be the source for the EPR channelnon-locality phenomenon. Lets return to the experimental setup again. Suppose we have the following: -The entangled particles, Particles 2 and 3, are collectively described by: -The newly introduced particle, Particle 1 (a qubit) is decribed like we already saw in equation 1, thus by: Also note the subscripts, which may help in distinguishing the particles. At a certain moment, when particles 1 and 2 are really close, (as in subfigure 4 of the figure above), we have a 3 particle system, which have to be described using a product state . as in: 952 123 966 1 8855 Psi 2,3 (equation 6) Such a product state, does not imply a strong measurement or interaction, so the entanglement still holds. Remember, we are still in the situation as depicted in subfigure 4 of the figure above. We now try to rewrite our product state in a more convienient way. If the product is expanded, and some some re-arrangements are done, we get an interresting endresult. Its quite a bit math, and does not add value to our understanding, I think, so I will represent this endresult in a sort of pseudo Ket equation: Note the factor Phi 12 . We have managed to factor out the state of particles 1 and 2 into the Phi 12 term. At the same time, the state of particle 3 looks like a superpostion of four qubit states.Indeed. Actually, it is a superposition. Now, Alice performs a masurement on particle 1 and particle 2. For example, she uses a laser, or EM radiation to alter the state of Phi 12 . This will result in the fact that Phi 12 will collapse (or flip) into another state. It will immediately have an effect on Particle 3, and Particle 3 will collapse (or be projected, or flip) into one of the four qubit states as we have seen in equations 7 and 8 above. Ofcourse, the Entanglement is gone, and so is the EPR channel. Now note this: While Alice made her measurement, a quantum gate recorded the resulting classical bits that resulted from that measurement on Particles 1 2. Before that measurement, nothing was changed at all. Particle 1 still had its original ket equation 966 1 a1 b0 We only smartly rearranged equation 6 into equation 7 or 8, thats all. Now, its possible that you are not aware of of the fact that quantum gates do exists, which functions as experimental devices, by which we can read out the classical bits that resulted from the measurement of Alice. This is depicted in subfigures 5 and 6 in the figure above. These bits can be transferred in a classical way, using a laser, or any sort of other classical signalling, to Bobs Lab, where he uses a similar gate to reconstruct the state of Particle 3, exactly as the state of particle 1 was directly before Alices measurement. Its an amazing experiment. But it has become a reality in various real experiments. -Note that such an experiment cannot work without an EPR channel, or, one or more entangled particles. Its exactly this feature which will see to it, that Particle 3 will immediately respond (with a collapse), on a measurement far away (in our case: the measurement of Alice on particles 1 2). -Also note that we need a classical way to transfer bits, which encode the state of Particle 1, so that Bob is able to reconstruct the state of Particle 3 into the former state of Partcle 1. This can only work using a classical signal, thus QT does NOT breach Einsteins laws. -Also note that the no cloning theorem was also proven here, since just before Bob was able to reconstruct the state of Particle 1 onto Particle 3, the state of the original partice (particle 1) was destroyed in Alices measurement. -Again, note that both a classical- and a nonclassical (EPR) channel, are required for QT to work. Chapter 2, simply fits quite well in the central theme of this note. It seems that QT does not work without strong entanglement between member particles. Whether this is really proof of non-locality. it seems quite likely. Chapter 3. A nano introduction to QM, with regards to operations and notations. I think that it might be useful to provide for a nano-introduction into QM. So, only some of the very basic subjects will be shown in this chapter. Nevertheless, using the whole of this text, including this chapter, might help if you like to try the professional articles on EPR and related subjects. I will start with describing the wavefunction, which was the orginal form to represent entities like particles having position and momentum. This is still heavily used today. However, Diracs ket vector notation fits vector spaces (Hilbert spaces) rather well, and using it, seems to be the favourite practice among physicists (since quite some time). 3.1 The Wavefunction and Probabilities. 3.1.1 Description of the Wavefunction: Near the end of the 1800s and the early 1900s, some amazing experiments were performed. While the classical theories (electrodynamics, classical mechanics), made a clear distincion between particles and waves, some experiments pointed towards a more dualistic character of entities. For example, particles that were beamed through a double slit, created an interference pattern , a phenomenon of which it was formerly thougth, that it could only be produced by waves like electromagnetic radiation. As another example, the photoelectric effect showed that light, at certain circumstances, behaved like particles, like transferring momentum to a real particle (such as an electron). So, at certain observations, particles could behave like waves, and the other way around, what was traditionally seen as waves, could behave like a particle. Planck and Einstein found in such observations, that radiation thus exibited a corpusculair character. Also was discovered that radiation seemed to be emitted (or absorbed) in quanta (discrete energy packets). And, as already noted, it seemed that those quanta possessed momentum too, as was observed in certain experiments. Many other observations, and theoretical considerations, lead to the start of a new theory, which was able to accurately describe microscopic entities and events. Indeed, this was the primordial form of Quantum Mechanics. Around 1924, De Broglie showed, that there exists a relation between momentum (p) and wavelength ( 955), in a universal way. In fact, its a rather simple equation (if you see it), but with rather large consequences. Its this: p 8462 955 (equation 9) where h is Plancks constant. Now, momentum, at that time, was considered to be a true particle-like property, while wavelength was understood to be a typical wave-like property, which for example stuff like light and radio waves have. The formula of De Broglie, is quite amazing really. The conseqence is thus, for example, if you have a particle like an electron flying around, you can associate a wavelenght to it. So, whats going on here Do we have a matterwave or something Intuitively, when we now think of a particles position, we might visualize a certain spreading in that position. This and many more considerations, lead some people to introduce the wavefunction 936(r,t). to describe observables of an entity. More specifically, it can be used particularly well to describe the position of a particle. Or formulated in a more general way: a wave function in QM is a description of the quantum state of a system. Around the twenties of the former century, more and more physicists started to describe states in terms of the wave function. What fitted the bill quite well, was that around 1926, Erwin Schrodinger published a mathematical equation about the evolution, that a quantum system undergoes with respect to time, when that system happen to be in some sort of force field. So, a mathematical description was sought, to find a wave-like equation, that could for example describe the position of a particle, and which also adheres to Schrodingers equation. A flat plane wave, like: 936(r,t) e i(kr-969t) (equation 10) does not normalize, since its just a flat front expanding in space and time. By the way, equation 10 very much resembles a classical flat wave equation. But a superposition of multiple modes, can normalize, if the modes (defined by k) are also multiplied by a sort of gaussian distribution function. The result then is a wave packet, localized around some maximum, and quickly lowering amplitude of we observe positions away from that maximum. The addition of waves to the packet, is like adding harmonics with a mode k (related to the frequency), each with a relative amplitude g(k). The more wave components you would add, the more the particle gets localized. In the limit, the summation would become an integral. In the example below, we use one dimension x only, and we also consider the time-independent solution as well. 936(x) 18730 (2960 8463) 8747 -8734 8734 g(k) e ikx dk (equation 11) If you would really use an infinite summation, thus really an a integral, the equation above would represent a very localized position x. If it is a summation, then a spreading around a maximum, would be expected. A physical interpretation of this, is that any wavefunction 966(x) can be expressed as a superposition of states e ikx8463 In case of a limited superpostion, and discrete k, we then can write equation 10 as: 936(x) 18730 (2960 8463) 931 A n e ikx (equation 12) Maybe, in using the original QM language, its better to say that a particle does not have classical properties like position or momentum. It might be better to say, that the wavefunction must be used to calculate probabilities of such observables. Thats probably right So now we are going to see how to do that. 3.1.2. Probabilities: Its often called the Borns rule, since Max Born was the first who proposed how to calculate probabilities when using the wavefunction. Specifically, it can be used (and easily be visualized) when discussing the position of a particle represented by the wavefunction 936(r,t). Although the Dirac notation is more often often used, the orthodox wavefunction representation is rather equivalent to Diracs notation, although Diracs Bra Ket vectors, nicely and directly relate to mathematical vectorspaces. Now, how can we use 936(r,t) to calculate probabilities It can be made plausible, by going back for a while to traditional electrodynamics . Its really true, that if you would have a classical wave 966, then the energy density of that wave is proportional to 936 2. So, the energy (per unit volume) of a light wave is: E 8765 936 2 Born applied that same reasoning to the QM wavefunction probability density. However, the wavefunction is a complex valued function, so the QM probability density is: 961(r,t) 936(r,t) 936(r,t) 936(r,t) 2 (equation 13) where 936(r,t) is the socalled complex conjugate. Now, from complex number theory, we know that z z z 2 , so then equation 13 is correct. This knowledge, must imply that, the probability P to find a particle in a certain region is: P(somewhere on x-axis) 8747 -8734 8734 936(x,t) 2 dx 1 (equation 14) P(x in a,b) 8747 a b 936(x,t) 2 dx (equation 15) where I only used one dimension x, instead of 3 dimensional space r . Equation 14 is logical, since the probability to find the particle somewhere on the x-axis, must be 1 (100). Rewriting that for R 3 would then result in: P(somewhere in Space) 8747 -8734 8734 936( r ,t) 2 d r 1 (equation 16) 3.2 Schrodingers equation. We now know what the wavefunction represents. Ofcourse, its gearded towards position and momentum, of an entity like a particle. In short: The wavefunction 936(r,t) represents the probability distribution of finding the particle. But how does it behave in time The evolution of the wavefunction, is described by Schrodingers equation (1925). It sort of describes the movement of the wave, due to a force acting on it. This is the case if the particle has a potential, due to some field. Its a differential equation, relating change in time (movement) to change in Energy (change in potential). Its often notated as: You can read it as: Total Energy term Kinetic Energy term Potental Energy term. If you would take time for it, and like to try some math, you would see that if you would try equation 10 in Schrodingers equation, you would see that it indeed is a solution. You can put equation 10 into the differential equation, and it works. Therefore, a superposition of equations like equation 10, is a solution too Formally, the differential equation above, is also called the time dependent Schrodinger equation. A simpler variant is the time-independent Schrodinger equation, since it effectively does not take the differential in time, into consideration. Sometimes this is indeed allowed, if a system does not fundamentally change in time. For example, the particle in a box problem can be solved using a time-independent Schrodinger equation. The particle in a box problem, is, viewed from a timeline perspective, a stationary problem. However, you can always simply start (or try) using the time-dependent Schrodinger equation. We are going to try that as an example. Example: The particle in a box problem (1 dimensional, x-axis). For about the math in this eaxmple, I think it is not relevant at all. But, it would be really nice, if you just follow the general argument here. Just interpret the problem as a wavefunction in a 2 dimensional box, that is, a horizontal x-axis of length L, with vertical boundaries at x0 and xL. Along the y-axis, we may visualize the Amplitude of the wavefunction. As a little abstraction of the situation, lets suppose that: -at the vertical boundaries of the box, the potential V is 8734. -for 0 2 8706 2 -------- 2m 8706 x 2 Since, if we look at equation 18, here we simply have V0, and 916 means 8706 2 8706 x 2 , in a one dimensional situation. Although its not important to follow the math here, its really possible to rewrite equation 19 into: The equation above, is actually a well-known differential equation, for which mathematicians and physicists immediately have general solutions. Again, this note is not a math textbook, and here it is not important at all, so I will simply say that a solution can be: 936 N (x) C e ik N x Or, you might also say that you have found a classical-like, harmonic solution as If a mathematician or physicists, use the boundary conditions in a proper way, a good solution then is: where P is some constant, and L is the interval on the x-axis where V0. Remember too, that k N 2 2m E N 8463 2 What we see here, is that we have an integer N, determining the eigenstates of the equation, or in other words, we have found sine-like wavefunctions, each respectively with a higher frequency, and a higher energy E N too. They all obey the Schrodinger equation, as expressed in equation 19, for this particular situation. 3.3 Common operations and notations. In order to be able to understand professional articles, you need to know at least the meaning of the operations and notations listed below. This section will be very short, and very informal, with the sole intention to provide for an intuitive understanding of such common operations and descriptions. If we consider, for a moment, vectors with real components (instead of complex numbers), some notions can be easily introduced, and get accessible to literally everyone . As a basic assumption, we take following representation as an example of a state 966 931 a i u i , like for example 966 a 1 u 1 a 2 u 2 a 3 u 3 . Especially, I like to give a plausible meaning to the following notations: (1): 60 AB . The inproduct, or inner product, of vectors, usually interpreted as the projection of B on A (or rather the ket B casted on the bra 60 A). (2): B 60 A. usually corresponds to a matrix or linear operator. (3): 966 60 966. corresponds to the Density matrix of a pure state. (4): 60 966 O 966 : corresponds to the expectation value of the observable O. (5): The Trace of an Operator Tr(O) 931 60 i O i Proposition 1: (1): 60 AB . is the inner product, of vectors or Kets. -Inner product of two kets 60 AB If we indeed use the oversimplification in R 3. then a (regular) vector or Ket) B can be viewed as a column vector: Note the elements b i of such a vector. We know that we can represent a vector as a row vector too. In QM, it has a special meaning, called a Bra, as to be the row vector with complex conjugate elements b i . Lets not worry about the term complex conjugate, since you may view it as a sort of mirrored number. And if such an element would be a real number, then the complex conjugate would be the same number anyway. The Bra 60 B can be viewed as a row vector: The inner product, as we know it from linear algebra, operates in QM too. It works the same way. The inner product of the kets A and B (as denoted by Dirac) is then notated as 60AB. From basic linear algebra, we usually write it as A 183 B . or sometimes also as ( a , b ). However, we stick to the braket notation: Which is a number, as we also know from elementary vector calculus. Usually, as an interpretation, 60 AB can be viewed as the length of the projection of B on A. Or, since any vector can be represented by a superposition of basis vectors, then 60966 i 934 represents the probability that 934 collapses (or projects or change state) to the state 966 i . -Inner product of a ket, with a basivector: 60u i 966 As another nice thing to know is, is that if you calculate the inner product of a (pure) state, like: 966 a 1 u 1 a 2 u 2 a 3 u 3 with one of its basis vectors, say for example u 2 (and this set of basis vectors is orthonormal), then: 60u 2 966 a 1 60u 2 u 1 a 2 60u 2 u 2 a 3 60u 2 u 3 a 2 -Operators: The operator O, as in OB C , meaning O operating on ket B , produces the ket C Ofcourse Operators (mappings) are defined too in Hilbert spaces. Here, they operate on Kets. Indeed, linear mappings, or linear operators, can be associated with matrices . This is no different from what you probably know of vector calculus, or linear algebra. Here is an example. Suppose we have the mapping O, and ket B. Then in many cases the mapping actually performs the following: meaning that the columnvector (ket) B is mapped to columnvector C. Or, simply said, the operator O maps the ket B to ket C We can write that as: Above, we see an example of how to multiply a column vector with a row vector, which is a common operation in linear algebra. It simply takes the syntax and outcome as you see above. So, proposition 2 seems to be plausible, since it follows that B 60 A is a matrix. Proposition 3: 966 60 966. corresponds to what is called the Density matrix of a pure state. In proposition 2, we have seen that B 60 A usually produces a matrix. Now, if we take a ket 966 and multiply it with its dual vector, the bra 60 966, as in 966 60 966, then ofcourse it is to be expected we get a matrix again. However, the elements of that matrix are a bit special here, since the elements tell us something about the probability to find that pure state in one of its basis states. In a given basis, the diagonal elements of that matrix, will always represent the probabilities that the state will be found in one of the corresponding basis states. In its most simple form, where we for example have that 966 u 1 u 2 , the density matrix would be: 9484 189 0 9488 9492 0 189 9496 The density matrix is more important, as a description, when talking about mixed states. Proposition 4: 60 966 O 966 : corresponds to the expectation value of the observable O. -Trying to make that plausible using simple vectors: We can make that plausible in the following way: We have associated a certain observable (such as momentum, position etc..) with a linear operator O. Now suppose for a moment that we have diagonalized the operator, so the only diagonal elements of the matrix, are not 0, and represent the eigenvalues. Then we may use an argument like so: where u i are basis vectors. We can write it as a columnvector too (in our simplification): We are going to show that 60 966 O 966 is the expectation value of O, by making it plausible for a simple case, thereby hoping that you will agree that it is true in general as well. Now suppose O is represented by the matrix: 9484 0 0 0 9488 9474 0 0 0 9474 9492 0 0 1 9496 which result can be read as the weighted average of the eigenvalues. Thus we say that its the expectation value of O. I hope you can see some logic in this. Proposition 4 is however, valid for the general case too. -Using the usual argumentation in QM: We know that for the usual wavefunction (over x, not considering t), the probability distribution (density), can be expressesd as: 961(x) 936 (x)936(x) which resembles equation 13 (but in this example for one dimension only). For the expectation value 60x, that is, the best average of finding 936 to be at a certain position x, considering all possible x (all x Space), then may be expressed as: 60 x 8747 -8734 8734 936 (x)936(x) d x If we now have a certain observable Q of 936, and we like to know the expectation value 60 Q of that observable, then we can use the integral shown above, for the determination of the average of Q: 60 Q 8747 -8734 8734 936 (x) Q(x) 936(x) d x (equation 21) We need to multiply the probability distribution 961(x) of 936, with Q(x), and integrate that over all space (here we only have considered one dimension). Proposition 5: The Trace of an Operator is: Tr(O) 931 60 u i Ou i . The trace of an Operator, or matrix, is the sum of the diagonal elements. With respect to pure- and mixed states, it has a different outcome (namely 1 or 3 (thus real numners only). In R 3. we can have the following set of orthonormal basisvectors: 9484 1 9488 9474 0 9474 9492 0 9496 9484 0 9488 9474 1 9474 9492 0 9496 9484 0 9488 9474 0 9474 9492 1 9496 You may say that those basis vectors corresponds to u 1 , u 2 , u 3 , like in our usual ket notation. If we consider the rightside of the expression 931 60 u i Ou i , then we have Ou i . We can interpret this as that O operates on a basisvector u i . Suppose that i1, meaning that it is our first basis vectors, just like the set of basisvectors of R 3 , as was listed above. Lets operate our matrix of O, operate on our basisvectors. I will do this only for the (1,0,0) basisvector (i1). For the other two, the same principle applies. So, this will yield: 9484 a b c 9488 9474 d e f 9474 9492 g h i 9496 9484 1 9488 9474 0 9474 9492 0 9496 9484 a1b0c0 9488 9474 d1e0f0 9474 9492 g1h0i0 9496 9484 a 9488 9474 d 9474 9 492 g 9496 Well, this turns out to be the first column vector of the matrix O. Lets call that the vector A (8224). Next, lets see what happens if we perform the leftside of 931 60 u i Ou i . We already had found that the vector A corresponds with Ou i . Using the leftside, we have 60 u i A . This is an inner product, like: 9484 1 9488 9474 0 9474 9492 0 9496 9484 a 9488 9474 d 9474 9492 g 9496 Note that this number a, is the top left element of the matrix O. Since Tr(O) 931 60 iOi , it means that we repeat a similar calculation using all basisvectors, and add up al results. Hopefully you see that this then is the sum of the diagonal elements. I already proved it for the first diagonal element (a), using the first basis vector. The 2 vectors remaining, to be used for a similar calculation, will then produce b and c. In this simple example, we then have Tr(O) a b c . Note that in general Ou i produced the i th column of O (see 8224) above. In the exceptional case where Ou i produces au i , thus a scalar coefficient a times a basisvector, thus Ou i a u i , then in our simple example in R 3. we would have: And, keeping in mind that Ou i the i th column of O (see 8224), then we would have a matrix with only the diagonal elements which are not null, and all others (off diagonal elements), which would then be nul. In such case, it is often said that the elements a, b, and c are the eigenvalues of the operator O. Its absolutely formulated in Jip Janneke language, but I hope you get the picture. 3.4 A few words on Diracs notation 3.4.1 Some general observations: In general, we may write a state vector, or a ket, as expanded (as a superposition) of basis states u i : 966 931 c i u i c i 60 u i 966 Thus each number c i is the inner product of 966 and u i 966 represents a quantum system to be in the state 966 and is called the state vector. Operations on 966, or its basis states, goes very similar as to what you may know from vector calculus. Indeed, for example, in section 3.3 we have seen the inner product of two kets. And the following are kets too, like x or p, representing the position and momentum respectively. In some (exceptional) cases, you might even use definitive kets, like x2, meaning the position of the quantum system (like a particle), is at x2. Its true that we often treat kets, like a statevector, where we speak of a certain probability of finding the statevector to be in the basis state (eigenstate) u i , after some measurement is done. That is true, but it can also be that the state is in a certain value (or certain basis state), for example, after an observation has been performed. Some notations we already are familiar with, can be written in a slightly different (but equal) form. Just take a look at this. We already know our qubit: Now suppose we have a 2 dimensional Hilbert space. An arbitrary vector can be expressed as a linear combination of the unit vectors or basis vectors. Suppose the vectors i and j form such a basis. An arbitrary vector v, or ket, might be written as: v c 1 i c 2 j 60 i v i 60 j v j i 60 i v j 60 j v Its a slightly different format, but its really the same as c 1 i c 2 j . Since v v , the above equation also means that i 60 i v j 60 j v (i 60 i j 60 j) v . Thus: i 60 i j 60 j 1 You can also express it in a matrix, since we already know from the former section that, in general, A 60B is a matrix. So: 9484 1 09488 9492 0 19496 i 60 i j 60 j Where I is the identity matrix or identity operator. Above, we considered the 2 dimensional case. In general, in dimension n, we may say that: 931 u i 60 u i I If u i represents a complet set of orthonormal basis states, of that n dimensional space. 966 931 c i u i But we may thus also write: 966 931 u i 60 u i 966 3.4.2 Interpretations of the Bra: We know what a ket represents, which is not too hard to visualize. Its simply a vector. But how can we interpret a bra 1. Informal interpretation: This interpretation is almost true, but it can be used as a very close, and good pictorial interpretation. In proposition 1, of the former section, we saw that if we represent A as a column vector, then 60 A is the corresponding row vector, with complex conjugate elements. If needed, take a look at proposition 1 again, of section 3.3. So, its actually a nice interpretation (I think). If you have the ket 966, then 60 966 represents the same state, but as a mathematical object, its the transpose of 966, that is, this time a row vector. Since the elements are complex numbers, 60 966 uses complex conjugate elements. And whats really quite the same, note that: 60 966 ( 966 ) 8224 Where the dagger symbol 8224, stands for conjugate transpose or Hermitian transpose. Here it means reversing the column elements to row elements (or vice versa), and taking the complex conjugate of those elements. 2. More abstract (and formal) interpretation: It fits mathematically well, to say that bras are elements of the dual Hilbert space. Or, the bras are vectors of that dual Hilbert space. What this also means, is that bras are functionals on the ket, which produce a (complex) number. Thats fine, since we already know that for example: 60 u i 966 c i Thus, here we can view it as the functional 60 u i , operating on 966 , producing the number c i . But if we consider a certain ket, 966 , then its corresponding bra, is unique 60 966. There is always a one-to-one correspondence from a certain ket, to its unique bra. 3. The bra as the projection, or final state: If we look at (2) again, above, we can also say that 60 u i 966 represents the projection of 966 , on 60 u i . Note that we here talk about a different bra, and a different ket (not a certain ket with its unique corresponding bra). So, if in an observation (or experiment), it turns out that we find the value c i , then we can also say (or interpret it), as that 966 was projected (or cast) on 60 u i , or you may also say that 966 was projected (or cast) on u i . 3.4.3 Some well known states you may see in articles: 936 a1 b0 936 a0 b1 c2 - Product state, or outer product of two qubits: 936 966 1 8855 966 2 a 00 00 a 01 01 a 10 10 a 11 11 - General Product state, or outer product of three qubits: 936 a 1 000 a 2 001 . a 8 111 -Specific example of the product state of three qubits: 936 187308 000 187308 001 187308 010 187308 100 187308 110 187308 011 187308 101 187308 111 -Bell entangled states of 2 qubits: 936 1 187302 ( 00 11 ) 936 2 187302 ( 00 - 11 ) 936 3 187302 ( 01 10 ) 936 4 187302 ( 01 - 10 ) - (Entangled) Singlet state: 936 187302 ( 01 - 10 ) -A Greenberger-Horne-Zeilinger (GHZ) entangled state, of three qubits: GHZ 187302 ( 000 - 111 ) Often, this is the entangled 3 qubit state which is used in experiments or theoretical conjectures, quite similar to our familiar bi-particle Bell states, or the Singlet state. 3.4.4 Some more words on Operators, Matrices, Observables: Its probably really neccessary, I believe, that you have seen section 3.3, before turning to this one. I (like to) think that 3.3 is a quite gently introduction on how we use matrices in QM. (1). One of the rules in QM is that with each measurable observable of a quantum system, is associated a quantum mechanical (linear) Operator . If you read that literally, then the Operator defines (so to speak), the observable. We know that the wavefunction, or statevector, represents the probability amplitude of finding the system in a certain state of the observable. (2) You may also rephrase (1) to this: A wavefunction (Schrodinger like) or statevector (like Dirac formulated it), describes the observable quantity, while the Operator acts on the wavefunction (or statevector). It is true that a measurement can be formulated as an Operator acting on the wavefunction (or statevector). This often then means a projection on one of the eigenstates, with a certain associated probability of finding that particular eigenstate. With this view in mind, we can indeed say that the Operator is associated with the observable. Formally, in the sense of regular vector calculus, the Operator A working on the ket 936, produces another ket 966. So: A 936 966 (equation 22) Equation 22 is very general. That is, the ket 936 is mapped to 966, just as we know from regular vector calculus. However, if we would have: A 936 a 936 (equation 23) then 936 is called an eigenstate or eigenket of the operator A, and a is called an eigenvalue. As another thing, since QM is about physical systems, for A it is also required that: 60 966 60 936 A 8224 (equation 24) where 8224, stands for conjugate transpose or Hermitian transpose. Remember that it is formally defined, that we have a dual Hilbert space of bras (H ), associated with every ket of the regular Hilbert space H. Dont worry about these (possibly confusing) statements. It simply means that there are certain requirements on such Operators. We know that an Operator can be viewed as a matrix, especially when using the Dirac notation. So, there are certain requirements on those matrices too. Taking the conjugate transpose of a matrix A, means that we switch to rows and colums, and take the complex conjugate of all the matrix elements. Its also rather neccessary to have A A 8224 (equation 25) An important reason is, that the expectation value of A, meaning 60 966 A 966 , must produce a real number . and not a complex number. Yes indeed. In real experiments we simply must find real numbers, although QM entities uses complex numbers. If you like, you can write it out, using the examples of section 3.3, and see that this requirement really is rather acceptable. When A A 8224. then we talk about self adjoined Operators. So it simply means that taking the conjugate transpose of a matrix A, results into the same matrix. Note also that in section 3.3, proposition 2, it was made plausible that: represents a matrix, as a very general statement, for two kets A and B . If needed, you can take a look again at section 3.3, to verify that statement, and also some example operations by some Operator O, on kets. Stated in some more formal terms: Given vectors and dual vectors, we can define operators O (i.e. linear mappings from H to H), in the format of: As already said above, we usually want Hermitian or self-adjoined Operators, garanteeing that the expectation value of the Operator, is a real (and not complex) number. Consider the following Operator (and matrix) A: 9484 0 -i9488 9492 i 0 9496 Is this a self adjoined Operator Here it is true that A A 8224. since if you take the conjugate transpose, that is, switch the rows and columns, and take the complex conjugate of each matrix element, you will see that this is indeed true. Note: - the complex conjugate of a real number, like 1, is the same real number again (1). - the complex conjugate of an imaginary number like i, is -i and vice versa. Maybe you have still some questions right now. If so, take a look at section 3.3 again. It will show some foundations of matrices, which may help. Chapter 4. Meaning of pure states and mixed states. 4.1 A few words on pure states: While you might think that a completely defined state as 0 , is pure, it holds in general for our well known superpositions . An example of an superpositional state, can be this: You may also view a pure state as a single state vector . as opposed to a mixed state. So, even at this stage, we already may suspect what a mixed state is. Thus pure states: We have seen them before in this note, sofar. A mixed state is a statistical mixture of pure states, while superposition refers to a state carrying some other states simultaneously. Although it can be confusing, the term superposition is sort of reserved for pure states. So, our well-known qubit is a pure state too: 966 a 0 b 1 Or as a more general equation, we can write: 966 931 a i u i (equation 26) This is a shorthand notation. Then i runs from 1 to N, or the upper bound might even be infinite. Usually, such a single state vector 966, is thus represented by a vector or ket () notation, and is identified as a certain unknown observable of a single entity, as a single particle. So, a pure state is like a vector (called ket), and this vector be associated with a state of one particle. A pure state is a superposition of eigenstates, like shown in equation 29. Other notes on pure states: Such vectors are also normalized, that is, for the coefficients (a 1 . a 2 . etc..), it holds that a 1 2 a 1 2 . 1 Its also often said that a pure state can deliver you all there is to know about the quantum system, because the systems evolution in time can be calculated, and Operators on pure states work as Projection operators. In sections above, we have also seen that the coefficient a i can be associated with the probability of finding the state to be in the ua i eigenstate (or basisvector) after a measurement has been performed. In general, an often used interpretation of 966, is that it is in a superposition of the basis states simultaneously.Then, the keyword here is simultaneously. However, this interpretation depends on your view of QM, since many interpretations of QM exist. But superpostion will always hold, and is a key term of a pure state (like equation 29). When you would insist on the qualifying phrase a pure state gives us all there is to know , then probabily (or maybe) known coefficients are required too, like for example with: Note that some authors treat it that way. But in general, undetermined coefficients are OK too. As long as we can talk of a ket, we have a pure state. Furthermore, it is required that the inner product of 966 with its associated bra, is normalized, that is, has a unit length. That is, the inner product returns the value 1. Thus: 4.2 A few words on mixed states: A mixed state, is a mix of pure states. Or formulated a little better: a probability distribution of pure states, is a mixed state. Its an entity that you cannot really describe, using a regular Ket statevector. You must use a density matrix to represent a mixed state. Another good description might be, that it is a statistical ensemble of pure states. So we can think of mixed state as a collection of pure states 966 i , each with associated probability density 961 i . where 0 8804 961 i 8804 1 and 931 961 i 1. It cannot be stressed enough, that a linear superposition is not a mixture. Mixed states are more commonly used in experiments. For example, when particles are emitted from some source, they might differ in state . In such a case, for one such particle, you can write down the state vector (the Ket). But for a statistical mix of two or more particles, you cannot. The particles are not really connected, and they might individually differ in their (pure) states. What one might do, is create a statitistical mix, what actually boils down in devising the density matrix. The statistical mix, is an ensemble of copies of similar systems, or even an ensemble with respect to time, of similar quantum systems So, you can only write down the density matrix of such an ensemble. In equation 3, we have seen a product state of two kets. Thats not a statistical mix, as we have here with a mixed state. In a certain sense, a mixed state looks like a classical statistical description, of two pure states. When particles are send out by some source, say at some interval, or even sort of continuously, its even possible to write down the equation (density matrix) of two such particles which were emitted at different times. This should illustrate that the component pure states, do not belong to the same wave function, or Ket description. You might see a bra ket-like equation for a mixed state, but then it must have terms like 966 60 981 . which indicate that we are dealing with a density matrix. In general, the density matrix (or state operator) of a (totally) mixed state, should have a format like: Hopefully, you can see something that looks like a statistical mixture here. Here is an example that describes some mix of two pure states a and b : 961 14 a 60 a 34 b 60 b (equation 27) Note that this not an equation like that of a pure state. Ofcourse, some ket equations can be rather complex, so not all terms perse need to have to be in the form 966 60966 . Especially intermediate results can be quite confusing. Then also: by no means this text is complete. Thats obvious ofcourse. For example, partial mixed systems exist too, adding to the difficulties in reckognizing states. A certain class of states are the socalled pseudo-pure families of states. This refers to states formed by mixing any pure state . with the totally mixed state . So, please do not view the discussion above, as comprehensive description of pure and mixed states, which is certainly not the case here. 4.3 What about our entangled two partice system: Equation 4, which described an entangled bipartice system, is repeated here again: 936 187302. ( 01 10 ) Note that this is a normal ket equation, and it is also a superposition. We do not see the characteristic 60 terms which we would expect to see in a mixed state. Therefore, its a pure state There are several perculiar things with such entangled states. We already have seen some in section 1.2, where Alice and Bob performed measurements on the member particles, in their own seperate Labs. Another perculiar thing is this: I will not illustrate it further, but using some mathematical techniques, its possible to trace out the state of one particle, from a two-particle system. -For example, if you would have a normal product state like equation 3, then tracing out particle, like particle 2, just gives the right equation for particle 1. This was probably to be expected, since the product state is seperable. - If you would do the same for an entangled system, then if you try to trace out a particle, then you end up with a mixed state, even though the original state is pure. Thats is really quite remarkable. Later more in this. For now, lets go to the next chapter. Chapter 5. The inequalities of Bell, or Bells theorem. 5.1 The original formulation. The famous Bell inequalities (1964), in principle, would make it possible to test if a local realistic theory, like the Local Hidden Variables (LHV) theory, could produce the same results as QM. Or, in stated somewhat differently: No theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics. Or again stated differently: There is no underlying classical interpretation of quantum mechanics. For about the latter statement, I would like to make a small (really small) reservation, since, say from 2008 (or so), newer parallel universe theories have been developed. Although many dont buy them, the mathematical frameworks and ideas are impressive. In chapter 8, I really like to touch upon a few of them. The Bell theorem was revised at a later moment, by John Clauser, Michael Horne, Abner Shimony and R. A. Holt, which surnames were used in labeling this revision to the CHSH inequality. The CHSH inequality can be viewed as a generalization of the Bell inequalities. Probability, and hidden variables. To a high degree, QM boils down to calculating probabilities of certain outcomes of events. Most physicist, say that QM is intrinsically probabilistic. This weirdness is even enhanced due to remarkable experiments, like the one as decribed in section 1.2. It is true that the effects described in section 1.2, are in conflict with local realism, unless factors play a role of which we are still fully unaware of, like hidden variables. We may say that Einsteins view of a more complete specification of reality, related to QM, is our ignorance of local pre-existing, but unknown, variables. Once these unknown hidden variables are known, the pieces fall together, and the strange probabilistic behaviour can be explained. This then includes an explanation of the strange case as described in section 1.2 (also called the EPR paradox). This is why a possible test between local realism, and the essential ideas of QM, is of enormous importance. It seems that Bell indeed formulated a theoretical basis for such test, based on stochastic principles. I have to say that almost all physicist agree on Bells formulation, and real experiments have been executed, all in favour of QM, and against (local) hidden variables theories. What is the essence of the Bell inequalities In his original paper (Physics Vol. 1, No. 3, pp. 195-290, 1964), Bell starts with a short and accurate description of the problem, and how he wants to approach it. Its really a great intro, declaring exactly what he is planning to do. I advise you to read the secions I and II of his original paper (or read it completely, ofcourse). You can find it here: Bells Theorem, or more accurately, the CHSH inequality, has been put to the test, and also many theoretical work has been done, for example, on n-particle systems, and other more complex forms of entanglement. On the Internet, you can find many (relatively) easy explanations of Bells Theorem. However, the original paper has the additional charm that it explicitly uses local variables, like 955, which stands model for one or more (possibly a continuous range) of variables. His mathematics then explicitly uses 955 in all derivations, and ultimately, it leads to his inequalities. If we consider our experimental setup of section 1.2 again, where Alice and Bob (both in remote Labs), perform measurements again on their member particles, then one important assumption of local realism is, that the result for particle 2 does not depend on any settings (e.g. on the measurement device) in the Lab of particle 1, or the other way around. In both Labs, the measurement should be a local process. Any statistical illusion would then be due, to the distribution of 955, in the respective Labs, as prescribed by a Local Hidden variable theory. The Bell inequalities provide a means to statistically test LHV, against pure QM. In effect, experimental tests which violate the Bells inequalities, are supportive for QM non-locality. Sofar, this is indeed what the tests have delivered. Some folks see the discussion in the light of two large believes: or you believe that signalling is not limited by c, or you believe in super determinism. Super determinism then refers to the situation where any evolution of any entity or process is fully determined. So to speak, as of the birth of the Universe, from where particles and fields snowed from the false vacuum. Interestingly, all particles and other stuff, indeed have a sort of common origin, and thus may have given rise to a super entanglement of all stuff in the Universe. Still unkown variables have then sort of fixed everything, thus a sort of super determinism follows. Personally, I dont buy it. And it seems too narrow too. There are also some newer theories (Chapters 7 and 8) which do not directly support super determinism. 5.2 Newer insights on the Bell inequalities and LHVs. -Simultaneous measurements vs non-Simultaneous measurements. Since the second half of the 90s (or so), it seems that newer insights have emerged on Bells Theorem, or at least some questions are asked, or additional remarks are made. One such thought is on how to integrate the Heisenberg relations into the Theorem, and the test results. Here is a good example of such an article: The authors state that near simultaneously measurements, implicitly relies on the Heisenberg uncertainty relations. This is indeed true, since if Alice measures the spin along the z-direction and if she finds up, then we may say that if Bob would also measure his member particle along the z-direction too, then he will certainly find down. Therefore, the full experiment will use (also) axes for Alice and Bob which do not align, but have a variety of different angles. Then, afterwards, all records are collected, and correlations are established, and then using Bells inequalities, we try to see if those inequalities are violated (in which case LHV gets a blow, and QM seems to win). The point of the authors is however, that the measurements will occur at the same time. If now a time element is introduced in the derivation of Bells theorem, a weakening of the upper bound of the Theorem is found. As the main cause of this, the authors make it clear that second-order Broglie-Bohm type of wavefunctions may work as local operators in the Labs of Alice and Bob. I personally cant really find mistakes, apart from the fact that Broglie-Bohm is actually another interpretation of QM, which might not have a place in the argument. However, I am not sure at all. By the way, the Broglie-Bohm pilot wave interpretation, is a very serious interpretation of QM, with many supporting physicists. However, the main point is that the traditional Bell inequalities (or CHSH inequality) in combination with the experimental setup, is not unchallenged (as good physics should indeed operate). -Werner states. Amazingly, as was discovered by Werner, there exist certain entangled states that likely will not violate any Bell inequality, since such states allows a local hidden variable (LHV) model. His treatment (1989) is a theoretical argument, where he first considers the act of preparing states, which are not correlated, thus not entangled, like the example in equation 3 which is a seperable product state. Next, he considers two preparing devices, which have a certain random generator, which makes it possible to generate states where the joint Expectation value . is no longer seperable or factorizable. His artice is from 1989, where at that time it was hold that systems which are not classically correlated, then they are EPR correlated. Using a certain mathematical argumentation, he makes it quite plausible to have a semi-entangled state, or Werner state, which has the look and feel of entanglement, and where a LHV can operate. He admits its indeed a model, but it has triggered several authors to explore this idea in a more general setting. The significance is ofcourse, to have non seperable systems, using a LHV. If you are interested, take a look at his original paper: -Countless other pros and contras: There are many articles, (somewhat) pro- or contra Bells Theorem. Many different arguments are used in the battle. You can found them easily, for example, if you Google with the terms criticism Bells theorem arxiv, where the arxiv will produce the free, uneditorial, scientific papers. Here is one that makes a strong point against LHV, and is very much pro QM: This article is great, since it uses a model of 2 entangled particles without a common origin . and thus this system is very problematic for any type of classical or LHV related theory. I am not suggesting that you should read the complete artice. Contrary, often only the introduction of such articles is good enough, since then the authors outline their intentions and arguments. The next article uses a truly different perspective. According to this author, we do not need non-locality and all strange observed effects, are simply due (under the hood) to the superpostion principle. Furhermore, he makes a case that QM simply does not give a complete view on reality, just like Einstein said. You can find that article here. However, the EPR experiments, and what we have seen in Quantum Teleportation, is probably hard to understand just by superposition alone, and not thinking in terms of non-locality. So, what do we have up to now Taken all together, you cannot say that there exists a full consensus among physicist on how to exactly interpret the EPR experiments, together with Bells inequalities. However, a majority of physicists still thinks in terms of non-locality, to explain the experimental results, and see sufficient backing of theoretical considerations, for their position. Sofar, what we have seen in section 1.2 (EPR entangled bi-particle experiment), and chapter 2, (Quantum Teleportation), is that something that behaves like an immediate action at a distance, seems to be at work. This does not suggest that any form of signallingcommunication exist, that surpasses the speed of light. As said in section 1.2, the no communication theorem states exactly that. However, not all folks would agree on this. By the way, the QT effect we saw in section 1.4, simply also needed a classical channel in order to transport the state of particle 1, to particle 3 at Bobs place. That is also supporting the view, that true information transfer does not go faster than c. There exists a number of interpretations of QM, like e.g. the Broglie-Bohm pilot-wave interpretation. Rather recently, also newer parallel universe models were proposed, with a radical different view on QM. For about the latter: you might find that strange, but some models are pretty strong. The most commonly used interpretation, is the one that naturally uses superpositions of states. That model works, and is used all over the World. For example, most articles have no problem at all in writing a state (Ket) as a superposition of basis states, like in a pure state, as we have seen in section 2.1. In fact, once describing QM in the framework of Hilbertspaces (which are vectorspaces), superpostion is then sort of imposed or un-avoidable. But ofcourse, the very first descriptions using wave-functions to describe particles and quantum systems in general, is very much the same type of formulation. And this vector formulation, fits the original postulates of QM, quite well. But it seems quite fair to say that it is actually just this principle of superposition . that has put us in this rather weird situation, where we still cannot fully and satisfactory understand, exactly why we see what think we see as was described in section 1.2 (or the lightgreen text above). Not all physicist like the non-locality stuff as displayed in the lightgreen text above. For quite a few, a Hidden Variable theory (or similar theory) is not dead at all. Although the experimental evidence using the Bell tests seems rather convincing, there still seems to exist quite some of counter arguments. For now, we stay on the pure QM path (superpositions, EPR non-locality, probabilities, Operators, Projectors etc..), and how most people then nowadays interpret Quantum Steering, Entanglement, and Bell non-locality. Lets go to the next section. Chapter 6. Steering, Entanglement, and Bell non-locality. 6.1 Some descriptions: Lets first try to describe steering: Quantum Steering: Quantum steering is the ability of Alice to perform a measurements on her local member of an entangled system, with different outcomes, and that leads to different states for the remote part of that entangled system (at Bobs Lab), independend of any distance between them. How did I came up with such a nice description Here you can find an article of the man who used such text for the first time (Schrodinger, 1935), as a response to Einsteins EPR paper: (in the document, of the url above, you might scroll down a bit, to view the article) If I may quote a nice paragraph from that article: (when he is dicussing two remote members of an entangled system, or entanglement in general. ) . It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenters mercy in spite of his having no access to it. This paper does not aim at a solution of the paradox, it rather adds to it, if possible. A hint as regards the presumed obstacle will be found at the end. Schrodinger already considered (or suspected) the case (as described in section 1.2), that the result that Alice measures, instantaneously steers what Bob will find. Althoug in section 1.2 we saw steering at work, I also like to try to discuss a modern test too, involving steering, and this all under the operational definitions as listed below. Many questions are left open at this point, among which are: - Can Alice steer Bob - Can Bob steer Alice - Does two-way steering exists - What is the difference when pure systems and mixed systems are considered - Does all types of entangled systems, enable steering We are not too far of from possible answers. Lets next try to describe entanglement and Bell non-locality. Entanglement: When 2 or more particles can be described as a product state (like equation 3), they are seperable. A measurement of an observable on one particle, is independent of the other particles. You can always seperate the original ket (of a certain particle) from the product state. In many cases however, two or more particles are fully intertwined (with respect to some observable), in such way, that you cannot seperate one particle from the other(s). A measurement on one particle, effects the other particle(s) too. A state as for example in equation 4, describes both particles (together in SpaceTime). They truly have a common (inseperable) state. A sort of key definition then seems to be: If you cannot write a combined system as a full product state, then its an entangled system. But then still various forms exist, like partially entangled systems (partly seperable), or maximally entangled systems (not seperable at all). Bell non-locality: This seems to apply for any situation, for which QM violates the Bell inequalities. So, it seems to be a very broad description. You might say that entangled states as in sections 1.2 and 1.4, fall under the non-locality description. How about steering Seems that this too, as a subset, is smaller than the notion of non-locality. But this is not correct. The exact difference, or applicability, between steering, entanglement, and Bell nonlocality, was not so much of a very hard issue in the minds of physicists, so it seems. We have to admit that steering, entanglement and Bell nonlocality, seemed to have much overlap in their meanings. Well, it proved to be not entirely true. Then, in 2006, the following article appeared: by Wiseman, Jones, and Doherty. They gave a pretty solid description for Steering, Entanglement, Nonlocality, in the sense of when such term applies . As the authors say themselves: they provided (sort of) operational definitions. The statements above with respect to the relative place (as subsets or supersets) of steering, entanglement, and nonlocality, were not corrects. As the article points out: Proposition 1: -We need entanglement to enable quantum steering. -But not all entangled systems provide conditions for quantum steering. The above sounds rather logical, since quantum steering, or EPR steering, is pretty much involved, and just seems to be a rather hard quality for true a non-classical phenomenon. The authors formulate it this way: Steerable states are a strict subset of the entangled states. So, if you would regard this from the perspective of Venn diagrams, then Steerable states lie within entangled states. Or, in other words: the existence of entanglement is necessary but not sufficient for steering. Thus: steering is deeper than just entanglement, although entanglement is required. Proposition 2: -Steering is a strict superset of the states that can exhibit Bell-nonlocality. This thus would imply that steering could happen in a local setting, which might be percieved as quite amazing. In other words: in a Bell local setting (thus NOT nonlocal), steering is possible too. Or, and this is important, some steerable states do not violate the Bell inequalities. As we shall see a while later, if we would only consider pure states . the original equivalence holds to a large extent. But considering mixed states too . leads to the propositions above. I recommend to read (at least) the first page of this article. True, all these sorts of scientific papers are rather spicy, but already on page one, the authors are able to explain what they want to achieve. 6.2 Entanglement Sudden Death: Maybe the following contributes to evaluating entanglement. or not. However, its an effect that has been observed (as of 2006) in certain situations. Early-stage disentanglement or ESD, is often called Entanglement Sudden Death in order to stress the rapid decay of entanglement of systems. It does not involve perse all types of quantum systems, which are entangled. Ofcourse, any sort of state will interact with the environment in time, and decoherence has traditionally been viewed as a threat, in for example, Quantum Computing. ESD however, involves the very rapid decay of the entangled pairs of particles, that is, the entanglement itself seem to dissipate very fast, maybe due to classical andor quantum noise. But the fast rate itself, which indeed has been measured for some systems, has surprised many physicists working in the Quantum field. Ofcourse, it is known that any system will at some time (one way or the other) interact with the environment. Indeed, a general phenomenon as decoherence is almost unavoidable. Its simply not possible to fully isolate a quantum system from the environment. This even holds for a system in Vacuum. Even intrinsic quantum fluctuations has been suggested as a source for ESD. However, many see as the source for the fast decay, the rather normal local noise, as e.g. background radiation. Yu and Eberly have produced quite a few articles on the subject. The sudden loss of entanglement between subsystems may be even explained in terms of how the environment seems to select a preferred basis for the system, thus in effect aborting the entanglement. Just like decoherence, ESD might also play a role in a newer interpretation of the measurement process. Whether it is noise or something else, its reported quick rate is still not fully understood. A good overview (but not very simple) can be found in the following article: To make it still more mysterious, an entanglement decay might be followed by an entanglement re-birth, in systems, observed in some experimental setups, with the purpose of studying ESD. A re-birth might happen in case of applying random noise, or when both systems are considered to be embedded in a bath of noise or other sort of thermal background. Many studies have been performed, including pure theoretical and experimental studies. A more recent article, describing the behaviour of entanglement under random noise, can be found below: As usual, I am not suggesting that you read the complete article. This time, I invite you to go to the Conclusion in the article, just to get a taste of the remarkable results. 6.3 Types of entanglement: Ofcourse, this whole text is pretty much lightweight, so if I cant find something, it does not mean a lot. So far, as I am able to observe, there is no complete method to truly systematically group entangled states into clear categories. There probably exist two main perspectives here. The perspective of formal Quantum Information Theory, in which, more than just occasionally, the physics is abstracted away. This is not a black-and-white statement ofcourse. Pure physics, that is, theoretical- and experimental research. Both sciences deliver a wealth of knowledge, and often must overlap, and often also are complementary in initiating ideas and concepts.So what types of entanglement, physicists have seen, or theoretici have conjectured How much the points below contribute to the understanding of entanglement, I do not dare to say. However, those point constitute knowledge, so at least they must have something to say to us. Anyway, lets see what this is about: 1. Pure- and mixed states can be entangled. For pure states, a general statement is, that an entangled state is one that is not a product state. Rather equivalent, is the statement: a state is called entangled if it is not separable. Mixed states can be entangled too. This is somewhat more complex, and in section 5.4 I will try do a lightweight discussion. 2. The REE distance, or strength of entanglement. Relative Entropy of Entanglement (REE) is based on the distance of the state to the closest separable state. It is not really a distance, but the relative entropy of entanglement . E R compared to the entropy of the nearest, or most similar separable state. In Physics Letters A, december 1999, Matthew J. Donalda, and Michal Horodecki, found that if two states are close to each other, then so are their entanglements per particle pair, if indeed they were going to be entangled. Over the years after, the idea was getting more and more refined, leading to the notion of REE. So, its an abstract measure of the strength of entanglement. Its an area of active research. Intuitively, its not too hard to imagine that for nonentagled states, E R 0, and for strong entangled states E R - 1. So, in general, one might say that 0 8804 E R 8804 1. You could find arguments that this is a way, to classify entangled states. 3. Bi-particle and Multi-particle entanglement. By itself, the distinction between a n2 particle system, and a n 2 system, is a way to classify or to distinguish between types of entanglement. Indeed, point 1 above, does not fully apply to multiparticle entanglement. In a n 2 system we can have fully separable states ofcourse, and also fully entangled states However, there also exists the notion of partially separable states. In ket notation, you might think of an equation like this: 936 966 1 8855 981 2,3 and suppose we cannot seperate 981 2,3 any further, then 936, which then is only separeated in the factors 966 1 and 981 2,3 , is a partially separable state . 4. Classification according to polytopes. When the number of particles (or entities) in a quantum system increases, the way entanglement might be organized, is getting very complex. While with n2 and n3 systems, its still quite manageble, with n 3, the complexity of possible entangled states, can get enourmous (exponentially with n). In 2012, an article appeared, in which the authors explicitly target multi-particle systems, which can expose a large number of different forms of entanglement. The authors showed that entanglement information of the system as a whole, can be obtained from a single member particle . The key is the following: The quantum correlation of the whole system N, affects the single- or local particle density matrices 961(1). 961(N) which relate to the reduced density matrices of the global quantum state. Thus using information from one member alone, delivers information about the entanglement of the global quantum state. From the the reduced density matrices, which thus also correspond to the density matrices 961(1). 961(N) of one member particle, the eigenvalues 955 N can be obtained. Amazingly, using the relative sizes of 955 N . a geometric polyhedron can be constructed which corresponds to an entanglement class. From this different geometric polyhedrons (visually like trapeziums) at least stronger and weaker entanglement classes can be calculated. Using a local member this way, you might say that this single member acts like a witness to the global quantum state. If you like more information, you might want to take a look at the original article of the authors Walter, Doran, Gross, and Christandl: Chapter 7. A few words on the measurement problem. This will be very short section. But I hope to say something useful on this extensive subject. Certainly, due to chapter 8, the role of the observer and the measurement problem, simply must be addressed. Its in fact a very difficult subject, and many physicists and philosophers broke their heads on this stuff. Its not really about in-accuracies in instruments and devices. One of core problems is the intrinscic probability in QM, and certain rules which have proven to be in effect, such as the Heisenberg uncertainty relations. And indeed, on top of this, is the problem of the exact role of the observer. Do not underestimate the importance of that last statement. There exists a fairly large number of (established) interpretations of QM, and the role of the observer varies rather dramatically over some interpretations. Whatever ones vision on QM is, its rather unlikely that it is possible to detach the observer completely from certain QM events and related observations, although undoubtly some people do believe so. So, I think there are at least five or six points to consider: Intrinscic probability of QM. Uncertainty relations, and non-commuting observables. Role of the observer. Disturbive (strong) measurements vs weak measurements. The quantum description of the measurement process. Decoherence and pre-selection of states nearin the measuring device. The problem is intrinsic to QM, and in many ways un-classical. 7.1 Decoherence:: For example point 6, if you are familiar with decoherence, then you know that a quantum system will always interact with the enivironment. And in particular, at or near your measurement device, decoherence takes place, and a process like pre-selection of states may take place. In a somewhat exaggerated formulation: the specific environment of your lab, may unravel your quantum system in a certain way, and dissipates certain other substates. Could that vary over different measurement devices, and with different environments What does that say, in general, about experimental results The subject is still somewhat controversial. A nice article is the following one. Its really quite large, but reading the first few pages already gives a good taste on the subject. You can also search for articles of Zurek and co-workers. 7.2 Role of the Observer: And ofcourse, the famous or infamous problem of the role of the observer. For example, are you and the measurement apparatus connected in some way Maybe that sounds somewhat hazy, but some folks even study the psychological- and physiological state of the observer, with respect to measurements. I dont dare to say anything on such studies, but you should not dismiss them. A comprehensive analysis of making an observation, and certain choices, is very complex, and is probably not fully undestood. But there exists more factual and mathematical considerations. For example, what is generally understood with the Heisenberg uncertainty relations 7.3 A few words on the Heisenberg uncertainty relations: Today, among physicists, a large degree of consensus exists on the following: The Heisenberg uncertainty relations, have nothing to do with disturbing effects of the art of making a measurement. Here too, these relations are truly intrinsic in QM, independent of any measurement. So why then a few words on the Heisenberg uncertainty relations In the early days of QM, especially Schrodingers approach, was popular, which essentially is integral and differential calculus. Diracs vector Ket (bra ket) notation emerged somewhat later. The initial way to describe a quantum system, was by using wave equations. Some time before, it was realized that even particles could exhibit a wave-like character, and that electromagnetic radiation, proved to have a particle-like character at certain observations. Since a particle is quite localized, but not exactly localized, is why physicists introduced the Wave Packet 966(r,t). Its a superpostition too, but for a free particle, the components add up in such a way, that the packet is grossly localized, like some sort of gaussian distribution, with a maximum, and it fastly diminishes at the fringes. The addition of waves to the packet, is like adding harmonics with a mode (related to the frequency) p, each with a relative amplitude g(p). The more wave components you would add, the more the particle gets localized. In the limit, the summation would become an integral like: (one dimension x only, time-independent) 966(x) 18730 (2960 8463) 8747 -8734 8734 g(p) e ipx8463 dp (equation 28) Now, the more modes you add, the narrower in position x, the gaussian bell shaped wave packet becomes. When the number of modes is really large, we get the integral over p, and a very localized x (position). This p is related to the frequency of such mode, and represents the momentum of that mode. So, with just a low number of ps, we have a very dispersed (wide) packet, in the x position. If we have a large number of ps, we have a very narrow packet, in the x position. In other words: With a large variation in p, we have a narrow x. With a large variation in x, we have a narrow p. Using Operators, or just wave functions and Fourier transforms, it can be formalized. But even in this stage, we can hopefully understand how Heisenberg came to his famous uncertainty principle with respect to momentum and position: 916 p x 916x 8805 84632 (only for x-dimension) 916 p 916 r 8805 84632 For these observables p and r, it must hold that no matter how small you try to make the variation, (the deltas), the product must always be larger than a certain number. This result is simply a consequence of the QM wave approach to the position and momentum of a particle. Using the notion of Observables, and corresponding OperatorsMatrices, it can be shown that for non commuting observables (or Operators), it is not possible to measure them simulteneously with unlimited precision. For example, if you would pinpoint the position very precisely, the momentum is very in-precise, and vice versa. The principle applies to all non-commuting Observables, like Energy E and time t, and for example also for the spin components along the x and z directions, usually notated as 963 x and 963 z . This is why the principle can be of relevance in any discussion about measurements: The principle is intrinsic to QM, however, the consequence (or collary) is that it is not possible to measure non-commuting observables simulteneously, with unlimited precision. 7.4 The strange case of section 1.2, and the role of measurements: The strange case of section 1.2, or the EPR paradox, represents one of the most fundamental, and strangest scientific debates for about 80 years (or so), since 1935. We can indeed read all about Bell, and experiments, and steeringentanglementnon-locality, and many more theoretical arguments, but the truth is. there still is no full consensus reached among physicists. Chapter 8. Some Many Worlds interpretations in QM. Some rather remarkable ideas have been published rather recently on new interpretations in QM. Although you cannot truly call them theories, its also true that you cannot call them speculations either, since the authors provide a conceptual and mathematical framework. So, in this text, I will simply call them theories. I like to spend a few words on a few of those rather remarkable theories, but I am afraid I cannot classify this in any other way than saying that it resides in my hobby sphere. So, if there was any reader at all. this stuff is one of my hobbies . so to speak. In this chapter, I like to spend a few words on rather new parallel universe theories, and, in chapter 8, a few words on recent resarch on micro blackholeswormholes, at the Planck scale. However, I will start with the Grandfather theory of Many Worlds in QM, that is, Everetts MWI theory, which was published in 1957. Much later, around 2010 and 2014, newer models were introduced. 8.1 Some important Interpretations of QM: I should start, with a sort of listing of established interpretations. There are quite a few of them, and a few represent a rather dramatic deviation from the (more or less) standard Copenhagen interpretation. Here are a few of them: The Copenhagen interpretation (or in a new jacket). QM with Hidden variables and local realism. QM with non-locality. Decoherence as a successor to the specific Copenhagen collapse. Everetts Many Worlds Interpretation (MWI). The Many Minds Interpretation. Poiriers Waveless Classical Interpretation (MIW). Quantum hydrodynamics and Trajectory analysis (e.g. Madelung, Holland). The transactional interpretation. Time symmetric theories. de Broglie - Bohm Pilot Wave Interpretation (PWI). possibly also the Two State Vector Formalism. I would not dare to say that this listing is complete. Furthermore, some of them quite overlap, but at certain issuess, there are fundamental differences (like nonlocality, and local realism).There are also quite subtle differences. For example, a Ket vector, or wavefunction, can be interpreted as just a working vehicle, but its true existence might be doubted. Some other interpretations see them as a real representation of entities and properties. There are no polls of who likes which the most, but as I observe it, it seems as if a lot of folks are attracted by some features of the Broglie - Bohm Pilot Wave Interpretation. 8.2 The Grandfather theory of Parallel universes in QM: Everetts MWI (1957). Its impossible not to spend a few words on the original Parallel universe theory in QM. It was formulated by Everett, in 1957, although Schrodinger in 1952 already hinted to such an idea, in the form of simultaneous existing outcomes. Everett published his theory as a Phd thesis, called the Relative State Formulation of Quantum Mechanics, under a certain degree of guidance by his mentor Wheeler. Slightly later, also due to the promotional work of DeWitt, it became know as the Many Worlds Interpretation or MWI. In this theory, the wavefunction is a real existing entity . and forms the basis for all entities. The key of his idea is the following. Although in the simple example below, a ket vector notation is used, its rather equivalent to a wave function setting. If you would consider the following state (rather equivalent to a wave function interpretation): and in an observation, you will find the state u 2 , then in the Copenhagen interpretation it is said that the state 966 collapsed (or was projected) to the state u 2 , with a probability that relate to a 2 . Its quite possible then, that you could have found, for example, u 3 , with a probability that relate to a 3 . However, you found one outcome, and the former state is destroyed. In Everetts theory, all 3 outcomes are realized. So, if you performed the measurement, then 3 different states forks off and undergo their further evolution. So, following the example above, the quantum superposition of the combined observer-state and observed object-state wavefunction, will resolve into three relative states, completely independed from each other. Hence the notion of branched off worlds, or Many Worlds. If you want to know more, then here is a (new version) pdf of Everetts original article: Indeed, its not a strange framework at all. Although his idea did not find any support at first, it is true that as from the late 60s, up to the 90s, it became quite popular, and many physicists considered it to be a viable interpretation. Some even considered it to be the best interpretation thus far. Note also that Everetts theory is a no-collapse formulation of QM, quite unlike the Copenhagen interpretation. -However, the popularity declined over later years, and quite a few articles expressed to have found some (supposedly) inconsistencies in MWI. As is rather usual in science, not all physicist turned around, and still some are very sceptical on those (supposedly) inconsistencies in MWI. Below I provide some links to arxiv articles, pro- and contra MWI, which will demonstrate some of those inconsistencies. -New theoretical paths in general in physics, and in QM specifically, probably did not helped MWI much. For example, around the 80s, the theoretical principle of decoherence was discovered (Zeh, Zurek), which provided a new way on how a wavefunction would interact with its environment, like a measuring device, or the environment in general. But it must be said that decoherence shows some important similarities with MWI, but then ofcourse without the branching into other Worlds. But even up to this very day, we have physicist who publish arguments in favour of MWI. Its still a valid interpretation of QM, and as usual, some folks like it, and some dont. I highly recommend to invest some time to explore some great articles in MWI. The following short article tells us about some inconsistensies in MWI, which ofcourse, you dont need to take for granted. But this is an exceptionally nice and sometimes humouristic text, and I am sure you like it. By the way, you will absolutely learn a lot of MWI, by reading this article. However, the same author, at another moment, produces an article which you can call rather pro MWI. Since this one explains MWI rather well, I like to list it here too: The next article is from Vaidman, and is rather against QM non-locality, and the article expresses lots of motivation, why MWI should the theory of choice. 8.3 Modern Parallel Universe theories in QM: beyond 2005 In 2004, a rather remarkable article appeared, from P. Holland, titled: Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation. It looks quite non-quantum to talk about precisely defined spacetime trajectories. At least, this is a common view among some people in the field. Also, the common interpretation of the wavefunction, works against such a view too. However, plenty of physicists would not not agree to such statements. Ofcourse, it also depends on which particular Interpretation of QM, you favour the most. For example, the Broglie-Bohm Pilot Wave Interpretation seems to re-introduce some classical features again. Not literally, but essentially the theory states that we have a real particle, with an accompanying pilot wave . which can be regarded as a velocity field guiding the particle. The pilot wave then (sort of) has the look and feel of the Copenhagen wavefunction. One remarkable thing is, that if you would have the initial conditions, you can further calculate future positions precisely. Hollands article is not part of my intended discussion, but ofcourse the article is great. In short, his article seem to deal about the following: Holland notes that in QM, in certain cases, certain results of hydrodynamics and fluid mechanics, can help in calculating particle positions. Such an idea was first pointed out by Madelung. In such approaches, the QM probability density then looks similar to the fluid density. Then he works out all sorts of calculations, using classical Euler-Lagrange equations to calculate the trajectories of the fluid particles as functions of their initial coordinates, as if you would regard such initial ensemble as the initial wavefunction. In 2010, Poirier published a remarkable article too, called Bohmian mechanics without pilot waves . It was first published in Chemical Physics, Volume 370, May 2010 . The amazing thing is, that Poirier has arguments to strip away the Pilot wave, from the particle, when considering Bohemian mechanics (Broglie-Bohm Pilot wave mechanics). As he says: can a trajectory ensemble itself, be possible too, in the sense that it obeys Schrodingers equation, and therby making no explicit or implicit reference to an external pilot wave Then, using kinematics and rather classical looking differential equations, and using results from (especially) Hollands work (I think), he arrives at the remarkable proposition: It seems possible to replace the wavefunction by a real-valued trajectory field, furher defined by a trajectory density weighting function. The orthodox complex valued wavefunction is unneccessary. Note this line of thinking seems like a reduction of QM, to a semi-classical statistical Mechanics. From this on, he is not so far away anymore from his final MIW interpretation, that is, the Many Interacting Worlds (MIW) interpretation. The Many Interacting Worlds (MIW) interpretation. It seems that Poirier and Schiff were first (2012) in devising a true MIW model. However, in 2014, Wiseman, Hall, and Deckert used a slightly other approach to reach (I think) a better model, since they can avoid a continuum of many interacting worlds. Generally, MIW essentially means this. The proposition is essentially, to have n Classical parallel universes, where in one such Universe, an entity (like a particle), has well defined (sharp) observables (like position), all the time. However, there exists a variation over such observables, in the various different Universes, in such a way, that an observer experiences the illusion of QM, like the implicit spreading of the wavefunction (which leads to an uncertainty in e.g. the position). So, each World it has its own private sharp values for that observable, but since there exists variation between Worlds, we percieve the illusion of a composed State vector where the same experiment might show different values all the time. In fact, in MIW, there is no state vector, no kets, no wavefunctions. This is a remarkable different view, compared to Everetts MWI. Most notably, the differences are: In Everetts MWI, the wave function is real and plays a central role. In MIW, the wave function does not exist, and in fact MIW is a sort of QM without the wavefunction. In Everetts MWI, universes branches off due to the fact that superimposed waves each interact with the environment. In Poiriers MIW, there is no branching, no wavefunctions, but the many worlds may interact. Everetts MWI uses standard Quantum Mechanics (wavefunctions, probabilities etc..), however Poirier uses established rules from many parts of physics, whereas many are simply familiar classical, like trajectories, positionimpuls, Lagrangian etc. where paths over different trajectories relate to the Many Interacting Worlds. A short survey of one of the MIW theories: I think the approach of Hall, Deckert, and Wiseman is the best one, up to this day, for constructing a MIW theory. Their original article, published in 2014, is the following: As usual, I advice to read the whole article. However, even only reading the first 2 pages (the Introduction), will tell you what Poiriers MIW theory tried to prove, and what the plan of the authors is, to provide for a better alternative derivation, in order to arrive to an (almost) equivalent MIW theory. In this paragraph, I will try to say something useful on this improved MIW theory. Essentially, it goes like this: If you would have n particles, then you might define a (single) configuration space for that set of particles, which space can be described by the vector: The number kdepends on which number of dimensions, and which attributes, you would take into consideration. The vector Q . then describes the set of particles, for example, with respect to position. The authors also see this vector as to be equal to the configuration space of the set of particles, and also called the world space for that set. For Q . at a certain t, you could define a position as Q R . Sofar, there is nothing too special here. As an nterim object its possible to define a Probability Density P( Q R ) 966( Q R ) This would then express the distribution of the positions of the particles . Next, they make the assumption that N of such worldspaces, thus Q 1 . Q N could exist, or if you like, could be postulated. If we would find that a credible assumption, we then would have N configuration spaces for those n particles, expressed as the collection Q 1 . Q N . N is not aboslutely fixed, and might even go to nfinity. The remarkable thing then is, and what is not showed here (see the article above), that a sort of local repulsive force can be derived, between those worldspaces Q i . A very important aspect of that local force is, that it is only of relevance when Q l and Q m are close, with respect 966( Q R ). If the worldspaces Q i are now viewed as seperate Universes, which might be percieved as a rather bold proposition, then at least the force between close Universes would be repulsive. The authors then apply this theory to e.g. The Ehrenfest theorem, spreading of the wave packet, tunneling, and a few other QM effects. Remarkably, they indeed seem to succeed in their examples. For example, the spreading of a particle over configuration spaces (the Universes), gives the illusion of a wave packet. However, the Universes themselves, are fully classical, and all observables have sharp values at all times. Other Multiple Worlds theories in the scope of QM: - Many Minds theory: Everetts MWI, and Poirier MIW, are not the only propositions of Many Worlds, in the scope of QM. For example, the Many Minds theory of Dieter Zeh, is one important line of thinking. This theory, actively integrates the observer (or the mind of the observer) in the process of observation, and interpretation. It maybe of interest of people working in the field of psychology, or any sort of neural science. Other Multiple Worlds theories not directly in the scope of QM: Outside the immediate scope of QM, some other Parallel Universe models were deviced. Here, you might think of Hubble Volumes, or Chaotic Inflation and some other models. Some time back, I tried to make an overview in some sort of note. I you want to try it, you can use this link. Even if you do not buy these classes of theories. they certainly gives us a different perspective on QM, and thats valuable anyway. Chapter 9. A few words on entropy. 9.1 What is entropy Entropy is a physical quantity, which is especially used in statistical mechanics, and thermodynamics. Originally, you might say that it especially useful in macroscopic systems, having many entities, like gasmolecules in a room, or cilinder etc.. In ordinary statistical mechanics, the entropy (S) provides us a pointer to the measure of the multiplicity of microstates that sits behind a particular macrostate. For example, you have a large number of gas atoms, where all of them are somehow magically placed in one single corner of a vacuum room, and you release the ban, you would observe that immediately the gas will be distributed all over that volume of that room. In this example, we can talk about on how many different ways the atoms or molecules can be arranged. Just after the ban was released, the number of arrangements increased enormously. When energy is increased into a system, or by adding new arrangements (microstates), entropy is increased. The second law of thermodynamics says that the entropy of a closed system, will never decrease. In fact, thats a remarkable law, and it even may suggest that Nature wants to promote disorder over order, in an isolated system. Ofcourse, we need to be very careful using such sort of statements. A mathematical relation has been found that relates the Entropy S, to the number of microstates W: S 8765 ln(W) (equation 29) In thermodynamics, chemistry, the usual equation is: S k B ln(W) (equation 30) where k B is Boltzmann constant. So, entropy in general seems to be a macroscopic quantity. True, but its applicability has turned out to be extremely general. For example, just think of the all of the subwaves that contribute (or sum up) to a wavefunction or wavepacket. So, entropy can be used here as well And it uses the same natural logarithm, that is, the ln function. Note that equations 29, or 30, have a certain resemblence to Shannons Law (1948) of data communication technologies, and related sciences: S B log 2 (1 SN) (equation 31) where C is the maximum attainable error-free data speed in bps that can be handled by a communication channel, B is the bandwidth of the channel in Hz, and SN is the signal-to-noise ratio of that communication channel. Also note that log N (x) is a logarithmic function, based on N, while ln is the natural logarithmic function, based on e. Although the above relation stems from 1948, modern theories consider variables, symbols, and distribitions. Indeed, entropy has a very central meaning as well in Information Sciences . 9.2 Why entropy can be expressed as a ln(W) function. Many processes in Nature can be expressed in the form of an e x function, or a ln(x) function. Both are indeed quite remarkable. -In the figure below, you see e x , which starts out rather slow, but as x increases, it almost explosively starts to climb upwards. - The inverse function of e x , is ln(x). This one is exactly the mirrored function of e x , with respect to the line yx. The function ln(x) starts out climbing extremely rapidly, but as x increases, it climbs less and less steeply, until it almost (but not quite) reaches a nearly horizontal slope. The function ln(x) is natural. For example, if you have a certain amount of radiactive material, then it will decay over time. The amount of active material thats left after some period t, can be expressed as a function of of the half-live or mean lifetime 964, that is ln(964). How can we make plausible, that ln(W) is indeed related to the entropy S of a system Suppose you add microstates to a system. Suppose you originally had 10 states. Adding for example 3 additional states, is a rather big increase in entropy. Indeed 103 is a relevant change. However, once you reach the number of 1000 states, and again add 3 microstates, then 10003 is not a large increase in entropy anymore. This behaviour reflects precisely the graph of an ln(W) function. The slope gets less and less, if W increases. 9.3 Entropy of a quantum system. The text below, perhaps, may strike you as a bit weird. By now, we know what we must understand about a pure- or mixed state. As it will turn out, the entropy of a pure state is zero. Thats remarkable, since we know that, in general, it is a superposition. First, in general, about the concept of entropy which is involved here, is the Von Neumann entropy, which is shown below. This framework was, and still is, accepted, since it works. Secondly, considering a pure state describing superposition, we talk about quantum superpostion Indeed, we can write a ket (statevector) as a sum of eigenstates (basis vectors), but we dont know really what it is in terms of classical physics. Now, you might say that this is an incorrect statement, since we can attain a probability of finding a certain eigenstate. That is still true, however, a quantum superpostion is still something very different from a statistical ensemble . which a mixed state is supposed to describe. Indeed, in case of a statistical ensemble, we have physical states . on which we can apply true classical statistics, since that statistical ensemble looks exactly like an ordinary statistical ensemble. Remember from chapter 4, that a mixed state is a statistical mixture of pure states . while superposition refers to a state carrying some other states simultaneously. It might be relatively hard to understand (or not). Lets also take a look at the statements below: -Statement: Entropy, as a quantity that in general relates to distinguishable microstates . applies well to mixed states. It simply looks like a classical statistical ensemble, on which we can apply the term entropy. -Statement: Entropy, as applied to quantum superposition, as is meant in a pure state, should indeed return 0, also since we cannot say anything definite of the state (unless we observe it). If we observe it, the former state is lost. After measurement, we simply have one eigenstate. -Statement: Since 1948, entropy started to be used in terms of mathematical communication theory, as information entropy . Folks started to think that the physical degree of distinguishable states of a system (statistical entropy), is related to its information (information entropy). For a pure state in superposition, we only know that 936 936. If we observe it (Copenhagen talk), we are left with a certain eigenstate. A mixed state is a true statistical ensemble. Hopefully, we are in the clear now, for how we must interpret entropy for pure- and mixed states. Lets see how this part of QM works. Chapter 10. EREPR models. Personally, I find these theories (or hypotheses) quite appealing. For example, do you remember the Inflationary Universe theorie(s) Those are regarded as the most plausible theories today, to explain (or describe) the origin and evolution of the Universe. At the tiniest fragment of the start of time, at the earliest phase (maybe around 10 -42 sec), a Quantum Fluctuation (according to the theory), gave rise to a pre-form of SpaceTime and a precursor of Gravity. Then, for very short period, an exponential inflation of SpaceTime took place. In case you are not familiar to Inflationary Universe theorie(s), you might want to do a websearch first on that topic. A rather bold assumption might be made: An embedded relation, even today, between SpaceTime components and for example entanglement, might be possible, according to these lines of thought. Thus the current theories try to establish plausible models for SpaceTime, Gravity, and indeed, QM effects, like entanglement. - Be warned though . that most physicists seem (or probably really are), very weary or sceptical on those models. Its true that only a rather select company of theoretical physicists are actively working on these models. - Be double warned . since the entities they try to study (from a theoretical framework), are on the smallest scale possible, that is, typically in the order of the Planck length. This scale is fully out-of-reach for direct experimental work, and the present day particle accelerators, lack many billions of orders of magnitude of Energy, to probe such small scales. Since this is really a fact, all work done up to now, is purely theoretical. - Or, be not warned at all . since science is just simply always in progress and sometimes we have an established theory, which does not hold up anymore to new experimental results, or new, sufficiently backed, theoretical considerations, especially if established theories fail in certain domains. 10.1 General overview. Lets start with a sort of overview, of which sort of ideas emerged, and when. In the second section, I will try to go somewhat deeper into the theories, but for now, having this sort of overview helps to put stuff in perspective. Entanglement seems to have (or might have) a rather large area of application. At least, that is how many theoretical physicists look at it nowadays, especially since the 2000s, and even more so since 2013. But long time ago, but after QM stood firmly in the physics books, science went on, ofcourse. All those years during the 50s, 60s etc. in the former century, enormous progress was made in astronomy, particle physics, theoretical physics etc.. Its impossible to say anything useful here, ofcourse, unless one is planning to write a book on the theme of progress in physics, during the 50s up to today. Its seems fair to say that string theory (since the 80s), AdSCFS, Quantum field theories, Quantum Gravity, SpaceTime models, Cosmological models etc. kept people busy for a long time. I cant say that those theories, ultimately, came together, but certain results from all of them, created an atmosphere (so to speak), to bring in entanglement into the picture. Holographic principle and Entanglement Entropy. Here is an article (2006) from Shinsei Ryu and Tadashi Takayanagi, where the authors link entanglement from a holographic perspective on entropy from AdSCFT: This link above, is for illustrational purposes only. You can read it ofcourse, but its rather involved. I simply only like to create a (although on a nano-scale) small historical perspective too. Thats why I listed the article above. AdSCFT is a specific SpaceTime model, and in some respects compatible with string theory, in the sense of a certain correspondence. The article seems to succeed in deriving entanglement entropy from minimal surfaces (one dimension less), in some form of AdS space of certain dimensions. Its all highly theoretical, but its getting quite concrete in using entanglement in SpaceTime models. Entanglement and the creation of SpaceTime geometries. More importantly is the following article. And this time, you are encouraged to read (or browse through) it. Possibly, its the first concrete article, postulating entanglement as the cement in the SpaceTime fabric. Its the classic article from M. van Raamsdonk (2010), and you can find it here: Ofcourse, the article is partly inspired from former work, like e.g. articles of Maldacena, but nevertheless, as far as I know, it explicitly uses quantum entanglement to build geometries of SpaceTimes. Ofcourse, a various ideas on the dicreteness of SpaceTime, spin-networks, loops, already circulated quite some time before (e.g. some ideas of Penrose and others). But finding entanglement as the fundamental sculpter of the geometry of SpaceTime, is quite new (or new). Essentially, using the methods from Ryu and Takayanagi, Raamsdonk shows that if you would slowly tear down entanglement from a certain AdS model of SpaceTime, then when entanglement is finally reduced to nothing, this SpaceTime will be no more than fully disjoint parts of SpaceTime. Hereby, making plausible that the principle of entanglement, creates SpaceTime. Remember, this was only 2010. The methods of Ryu and Takayanagi stems around 2006. Some fundamental ideas of Malcedena were from (about) 1997. For (about) 2010 to today . the ideas are really alive throughout the community of theoretical physicists, and many refinements and explorations were made. Entangled Black Holes. As a slightly other line of thought, theoretical explorations were done on the subject of entangled black holes. Although Einsteins Relativity theory allows for the principle of Wormholes (also called the Einstein-Rosen bridge, or the ER bridge), Juan Maldacena and Leonard Susskind, introduced the idea of applying entanglement on pairs of Black Holes. One of their articles is the following: In this facinating article, the authors explore the idea that an Einstein Rosen bridge between two black holes, might be very similar to the EPR-like correlations as seen in many applications and experiments in QM. In other words: entangled Black Holes. In the same article, the authors say that its tempting to suspect that any correlation through entanglement, has its roots in ER bridges (or wormholes) on a microscopic scale. Hereby, they rooted the idea of EREPR, which made quite a few folks enthousiastic for that concept. The article is quite spicy, if not at least some core ideas are introduced. Thats my challenge for the sections below. I am afraid that it all will be a bit lengthy. 10.2 AdS, Strings, Entropy, Holographic picture. 10.3 EREPR wormholes. 1057 PC Amsterdam The Netherlands KvK: 37125573 tel: Int: (0031)(0)6 2060 4148 NL: 06 2060 4148 mail: albertvanderselgmail absrantapex.org Any questions or remarks Then contact me at. albertvanderselgmail Site maintained by: Albert van der Sel last update: 16 Februari, 2017 Nederlandstalige paginas: Klik aub hier voor enkele andere Nederlandstalige paginas.Welcome to Antapex. EPR and Steering in Quantum Mechanics (QM). Version: 16 Februari, 2017. Status: Almost done. Some ideas on Quantum Entanglement and non-locality were re-discovered the last 15 or 20 years (or so), partly based on ideas of Schrodinger on EPR steering, which were expressed in 1935. There indeed exists a subtle difference between, what we describe as entanglement, Bell non-locality, and Steering. I like to say something on those subjects, since its absolutely facinating stuff. So, in case you rather unfamiliar with such subjects, this note might be of interest. The first five chapters will describe some well-know effects of entanglement, that traditionally led to the socalled EPR paradox. So, these first five chapters are a bit old-skool, I think. While after many recent efforts were, and are, spend on finding the essence of steering . entanglement , and nonlocality . it now seems that the views that were deveoped in the years before the 90s, probably needed quite some revision, especially due to all the research since the 2000s. However, the first five chapters will present the (pre 90s) old-skool ideas first, since this way probably still remains the best practice to present such material. Chapter 1 is about the famous EPR paradox, and that phenomenon is a important theme for this entire text. In this starting chapter, entanglement, and possibly even steering, will be presented in a strange case. Then, chapter 2 will spend a few words on Quantum Teleportation (QT), which is a real effect. Here too, entanglement and possibly non-locality are demonstrated. Chapters 3 and 4 might be viewed as a nano introduction to QM, showing the most basic concepts and notations used in QM. This might help if you like to read professional articles on the various subjects. Chapter 5 will be a short intro to the Bell inequalities, which in fact are stochastic criteria to decide if measurement results are due to non-local effects of QM, are are due to local mechanisms, like the Hidden variables, which in fact represent a local realistic view on reality. In chapter 6, I will try to decribe a few specifics of EPR steering, but it will be of a very lightweight nature, and can only be of interest if you are really unfamiliar with the subject. Inevitably, a problem pertinent to any interpretation of QM must be addressed: In chapter 7, I will try to say something useful on the measurement problem, and the role of the observer. In chapter 8, I like to touch upon some quite radical ideas on the interpretation of QM, namely some new parallel theories like MIW (and ofcourse also the older Everetts MWI theory). Chapter 10, is a very simple intro on some newer ideas, that SpaceTime itself can be build from entanglement, as well as that entanglement might play a large role in new gravitational models. Here, one central theme is the EREPR paradigm. I personally think that this is a promising path in physics. Whatever you might think of it. it certainly is quite spectecular. Due to my own limitations, the whole text is quite elementary, and it almost exclusively uses (as we say in Dutch) a sort of Jip en Janneke languange. Sorry for that So, you are warned But if you want to try it anyway. then lets see what this is all about. Contents at a glance: Chapter 1. Introduction to the EPR paradox. Chapter 2. A few words on the original Quantum Teleportation article. Chapter 3. A nano intro to QM: some important concepts and notations in QM. Chapter 4. A nano intro to QM: pure states and mixed states. Chapter 5. The inequalities of Bell, or Bells theorem. Chapter 6. A few words on Steering, Entanglement, and Bell non-locality. Chapter 7. A few words on the measurement problem. Chapter 8. Some Many Worlds interpretations in QM. Chapter 9. A few words on entropy. Chapter 10. Newer EREPR models and ideas. Short Appendices: appliances A1: Quantum Radar. Chapter 1. Introduction to the EPR paradox. 1.1 Introduction and Some background information. We know that it is not allowed for information, to travel faster than c. However, there exists certain situations in Quantum Mechanics (QM), where it appears that this rule is broken. I immediately haste to say, that virtually all physicists believe that the rule still holds, but that something else is at work. What that something else precisely is, is not fully clear yet, although some well-funded ideas, do exist. QM uses several flavours to mathematically describe entities (e.g. a particle), properties (e.g. position, spin), and events. One such flavour which is often used is the Dirac (vector) notation. In chapter 3, for folks who are not too aquinted with QM, I will introduce some highlights of QM, and the wavefunction, and something on the Dirac notation. For now, however, even if you would have only a basic notion of vectors, then we will come a long way in this chapter. Suppose we have a certain observable (propery) of a particle. Suppose that this observable indeed can be measured by some measuring device. For the purpose of this sort of text, the spin of a particle is often used as the characteristic observable. This spin, resembles an angular magnetic momentum, and can be either be up (or often written as or 1), or down (or written as - or 0), when measured along a certain direction (like for example the z-axis in R 3 . Most physicists agree on the fact that the framework QM is intrinsically probabilistic. It means that if the spin of a particle is unmeasured, the spin is a linear combination of both up and down at the same time . Actually, it resembles a vector in 2 dimensional space (actually a 3D Bloch sphere), so in general, such a state can be written as: 966 a1 b0 (equation 1) ( Note: arguments can be found to speak of a 3D Bloch sphere, but we dont mind about this, at this moment. ) where 1 and 0 represents the basis vectors of such superposition. This is indeed remarkable by itself But this is how it fits in the framework of QM. There is a certain probability of of finding 9661 and a certain probability of finding 9660 when a measurement is done. For those probabilities, it must ho ld that a 2 b 2 1, since the total of the probabilities must add up to one. By the way, the system described in equation 1, is often called a qubit, as the Quantum Mechanical bit in Quantum Computing. Similarly, a qutrit can be written as a linear combination of 0 and 1 and 2, which are three orthogonal basis states. It all really looks like it is in vector calculus. The qutrit might thus be represented by: 966 a0 b1 c2 (equation 2) However, in most discussions, the qubit as in equation 1, plays a central role. Now, suppose we have two non-interacting systems of two qubits966 1 and 966 2 (close together). Then their combined state, or product state (that is: when they are not entangled), might be expressed by: 936 966 1 8855 966 2 a 00 00 a 01 01 a 10 10 a 11 11 (equation 3) Such a state is also called seperable, because the combined state is a product of the individual states. If you have such a product state, its possible to factor out (or seperate) each individual system from the combined equation. Note: When we say of equation 1, 966 a1 b0, that it means that it is a linear combination of both up and down at the same time . thus simultaneously, then this statement is a common interpretation in QM. Most folks do not question this interpretation . however, some folks still do. In some cases, such an equation as equation 3, does not work for a combined system of particles. In such case, the particles are fully intertwined, and in such a way, that a measurement on one, affects the state of the second one . The latter statement is extremely remarkable, and is what people nowadays would call steering, as a special subset of the more general term entanglement . Suppose we start out with a quantum system with spin 0. Now suppose further that it decays in two particles. Since the total spin was 0, it must be true that the sum of the spins of the new particles is zero too. But, we cannot say that one must have spin up, and the other must have spin down. However, we can say that their combination carries zero spin. It may appear strange, but a good way to denote the former statement, is by the following equation: 936 187302. ( 01 10 ) (equation 4) Note that this is a superpostion of two states, namely 01 and 10. In QM talk, we say that we have a probability of 189 to measure 01 for both particles, and likewise, a probability of 189 to measure 10 for both particles. That is, after measuerement. Note that an expression as 10 actually seems to say that one particle is found to be up, while the other is found to be down. But the superposition means, that both particles can be in any state, at the same time. Before measurement, we simply do not know. We only know that 936 is 936. 1.2. Describing the apparent strange case of the EPR paradox. A more historical perspective is presented in section 1.3. But I like to jumpstart directly to what is known as the EPR paradox. The following case uses two persons, namely Alice and Bob, each at seperate remote locations. Each have one member particle, of an entangled system (like equation 4), in their labs. The description of the case below, is not without critism: -One important question is, if one would consider the case using a pure states interpretation, or a mixed state interpretation. To better understand that, we should already know have studied chapter 4. For now, however, we take for granted that this case deals with one pure state, as decribed by equation 4. However, in a real experimental setup . we would have many experiments, and so introducing an ensemble anyway. This also means that to certain degree, statistics and correlations comes into play too. -Furthermore, as the spacetime seperation increases, some may also express doubts as to how real (or effective) a decription as equation 3 remains to be true. However, this argument is rather weak. Entanglement is really quite established, and confirmed, even over large distances like in Quantum Teleportation experiments. Only dissipation of the entangled state, due to interactions with the environment (decoherence), might weaken or destroy the entanglement (possibly even very abruptly). -Also, one may argue that both Alice or Bob will simply measure up or down for their member particle, and no strings attached. A down to Earth vision states that, nothing what Bob does, or what Alice does (or measures), will change a thing on their private measurements. One might suspect, that only if Alice informs Bob, or the other way around, one might find correlations. Such viewpoints probably complicates on how to interpret the results. However, the steering of Alices findings, on Bobs member particle, is believed to be true, since experimental results support this view. Whatever is true. or what we need to be careful of, I like to present the case in its original form. However, it will be a simplification of the original idea, and later experimental setups. Fortunately, it is quite accepted to present the case this way. Look again at equation 4. Both states, 01 and 10, seem to have an equal probability to be true or found, after an measurement has been performed. If any measurement is performed, the state reduces to 01 or 10, independend of any distance. This is the heart of the apparant problem. Note: The whole system seems to be pure, in a superposition, while each term seems to be mixed. The differences will be touched upon in chapter 4 (on pure- and mixed states). Suppose we have an etangled system again, which can be described by equation 4. Before we do any measurement, suppose we have a way to seperate the two particles. Lets say that the distance seperating the particles, gets really large. Alice is in location 1, where particle 1 is moving to, and Bob is in location 2, where particle 2 just arrived. Now, Alice performs a measurement to find the spin of particle 1. The amazing thing is, that if she measures up along a certain axis, then Bob must find down at the same axis. Do not think too lightly on this. We started out by saying that (in QM language), both states, 10 and 01, have an equal probability to be true. The total state, is always a superposition of 01 and 10 where each have an equal probability. How does particle 2 knows, that particle 1 was found by Alice, to be in the 1 state, so that particle 2 now knows that it must be 0 You might say that particle 1 quickly informs particle 2 of the state of affairs. But this gets very weird if the distance between both particles is so large, that only a signal (of some sort) faster than the speed of light, is involved. That is quite absurd ofcourse. (1). Equation 4 is not a socalled mixed state. Its a superposition, and a pure state. The probabilities calculated with mixed state, go a little different compared to true pure states. See chapters 3 and 4 for a comparison between mixed- and pure states. (2). The EPR paradox was first concieved by Einstein, Podolski and Rosen (1935). More information can be found in many sections below. The apparent paradox is that a measurement on either of the particles seems to collapse the state 936 187302. ( 01 10 ), thus of the entire entangled system, into 01 or 10. But the superposition (equation 4) was in effect, all the time . Why that collapse, which always determines the state of the second particle In effect, if you observe one particle along some measurement axis, then the other one is always found to be the opposite. This seems to happen instantaneously . for which we have no classical explanation. The effect has been experimentally confirmed, by Stuart Freedman (et al) in the early 70s, and quite famous are are the Aspects experiments of the early 80s. However, since the experiments were statistical of character, they were not fully loophole free. Later more on this. By the way, the first loophole free experiments were done in 2015 (Delft), almost conclusively confirming the strange effect as described above. In this setting, it really looks as if Alice steers what Bob can find. - One (temporary) explanation with a certain consensus among physicists: It would not be good to keep the apparant paradox, fully open, at this point. Its true that much details must be worked out further, since all descriptions above, are presented in a very simple and incomplete manner. If the distance between Alice and Bob is sufficiently large, then if you would assume that the first measured particle, informs the second particle on which state it must take, then such signal must go faster than the speed of light. This is quite is unacceptable, for most physicists. In the thirties of the former century, several models were proposed (Einstein: see chapter 2), of which the (Local) Hidden variables theory was the most prominent one. Essentially it is this: At the moment the entangled pair (as in section 1.2) is created, a hidden contract exists which fully specify their behaviours in what only seems to be non-local events . Its only due to our lack of knowledge of those hidden variables, which make us think of a spooky action at a distance. In a way, this hypothesis is a return to local realism. - A Modern understanding (without full consensus among physicists): A modern understanding lies in the superposition of the entangled state as expressed by equation 4. Alice may measure her qubit, and she finds either up or down, each with a 50 probability. She knows nothing about Bobs measurement, if he indeed did something on that at all. One modern interpretation then says, that Alice does not know for certainty what Bobs finding is, or will be . unless Bobs does his measurement at his member particle (along the same direction). Now, the magic actually sits in the words unless Bobs does his measurement . which also implies that Alice and Bob (at a later time) compare their results. This magic thus sits in the entanglement, or non-locality, where both terms are rather similar, when considering pure states. Now, researchers are still faced with the astonishing inner workings of entanglement, non-locality, and steering, of which I hope that this simple note can shed some light on. Another, but related, puzzle: I also have to mention, that many researchers say, that to know that both particles truly form a pure state (like equation 4), you need access to both particles. If you can only observe one particle, you will effectively see a traced out entity, which, in case of a system like equation 4, will reduce to a mixed state. If one tries to seperate the equation for one particle, from the entangled pair, some weird results comes to surface. Using a well know mathematical operation, that is, tracing out one particle from equation 4, returns the reduced density matrix (see chapter 3 and 4): 961 189 ( 0 60 0 1 60 1 ) (equation 5) Although we havent covered how to do a partial trace yet, equation 5 above is a mixed state , meaning a statistical ensemble (see chaper 4). Say that we did the partial trace for Bobs particle, then the density matrix simply tells us there is 50 chance for finding 1 , and 50 chance for finding 0 . This fact is pretty amazing, since the probability calculation seems a local effect. These sorts of statements can be confusing, but thats also due to the fact that we still miss some important theory at this point. In later chapters, I will set things straight. -Other views: I also must make clear at this point, that some physicists do not see non-locality as the mechanism at work here. Some still stick to the socalled Hidden variables (see 1.3 and Chapter 6), and some even consider quite exotic theories like parallel World interpretations. First, we need some more information on the historical setting, and the spooky action at a distance, as it was percieved in the 30s, and even up to the 90s of the former century. So, there probably is a strange effect, and there is a consensus among many physicists, that entanglement, and non-locality (and steering), form the basis of the effect as described above, and which is rather quite un-like as the world we know from classical physics. And, as a sort of summery of critism: there is always a but. It must be stressed, that some physicists say that EPR may appear non-local . but since faster than light (FTL), is not possible, we dont really have non-locality, but there is something else going on. For example, hidden variables, or something else we are not aware of yet. Thus take notice that there are physicists who are quite weary of non-locality. It seems that those physicists, measure non-locality against the possibility of FTL. Since suggesting FTL is quite sinful in physics, non-locality must be false. However, please take notice of the fact that most physicists view non-locality as the best explanation. Still no definite answer here So lets proceed with some more historical background, and then focus on more modern insights. 1.3 EPR and possible alternatives. Quite a few famous scientists contributed to QM, roughly in the period 1890-1940. Ofcourse, also in later decades, countless refinements and discoveries took place. However, the original basic fundaments of QM were laid in the forementioned period. Eistein contributed massively as well. My impression is, that his original positivism towards the theory, slowly diminished to a certain extend, and mainly in the field of the interpretation of QM, and more importantly, to the question as to which extend the theory of QM truly represents reality. Together with a few colleques, in 1935, he published his famous EPR article: Can A Quantum-Mechanical Description of Physical Reality Be Considered Complete (1935). There are many places where you can find this classical article, for example: Even in this title you can already see some important themes which occupied Einstein: Physical Reality and Complete. There are several circumstances and theoretical QM descriptions, which troubled Einstein. Here, I like to describe (in a few words), the following four themes: (1): As an example of Einsteins doubts, may serve the determination of position and momentum, which are quite mundane properties in the Classical world. However, with socalled Quantum Mechanical non-commuting observables, it is not possible to measure (or observe) them simultaneously with unlimited precision. This is especially fairly quickly to deduce, using a wave-function notation of a particle. It is also expressed by one of Heisenbergs uncertainty principles: 916 p 916 v 189 8463 This relation actually says, that if you are able to measure the velocity (v) very precisely, then the momentum (p) will be (automatically) very imprecise. And vice versa. These sort of results of QM, made Einstein (quite rightfully) to question, as to how much reality we can attribute to these results of QM. (2:) Then we have the problem of local realism too. In a classical view, local realism is only natural. For example, if two billiard balls collide, then thats an action which causes momentum to be exchanged between those particles. As another example: a charged particle in an electric Field, notices the local effect of that field, and it may influence its velocity. As we have seen in section 1.2, the measurement of one particle of an entangled pair, seem to directly (instantaneously) have an effect on the measurement the other particle, even if the distance is so large that the speed of light cannot convey information of the first particle to the second one, in time. This is an example of non-locality. Many others had strong reservations to non-locality too. Quite a few conservative (in some respects) hypothesises emerged, most notably the Hidden variables theory. In a nutshell it means this: At the moment the entangled pair (as in section 1.2) is created, a hidden contract exists which fully specify their behaviours in what only seems to be non-local events . Its only due to our lack of knowledge of those hidden variables, which make us think of a spooky action at a distance. In a way, this hypothesis is a return to local realism. I must say that some alternatives to the Hidden variables existed too. In 1964, the physicist John Stewart Bell proposed his Bell inequality, which is a mathematical derivation which, in principle, would make it possible if a local realistic theory could produce the same results as QM. The Bell theorem was revised at a later moment, making it even a stronger argument for a conclusive test. Although Bells theorem is not controversial among physicists, still a few have reservations. The revised Bell inequality have indeed be put to the test in various experiments, in favour of QM. These tests seem to invalidate Local theories, like the Local Hidden variables, and promote the non-local features of QM. (3): The EPR authors also have some serious doubts on how to handle an entangled system, such a described above in 1.2. For example, if Alice would like to change the set of basic vectors, how would it affect Bobs system In fact, especially these doubts form the basis of Einsteins sceptism on the representation of QM on reality. (4): This theme is again about an entangled system. This time, the EPR authors considered entanglement mainly with respect to position and momentum. According to QM, both observables cannot be sharply observed simultaneously. The authors then provide arguments as to why QM fails to give a complete description of reality. Given the fact that QM was fairly new at that time, it seems to be a quite understandable viewpoint, although various physicists strongly disagreed with those arguments. As of the 90s, it seems to me that more and more people started to doubt the argumentation of the EPR authors, partly due to newer insights or theoretical developments. But, as already mentioned, also in the period of the 30s, some physicists fundamentally disagreed with Einsteins views (like for example Bohr). Before we go to EPR steering and some other great proposals, lets take a look at a nice example which has hit the spotlights the last few decades, namely Quantum Teleportation. I really do not have a particular reason for this example. But it exhibits strong characteritics of non-locality, and something which many folks call the EPR channel. And amazingly, we will see that we need to use classical bits and a classical channel too Chapter 2: a few words on the first article on Quantum Teleportation (QT) (1993). The following classical article, published in 1993: Teleporting an Unknown Quantum State via Dual Classical and EinsteinPodolskyRosen Channels (1993) by Charles H. Bennett, Gilles Brassard, Claude Crpeau, Richard Jozsa, Asher Peres, and William K. Wootters. really started to set the QT train in motion. Quantum Teleportation is not about the teleportation of matter, like for example a particle. Its about teleporting the information which we can associate with that particle, like the state of its spin. For example, the state of the system described by equation 1 above. A collarly of Quantum Information Theory says, that unknown Quantum Information cannot be cloned. This means that if you would succeed in teleporting Quantum Information to another location, the original information is lost . This is also often referred to as the no-cloning theorem. It might seem rather bizar, since in the classical world, many examples exists where you can simply copy unknown information to another location (e.g. copying the content of a computer register, to another computer). In QM, its actually not so bizar, because if you look at equation 1 again, you see an example of an unknow state. Its also often called a qubit as the QM representative of a classical bit. Unmeasured, it is a superposition of the basis states 0 and 1, using coefficients a and b. Indeed, unmeasured, we do not know this state. If you would like to copy it, you must interact with it, meaning that in fact you are observing it (or measuring it) . which means that it flips into one of its basis states. So, it would fail. Hence, the no-cloning theorem of unknown information. Note that if you would try to (stronly) interact with a qubit, it collapses (or flips) from the superpostion into one of the basis states. Instead of the small talk above, you can also formally work with an Operator on the qubit, which tries to copy it, and then it gets proven that it cant be done. One of the latest records in achieved distances, over which Quantum Teleportation succeeded, is about 150 km. What is it, and how does an experimental might look like Again, we have Alice and Bob. Alice is in Lab1, and Bob is in Lab2, which is about 100km away from Alice. Suppose Alice is able to create an entangled 2 particle system, with respect to the spin. So, the state might be written as 936 187302 ( 01 10 ), just like equation 4 above. Its very important to realize, that we need this equation (equation 4) to describe both particles , just as if they are melted into one entity. As a side remark, I like to mention that actually four of such (Bell) states would be possible, namely: 936 1 187302 ( 00 11 ) 936 2 187302 ( 00 - 11 ) 936 3 187302 ( 01 10 ) 936 4 187302 ( 01 - 10 ) In the experiment below, we can use any of those, to describe an entangled pair in our experiment. Now, lets return to the experimental setup of Alice and Bob. Lets call the particle which Alice claims, particle 2, and which Bob claims particle 3. Why not 1 and 2 Well, in a minute, a third particle will be introduced. I like to call that particle 1. This new particle (particle 1), is the particle which state will be teleported to Bobs location. At this moment, only the entangled particles 2 and 3, are both at Alices location. Next, we move particle 3 to Bobs location. The particles 2 and 3, remain entangled, so they stay strongly correlated. After a short while, particle 3 arrived at Bobs Lab. Next, a new particle (particle 1), a qubit, is introduced at Alices location. In the picture below, you see the actions above, be represented by the subfigures 1, 2, and 3. The particles 2 and 3, are ofcourse still entangled. This situation, or non-local property, is often also expressed (or labeled) as an EPR channel between the particles. This is presumably not to be understood as a real channel between the particles, like in the sense of a channel in the classical world. In chapter 2, we try to see what physicists are suggesting today, of which physical principles may be the source for the EPR channelnon-locality phenomenon. Lets return to the experimental setup again. Suppose we have the following: -The entangled particles, Particles 2 and 3, are collectively described by: -The newly introduced particle, Particle 1 (a qubit) is decribed like we already saw in equation 1, thus by: Also note the subscripts, which may help in distinguishing the particles. At a certain moment, when particles 1 and 2 are really close, (as in subfigure 4 of the figure above), we have a 3 particle system, which have to be described using a product state . as in: 952 123 966 1 8855 Psi 2,3 (equation 6) Such a product state, does not imply a strong measurement or interaction, so the entanglement still holds. Remember, we are still in the situation as depicted in subfigure 4 of the figure above. We now try to rewrite our product state in a more convienient way. If the product is expanded, and some some re-arrangements are done, we get an interresting endresult. Its quite a bit math, and does not add value to our understanding, I think, so I will represent this endresult in a sort of pseudo Ket equation: Note the factor Phi 12 . We have managed to factor out the state of particles 1 and 2 into the Phi 12 term. At the same time, the state of particle 3 looks like a superpostion of four qubit states.Indeed. Actually, it is a superposition. Now, Alice performs a masurement on particle 1 and particle 2. For example, she uses a laser, or EM radiation to alter the state of Phi 12 . This will result in the fact that Phi 12 will collapse (or flip) into another state. It will immediately have an effect on Particle 3, and Particle 3 will collapse (or be projected, or flip) into one of the four qubit states as we have seen in equations 7 and 8 above. Ofcourse, the Entanglement is gone, and so is the EPR channel. Now note this: While Alice made her measurement, a quantum gate recorded the resulting classical bits that resulted from that measurement on Particles 1 2. Before that measurement, nothing was changed at all. Particle 1 still had its original ket equation 966 1 a1 b0 We only smartly rearranged equation 6 into equation 7 or 8, thats all. Now, its possible that you are not aware of of the fact that quantum gates do exists, which functions as experimental devices, by which we can read out the classical bits that resulted from the measurement of Alice. This is depicted in subfigures 5 and 6 in the figure above. These bits can be transferred in a classical way, using a laser, or any sort of other classical signalling, to Bobs Lab, where he uses a similar gate to reconstruct the state of Particle 3, exactly as the state of particle 1 was directly before Alices measurement. Its an amazing experiment. But it has become a reality in various real experiments. -Note that such an experiment cannot work without an EPR channel, or, one or more entangled particles. Its exactly this feature which will see to it, that Particle 3 will immediately respond (with a collapse), on a measurement far away (in our case: the measurement of Alice on particles 1 2). -Also note that we need a classical way to transfer bits, which encode the state of Particle 1, so that Bob is able to reconstruct the state of Particle 3 into the former state of Partcle 1. This can only work using a classical signal, thus QT does NOT breach Einsteins laws. -Also note that the no cloning theorem was also proven here, since just before Bob was able to reconstruct the state of Particle 1 onto Particle 3, the state of the original partice (particle 1) was destroyed in Alices measurement. -Again, note that both a classical- and a nonclassical (EPR) channel, are required for QT to work. Chapter 2, simply fits quite well in the central theme of this note. It seems that QT does not work without strong entanglement between member particles. Whether this is really proof of non-locality. it seems quite likely. Chapter 3. A nano introduction to QM, with regards to operations and notations. I think that it might be useful to provide for a nano-introduction into QM. So, only some of the very basic subjects will be shown in this chapter. Nevertheless, using the whole of this text, including this chapter, might help if you like to try the professional articles on EPR and related subjects. I will start with describing the wavefunction, which was the orginal form to represent entities like particles having position and momentum. This is still heavily used today. However, Diracs ket vector notation fits vector spaces (Hilbert spaces) rather well, and using it, seems to be the favourite practice among physicists (since quite some time). 3.1 The Wavefunction and Probabilities. 3.1.1 Description of the Wavefunction: Near the end of the 1800s and the early 1900s, some amazing experiments were performed. While the classical theories (electrodynamics, classical mechanics), made a clear distincion between particles and waves, some experiments pointed towards a more dualistic character of entities. For example, particles that were beamed through a double slit, created an interference pattern , a phenomenon of which it was formerly thougth, that it could only be produced by waves like electromagnetic radiation. As another example, the photoelectric effect showed that light, at certain circumstances, behaved like particles, like transferring momentum to a real particle (such as an electron). So, at certain observations, particles could behave like waves, and the other way around, what was traditionally seen as waves, could behave like a particle. Planck and Einstein found in such observations, that radiation thus exibited a corpusculair character. Also was discovered that radiation seemed to be emitted (or absorbed) in quanta (discrete energy packets). And, as already noted, it seemed that those quanta possessed momentum too, as was observed in certain experiments. Many other observations, and theoretical considerations, lead to the start of a new theory, which was able to accurately describe microscopic entities and events. Indeed, this was the primordial form of Quantum Mechanics. Around 1924, De Broglie showed, that there exists a relation between momentum (p) and wavelength ( 955), in a universal way. In fact, its a rather simple equation (if you see it), but with rather large consequences. Its this: p 8462 955 (equation 9) where h is Plancks constant. Now, momentum, at that time, was considered to be a true particle-like property, while wavelength was understood to be a typical wave-like property, which for example stuff like light and radio waves have. The formula of De Broglie, is quite amazing really. The conseqence is thus, for example, if you have a particle like an electron flying around, you can associate a wavelenght to it. So, whats going on here Do we have a matterwave or something Intuitively, when we now think of a particles position, we might visualize a certain spreading in that position. This and many more considerations, lead some people to introduce the wavefunction 936(r,t). to describe observables of an entity. More specifically, it can be used particularly well to describe the position of a particle. Or formulated in a more general way: a wave function in QM is a description of the quantum state of a system. Around the twenties of the former century, more and more physicists started to describe states in terms of the wave function. What fitted the bill quite well, was that around 1926, Erwin Schrodinger published a mathematical equation about the evolution, that a quantum system undergoes with respect to time, when that system happen to be in some sort of force field. So, a mathematical description was sought, to find a wave-like equation, that could for example describe the position of a particle, and which also adheres to Schrodingers equation. A flat plane wave, like: 936(r,t) e i(kr-969t) (equation 10) does not normalize, since its just a flat front expanding in space and time. By the way, equation 10 very much resembles a classical flat wave equation. But a superposition of multiple modes, can normalize, if the modes (defined by k) are also multiplied by a sort of gaussian distribution function. The result then is a wave packet, localized around some maximum, and quickly lowering amplitude of we observe positions away from that maximum. The addition of waves to the packet, is like adding harmonics with a mode k (related to the frequency), each with a relative amplitude g(k). The more wave components you would add, the more the particle gets localized. In the limit, the summation would become an integral. In the example below, we use one dimension x only, and we also consider the time-independent solution as well. 936(x) 18730 (2960 8463) 8747 -8734 8734 g(k) e ikx dk (equation 11) If you would really use an infinite summation, thus really an a integral, the equation above would represent a very localized position x. If it is a summation, then a spreading around a maximum, would be expected. A physical interpretation of this, is that any wavefunction 966(x) can be expressed as a superposition of states e ikx8463 In case of a limited superpostion, and discrete k, we then can write equation 10 as: 936(x) 18730 (2960 8463) 931 A n e ikx (equation 12) Maybe, in using the original QM language, its better to say that a particle does not have classical properties like position or momentum. It might be better to say, that the wavefunction must be used to calculate probabilities of such observables. Thats probably right So now we are going to see how to do that. 3.1.2. Probabilities: Its often called the Borns rule, since Max Born was the first who proposed how to calculate probabilities when using the wavefunction. Specifically, it can be used (and easily be visualized) when discussing the position of a particle represented by the wavefunction 936(r,t). Although the Dirac notation is more often often used, the orthodox wavefunction representation is rather equivalent to Diracs notation, although Diracs Bra Ket vectors, nicely and directly relate to mathematical vectorspaces. Now, how can we use 936(r,t) to calculate probabilities It can be made plausible, by going back for a while to traditional electrodynamics . Its really true, that if you would have a classical wave 966, then the energy density of that wave is proportional to 936 2. So, the energy (per unit volume) of a light wave is: E 8765 936 2 Born applied that same reasoning to the QM wavefunction probability density. However, the wavefunction is a complex valued function, so the QM probability density is: 961(r,t) 936(r,t) 936(r,t) 936(r,t) 2 (equation 13) where 936(r,t) is the socalled complex conjugate. Now, from complex number theory, we know that z z z 2 , so then equation 13 is correct. This knowledge, must imply that, the probability P to find a particle in a certain region is: P(somewhere on x-axis) 8747 -8734 8734 936(x,t) 2 dx 1 (equation 14) P(x in a,b) 8747 a b 936(x,t) 2 dx (equation 15) where I only used one dimension x, instead of 3 dimensional space r . Equation 14 is logical, since the probability to find the particle somewhere on the x-axis, must be 1 (100). Rewriting that for R 3 would then result in: P(somewhere in Space) 8747 -8734 8734 936( r ,t) 2 d r 1 (equation 16) 3.2 Schrodingers equation. We now know what the wavefunction represents. Ofcourse, its gearded towards position and momentum, of an entity like a particle. In short: The wavefunction 936(r,t) represents the probability distribution of finding the particle. But how does it behave in time The evolution of the wavefunction, is described by Schrodingers equation (1925). It sort of describes the movement of the wave, due to a force acting on it. This is the case if the particle has a potential, due to some field. Its a differential equation, relating change in time (movement) to change in Energy (change in potential). Its often notated as: You can read it as: Total Energy term Kinetic Energy term Potental Energy term. If you would take time for it, and like to try some math, you would see that if you would try equation 10 in Schrodingers equation, you would see that it indeed is a solution. You can put equation 10 into the differential equation, and it works. Therefore, a superposition of equations like equation 10, is a solution too Formally, the differential equation above, is also called the time dependent Schrodinger equation. A simpler variant is the time-independent Schrodinger equation, since it effectively does not take the differential in time, into consideration. Sometimes this is indeed allowed, if a system does not fundamentally change in time. For example, the particle in a box problem can be solved using a time-independent Schrodinger equation. The particle in a box problem, is, viewed from a timeline perspective, a stationary problem. However, you can always simply start (or try) using the time-dependent Schrodinger equation. We are going to try that as an example. Example: The particle in a box problem (1 dimensional, x-axis). For about the math in this eaxmple, I think it is not relevant at all. But, it would be really nice, if you just follow the general argument here. Just interpret the problem as a wavefunction in a 2 dimensional box, that is, a horizontal x-axis of length L, with vertical boundaries at x0 and xL. Along the y-axis, we may visualize the Amplitude of the wavefunction. As a little abstraction of the situation, lets suppose that: -at the vertical boundaries of the box, the potential V is 8734. -for 0 2 8706 2 -------- 2m 8706 x 2 Since, if we look at equation 18, here we simply have V0, and 916 means 8706 2 8706 x 2 , in a one dimensional situation. Although its not important to follow the math here, its really possible to rewrite equation 19 into: The equation above, is actually a well-known differential equation, for which mathematicians and physicists immediately have general solutions. Again, this note is not a math textbook, and here it is not important at all, so I will simply say that a solution can be: 936 N (x) C e ik N x Or, you might also say that you have found a classical-like, harmonic solution as If a mathematician or physicists, use the boundary conditions in a proper way, a good solution then is: where P is some constant, and L is the interval on the x-axis where V0. Remember too, that k N 2 2m E N 8463 2 What we see here, is that we have an integer N, determining the eigenstates of the equation, or in other words, we have found sine-like wavefunctions, each respectively with a higher frequency, and a higher energy E N too. They all obey the Schrodinger equation, as expressed in equation 19, for this particular situation. 3.3 Common operations and notations. In order to be able to understand professional articles, you need to know at least the meaning of the operations and notations listed below. This section will be very short, and very informal, with the sole intention to provide for an intuitive understanding of such common operations and descriptions. If we consider, for a moment, vectors with real components (instead of complex numbers), some notions can be easily introduced, and get accessible to literally everyone . As a basic assumption, we take following representation as an example of a state 966 931 a i u i , like for example 966 a 1 u 1 a 2 u 2 a 3 u 3 . Especially, I like to give a plausible meaning to the following notations: (1): 60 AB . The inproduct, or inner product, of vectors, usually interpreted as the projection of B on A (or rather the ket B casted on the bra 60 A). (2): B 60 A. usually corresponds to a matrix or linear operator. (3): 966 60 966. corresponds to the Density matrix of a pure state. (4): 60 966 O 966 : corresponds to the expectation value of the observable O. (5): The Trace of an Operator Tr(O) 931 60 i O i Proposition 1: (1): 60 AB . is the inner product, of vectors or Kets. -Inner product of two kets 60 AB If we indeed use the oversimplification in R 3. then a (regular) vector or Ket) B can be viewed as a column vector: Note the elements b i of such a vector. We know that we can represent a vector as a row vector too. In QM, it has a special meaning, called a Bra, as to be the row vector with complex conjugate elements b i . Lets not worry about the term complex conjugate, since you may view it as a sort of mirrored number. And if such an element would be a real number, then the complex conjugate would be the same number anyway. The Bra 60 B can be viewed as a row vector: The inner product, as we know it from linear algebra, operates in QM too. It works the same way. The inner product of the kets A and B (as denoted by Dirac) is then notated as 60AB. From basic linear algebra, we usually write it as A 183 B . or sometimes also as ( a , b ). However, we stick to the braket notation: Which is a number, as we also know from elementary vector calculus. Usually, as an interpretation, 60 AB can be viewed as the length of the projection of B on A. Or, since any vector can be represented by a superposition of basis vectors, then 60966 i 934 represents the probability that 934 collapses (or projects or change state) to the state 966 i . -Inner product of a ket, with a basivector: 60u i 966 As another nice thing to know is, is that if you calculate the inner product of a (pure) state, like: 966 a 1 u 1 a 2 u 2 a 3 u 3 with one of its basis vectors, say for example u 2 (and this set of basis vectors is orthonormal), then: 60u 2 966 a 1 60u 2 u 1 a 2 60u 2 u 2 a 3 60u 2 u 3 a 2 -Operators: The operator O, as in OB C , meaning O operating on ket B , produces the ket C Ofcourse Operators (mappings) are defined too in Hilbert spaces. Here, they operate on Kets. Indeed, linear mappings, or linear operators, can be associated with matrices . This is no different from what you probably know of vector calculus, or linear algebra. Here is an example. Suppose we have the mapping O, and ket B. Then in many cases the mapping actually performs the following: meaning that the columnvector (ket) B is mapped to columnvector C. Or, simply said, the operator O maps the ket B to ket C We can write that as: Above, we see an example of how to multiply a column vector with a row vector, which is a common operation in linear algebra. It simply takes the syntax and outcome as you see above. So, proposition 2 seems to be plausible, since it follows that B 60 A is a matrix. Proposition 3: 966 60 966. corresponds to what is called the Density matrix of a pure state. In proposition 2, we have seen that B 60 A usually produces a matrix. Now, if we take a ket 966 and multiply it with its dual vector, the bra 60 966, as in 966 60 966, then ofcourse it is to be expected we get a matrix again. However, the elements of that matrix are a bit special here, since the elements tell us something about the probability to find that pure state in one of its basis states. In a given basis, the diagonal elements of that matrix, will always represent the probabilities that the state will be found in one of the corresponding basis states. In its most simple form, where we for example have that 966 u 1 u 2 , the density matrix would be: 9484 189 0 9488 9492 0 189 9496 The density matrix is more important, as a description, when talking about mixed states. Proposition 4: 60 966 O 966 : corresponds to the expectation value of the observable O. -Trying to make that plausible using simple vectors: We can make that plausible in the following way: We have associated a certain observable (such as momentum, position etc..) with a linear operator O. Now suppose for a moment that we have diagonalized the operator, so the only diagonal elements of the matrix, are not 0, and represent the eigenvalues. Then we may use an argument like so: where u i are basis vectors. We can write it as a columnvector too (in our simplification): We are going to show that 60 966 O 966 is the expectation value of O, by making it plausible for a simple case, thereby hoping that you will agree that it is true in general as well. Now suppose O is represented by the matrix: 9484 0 0 0 9488 9474 0 0 0 9474 9492 0 0 1 9496 which result can be read as the weighted average of the eigenvalues. Thus we say that its the expectation value of O. I hope you can see some logic in this. Proposition 4 is however, valid for the general case too. -Using the usual argumentation in QM: We know that for the usual wavefunction (over x, not considering t), the probability distribution (density), can be expressesd as: 961(x) 936 (x)936(x) which resembles equation 13 (but in this example for one dimension only). For the expectation value 60x, that is, the best average of finding 936 to be at a certain position x, considering all possible x (all x Space), then may be expressed as: 60 x 8747 -8734 8734 936 (x)936(x) d x If we now have a certain observable Q of 936, and we like to know the expectation value 60 Q of that observable, then we can use the integral shown above, for the determination of the average of Q: 60 Q 8747 -8734 8734 936 (x) Q(x) 936(x) d x (equation 21) We need to multiply the probability distribution 961(x) of 936, with Q(x), and integrate that over all space (here we only have considered one dimension). Proposition 5: The Trace of an Operator is: Tr(O) 931 60 u i Ou i . The trace of an Operator, or matrix, is the sum of the diagonal elements. With respect to pure- and mixed states, it has a different outcome (namely 1 or 3 (thus real numners only). In R 3. we can have the following set of orthonormal basisvectors: 9484 1 9488 9474 0 9474 9492 0 9496 9484 0 9488 9474 1 9474 9492 0 9496 9484 0 9488 9474 0 9474 9492 1 9496 You may say that those basis vectors corresponds to u 1 , u 2 , u 3 , like in our usual ket notation. If we consider the rightside of the expression 931 60 u i Ou i , then we have Ou i . We can interpret this as that O operates on a basisvector u i . Suppose that i1, meaning that it is our first basis vectors, just like the set of basisvectors of R 3 , as was listed above. Lets operate our matrix of O, operate on our basisvectors. I will do this only for the (1,0,0) basisvector (i1). For the other two, the same principle applies. So, this will yield: 9484 a b c 9488 9474 d e f 9474 9492 g h i 9496 9484 1 9488 9474 0 9474 9492 0 9496 9484 a1b0c0 9488 9474 d1e0f0 9474 9492 g1h0i0 9496 9484 a 9488 9474 d 9474 9 492 g 9496 Well, this turns out to be the first column vector of the matrix O. Lets call that the vector A (8224). Next, lets see what happens if we perform the leftside of 931 60 u i Ou i . We already had found that the vector A corresponds with Ou i . Using the leftside, we have 60 u i A . This is an inner product, like: 9484 1 9488 9474 0 9474 9492 0 9496 9484 a 9488 9474 d 9474 9492 g 9496 Note that this number a, is the top left element of the matrix O. Since Tr(O) 931 60 iOi , it means that we repeat a similar calculation using all basisvectors, and add up al results. Hopefully you see that this then is the sum of the diagonal elements. I already proved it for the first diagonal element (a), using the first basis vector. The 2 vectors remaining, to be used for a similar calculation, will then produce b and c. In this simple example, we then have Tr(O) a b c . Note that in general Ou i produced the i th column of O (see 8224) above. In the exceptional case where Ou i produces au i , thus a scalar coefficient a times a basisvector, thus Ou i a u i , then in our simple example in R 3. we would have: And, keeping in mind that Ou i the i th column of O (see 8224), then we would have a matrix with only the diagonal elements which are not null, and all others (off diagonal elements), which would then be nul. In such case, it is often said that the elements a, b, and c are the eigenvalues of the operator O. Its absolutely formulated in Jip Janneke language, but I hope you get the picture. 3.4 A few words on Diracs notation 3.4.1 Some general observations: In general, we may write a state vector, or a ket, as expanded (as a superposition) of basis states u i : 966 931 c i u i c i 60 u i 966 Thus each number c i is the inner product of 966 and u i 966 represents a quantum system to be in the state 966 and is called the state vector. Operations on 966, or its basis states, goes very similar as to what you may know from vector calculus. Indeed, for example, in section 3.3 we have seen the inner product of two kets. And the following are kets too, like x or p, representing the position and momentum respectively. In some (exceptional) cases, you might even use definitive kets, like x2, meaning the position of the quantum system (like a particle), is at x2. Its true that we often treat kets, like a statevector, where we speak of a certain probability of finding the statevector to be in the basis state (eigenstate) u i , after some measurement is done. That is true, but it can also be that the state is in a certain value (or certain basis state), for example, after an observation has been performed. Some notations we already are familiar with, can be written in a slightly different (but equal) form. Just take a look at this. We already know our qubit: Now suppose we have a 2 dimensional Hilbert space. An arbitrary vector can be expressed as a linear combination of the unit vectors or basis vectors. Suppose the vectors i and j form such a basis. An arbitrary vector v, or ket, might be written as: v c 1 i c 2 j 60 i v i 60 j v j i 60 i v j 60 j v Its a slightly different format, but its really the same as c 1 i c 2 j . Since v v , the above equation also means that i 60 i v j 60 j v (i 60 i j 60 j) v . Thus: i 60 i j 60 j 1 You can also express it in a matrix, since we already know from the former section that, in general, A 60B is a matrix. So: 9484 1 09488 9492 0 19496 i 60 i j 60 j Where I is the identity matrix or identity operator. Above, we considered the 2 dimensional case. In general, in dimension n, we may say that: 931 u i 60 u i I If u i represents a complet set of orthonormal basis states, of that n dimensional space. 966 931 c i u i But we may thus also write: 966 931 u i 60 u i 966 3.4.2 Interpretations of the Bra: We know what a ket represents, which is not too hard to visualize. Its simply a vector. But how can we interpret a bra 1. Informal interpretation: This interpretation is almost true, but it can be used as a very close, and good pictorial interpretation. In proposition 1, of the former section, we saw that if we represent A as a column vector, then 60 A is the corresponding row vector, with complex conjugate elements. If needed, take a look at proposition 1 again, of section 3.3. So, its actually a nice interpretation (I think). If you have the ket 966, then 60 966 represents the same state, but as a mathematical object, its the transpose of 966, that is, this time a row vector. Since the elements are complex numbers, 60 966 uses complex conjugate elements. And whats really quite the same, note that: 60 966 ( 966 ) 8224 Where the dagger symbol 8224, stands for conjugate transpose or Hermitian transpose. Here it means reversing the column elements to row elements (or vice versa), and taking the complex conjugate of those elements. 2. More abstract (and formal) interpretation: It fits mathematically well, to say that bras are elements of the dual Hilbert space. Or, the bras are vectors of that dual Hilbert space. What this also means, is that bras are functionals on the ket, which produce a (complex) number. Thats fine, since we already know that for example: 60 u i 966 c i Thus, here we can view it as the functional 60 u i , operating on 966 , producing the number c i . But if we consider a certain ket, 966 , then its corresponding bra, is unique 60 966. There is always a one-to-one correspondence from a certain ket, to its unique bra. 3. The bra as the projection, or final state: If we look at (2) again, above, we can also say that 60 u i 966 represents the projection of 966 , on 60 u i . Note that we here talk about a different bra, and a different ket (not a certain ket with its unique corresponding bra). So, if in an observation (or experiment), it turns out that we find the value c i , then we can also say (or interpret it), as that 966 was projected (or cast) on 60 u i , or you may also say that 966 was projected (or cast) on u i . 3.4.3 Some well known states you may see in articles: 936 a1 b0 936 a0 b1 c2 - Product state, or outer product of two qubits: 936 966 1 8855 966 2 a 00 00 a 01 01 a 10 10 a 11 11 - General Product state, or outer product of three qubits: 936 a 1 000 a 2 001 . a 8 111 -Specific example of the product state of three qubits: 936 187308 000 187308 001 187308 010 187308 100 187308 110 187308 011 187308 101 187308 111 -Bell entangled states of 2 qubits: 936 1 187302 ( 00 11 ) 936 2 187302 ( 00 - 11 ) 936 3 187302 ( 01 10 ) 936 4 187302 ( 01 - 10 ) - (Entangled) Singlet state: 936 187302 ( 01 - 10 ) -A Greenberger-Horne-Zeilinger (GHZ) entangled state, of three qubits: GHZ 187302 ( 000 - 111 ) Often, this is the entangled 3 qubit state which is used in experiments or theoretical conjectures, quite similar to our familiar bi-particle Bell states, or the Singlet state. 3.4.4 Some more words on Operators, Matrices, Observables: Its probably really neccessary, I believe, that you have seen section 3.3, before turning to this one. I (like to) think that 3.3 is a quite gently introduction on how we use matrices in QM. (1). One of the rules in QM is that with each measurable observable of a quantum system, is associated a quantum mechanical (linear) Operator . If you read that literally, then the Operator defines (so to speak), the observable. We know that the wavefunction, or statevector, represents the probability amplitude of finding the system in a certain state of the observable. (2) You may also rephrase (1) to this: A wavefunction (Schrodinger like) or statevector (like Dirac formulated it), describes the observable quantity, while the Operator acts on the wavefunction (or statevector). It is true that a measurement can be formulated as an Operator acting on the wavefunction (or statevector). This often then means a projection on one of the eigenstates, with a certain associated probability of finding that particular eigenstate. With this view in mind, we can indeed say that the Operator is associated with the observable. Formally, in the sense of regular vector calculus, the Operator A working on the ket 936, produces another ket 966. So: A 936 966 (equation 22) Equation 22 is very general. That is, the ket 936 is mapped to 966, just as we know from regular vector calculus. However, if we would have: A 936 a 936 (equation 23) then 936 is called an eigenstate or eigenket of the operator A, and a is called an eigenvalue. As another thing, since QM is about physical systems, for A it is also required that: 60 966 60 936 A 8224 (equation 24) where 8224, stands for conjugate transpose or Hermitian transpose. Remember that it is formally defined, that we have a dual Hilbert space of bras (H ), associated with every ket of the regular Hilbert space H. Dont worry about these (possibly confusing) statements. It simply means that there are certain requirements on such Operators. We know that an Operator can be viewed as a matrix, especially when using the Dirac notation. So, there are certain requirements on those matrices too. Taking the conjugate transpose of a matrix A, means that we switch to rows and colums, and take the complex conjugate of all the matrix elements. Its also rather neccessary to have A A 8224 (equation 25) An important reason is, that the expectation value of A, meaning 60 966 A 966 , must produce a real number . and not a complex number. Yes indeed. In real experiments we simply must find real numbers, although QM entities uses complex numbers. If you like, you can write it out, using the examples of section 3.3, and see that this requirement really is rather acceptable. When A A 8224. then we talk about self adjoined Operators. So it simply means that taking the conjugate transpose of a matrix A, results into the same matrix. Note also that in section 3.3, proposition 2, it was made plausible that: represents a matrix, as a very general statement, for two kets A and B . If needed, you can take a look again at section 3.3, to verify that statement, and also some example operations by some Operator O, on kets. Stated in some more formal terms: Given vectors and dual vectors, we can define operators O (i.e. linear mappings from H to H), in the format of: As already said above, we usually want Hermitian or self-adjoined Operators, garanteeing that the expectation value of the Operator, is a real (and not complex) number. Consider the following Operator (and matrix) A: 9484 0 -i9488 9492 i 0 9496 Is this a self adjoined Operator Here it is true that A A 8224. since if you take the conjugate transpose, that is, switch the rows and columns, and take the complex conjugate of each matrix element, you will see that this is indeed true. Note: - the complex conjugate of a real number, like 1, is the same real number again (1). - the complex conjugate of an imaginary number like i, is -i and vice versa. Maybe you have still some questions right now. If so, take a look at section 3.3 again. It will show some foundations of matrices, which may help. Chapter 4. Meaning of pure states and mixed states. 4.1 A few words on pure states: While you might think that a completely defined state as 0 , is pure, it holds in general for our well known superpositions . An example of an superpositional state, can be this: You may also view a pure state as a single state vector . as opposed to a mixed state. So, even at this stage, we already may suspect what a mixed state is. Thus pure states: We have seen them before in this note, sofar. A mixed state is a statistical mixture of pure states, while superposition refers to a state carrying some other states simultaneously. Although it can be confusing, the term superposition is sort of reserved for pure states. So, our well-known qubit is a pure state too: 966 a 0 b 1 Or as a more general equation, we can write: 966 931 a i u i (equation 26) This is a shorthand notation. Then i runs from 1 to N, or the upper bound might even be infinite. Usually, such a single state vector 966, is thus represented by a vector or ket () notation, and is identified as a certain unknown observable of a single entity, as a single particle. So, a pure state is like a vector (called ket), and this vector be associated with a state of one particle. A pure state is a superposition of eigenstates, like shown in equation 29. Other notes on pure states: Such vectors are also normalized, that is, for the coefficients (a 1 . a 2 . etc..), it holds that a 1 2 a 1 2 . 1 Its also often said that a pure state can deliver you all there is to know about the quantum system, because the systems evolution in time can be calculated, and Operators on pure states work as Projection operators. In sections above, we have also seen that the coefficient a i can be associated with the probability of finding the state to be in the ua i eigenstate (or basisvector) after a measurement has been performed. In general, an often used interpretation of 966, is that it is in a superposition of the basis states simultaneously.Then, the keyword here is simultaneously. However, this interpretation depends on your view of QM, since many interpretations of QM exist. But superpostion will always hold, and is a key term of a pure state (like equation 29). When you would insist on the qualifying phrase a pure state gives us all there is to know , then probabily (or maybe) known coefficients are required too, like for example with: Note that some authors treat it that way. But in general, undetermined coefficients are OK too. As long as we can talk of a ket, we have a pure state. Furthermore, it is required that the inner product of 966 with its associated bra, is normalized, that is, has a unit length. That is, the inner product returns the value 1. Thus: 4.2 A few words on mixed states: A mixed state, is a mix of pure states. Or formulated a little better: a probability distribution of pure states, is a mixed state. Its an entity that you cannot really describe, using a regular Ket statevector. You must use a density matrix to represent a mixed state. Another good description might be, that it is a statistical ensemble of pure states. So we can think of mixed state as a collection of pure states 966 i , each with associated probability density 961 i . where 0 8804 961 i 8804 1 and 931 961 i 1. It cannot be stressed enough, that a linear superposition is not a mixture. Mixed states are more commonly used in experiments. For example, when particles are emitted from some source, they might differ in state . In such a case, for one such particle, you can write down the state vector (the Ket). But for a statistical mix of two or more particles, you cannot. The particles are not really connected, and they might individually differ in their (pure) states. What one might do, is create a statitistical mix, what actually boils down in devising the density matrix. The statistical mix, is an ensemble of copies of similar systems, or even an ensemble with respect to time, of similar quantum systems So, you can only write down the density matrix of such an ensemble. In equation 3, we have seen a product state of two kets. Thats not a statistical mix, as we have here with a mixed state. In a certain sense, a mixed state looks like a classical statistical description, of two pure states. When particles are send out by some source, say at some interval, or even sort of continuously, its even possible to write down the equation (density matrix) of two such particles which were emitted at different times. This should illustrate that the component pure states, do not belong to the same wave function, or Ket description. You might see a bra ket-like equation for a mixed state, but then it must have terms like 966 60 981 . which indicate that we are dealing with a density matrix. In general, the density matrix (or state operator) of a (totally) mixed state, should have a format like: Hopefully, you can see something that looks like a statistical mixture here. Here is an example that describes some mix of two pure states a and b : 961 14 a 60 a 34 b 60 b (equation 27) Note that this not an equation like that of a pure state. Ofcourse, some ket equations can be rather complex, so not all terms perse need to have to be in the form 966 60966 . Especially intermediate results can be quite confusing. Then also: by no means this text is complete. Thats obvious ofcourse. For example, partial mixed systems exist too, adding to the difficulties in reckognizing states. A certain class of states are the socalled pseudo-pure families of states. This refers to states formed by mixing any pure state . with the totally mixed state . So, please do not view the discussion above, as comprehensive description of pure and mixed states, which is certainly not the case here. 4.3 What about our entangled two partice system: Equation 4, which described an entangled bipartice system, is repeated here again: 936 187302. ( 01 10 ) Note that this is a normal ket equation, and it is also a superposition. We do not see the characteristic 60 terms which we would expect to see in a mixed state. Therefore, its a pure state There are several perculiar things with such entangled states. We already have seen some in section 1.2, where Alice and Bob performed measurements on the member particles, in their own seperate Labs. Another perculiar thing is this: I will not illustrate it further, but using some mathematical techniques, its possible to trace out the state of one particle, from a two-particle system. -For example, if you would have a normal product state like equation 3, then tracing out particle, like particle 2, just gives the right equation for particle 1. This was probably to be expected, since the product state is seperable. - If you would do the same for an entangled system, then if you try to trace out a particle, then you end up with a mixed state, even though the original state is pure. Thats is really quite remarkable. Later more in this. For now, lets go to the next chapter. Chapter 5. The inequalities of Bell, or Bells theorem. 5.1 The original formulation. The famous Bell inequalities (1964), in principle, would make it possible to test if a local realistic theory, like the Local Hidden Variables (LHV) theory, could produce the same results as QM. Or, in stated somewhat differently: No theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics. Or again stated differently: There is no underlying classical interpretation of quantum mechanics. For about the latter statement, I would like to make a small (really small) reservation, since, say from 2008 (or so), newer parallel universe theories have been developed. Although many dont buy them, the mathematical frameworks and ideas are impressive. In chapter 8, I really like to touch upon a few of them. The Bell theorem was revised at a later moment, by John Clauser, Michael Horne, Abner Shimony and R. A. Holt, which surnames were used in labeling this revision to the CHSH inequality. The CHSH inequality can be viewed as a generalization of the Bell inequalities. Probability, and hidden variables. To a high degree, QM boils down to calculating probabilities of certain outcomes of events. Most physicist, say that QM is intrinsically probabilistic. This weirdness is even enhanced due to remarkable experiments, like the one as decribed in section 1.2. It is true that the effects described in section 1.2, are in conflict with local realism, unless factors play a role of which we are still fully unaware of, like hidden variables. We may say that Einsteins view of a more complete specification of reality, related to QM, is our ignorance of local pre-existing, but unknown, variables. Once these unknown hidden variables are known, the pieces fall together, and the strange probabilistic behaviour can be explained. This then includes an explanation of the strange case as described in section 1.2 (also called the EPR paradox). This is why a possible test between local realism, and the essential ideas of QM, is of enormous importance. It seems that Bell indeed formulated a theoretical basis for such test, based on stochastic principles. I have to say that almost all physicist agree on Bells formulation, and real experiments have been executed, all in favour of QM, and against (local) hidden variables theories. What is the essence of the Bell inequalities In his original paper (Physics Vol. 1, No. 3, pp. 195-290, 1964), Bell starts with a short and accurate description of the problem, and how he wants to approach it. Its really a great intro, declaring exactly what he is planning to do. I advise you to read the secions I and II of his original paper (or read it completely, ofcourse). You can find it here: Bells Theorem, or more accurately, the CHSH inequality, has been put to the test, and also many theoretical work has been done, for example, on n-particle systems, and other more complex forms of entanglement. On the Internet, you can find many (relatively) easy explanations of Bells Theorem. However, the original paper has the additional charm that it explicitly uses local variables, like 955, which stands model for one or more (possibly a continuous range) of variables. His mathematics then explicitly uses 955 in all derivations, and ultimately, it leads to his inequalities. If we consider our experimental setup of section 1.2 again, where Alice and Bob (both in remote Labs), perform measurements again on their member particles, then one important assumption of local realism is, that the result for particle 2 does not depend on any settings (e.g. on the measurement device) in the Lab of particle 1, or the other way around. In both Labs, the measurement should be a local process. Any statistical illusion would then be due, to the distribution of 955, in the respective Labs, as prescribed by a Local Hidden variable theory. The Bell inequalities provide a means to statistically test LHV, against pure QM. In effect, experimental tests which violate the Bells inequalities, are supportive for QM non-locality. Sofar, this is indeed what the tests have delivered. Some folks see the discussion in the light of two large believes: or you believe that signalling is not limited by c, or you believe in super determinism. Super determinism then refers to the situation where any evolution of any entity or process is fully determined. So to speak, as of the birth of the Universe, from where particles and fields snowed from the false vacuum. Interestingly, all particles and other stuff, indeed have a sort of common origin, and thus may have given rise to a super entanglement of all stuff in the Universe. Still unkown variables have then sort of fixed everything, thus a sort of super determinism follows. Personally, I dont buy it. And it seems too narrow too. There are also some newer theories (Chapters 7 and 8) which do not directly support super determinism. 5.2 Newer insights on the Bell inequalities and LHVs. -Simultaneous measurements vs non-Simultaneous measurements. Since the second half of the 90s (or so), it seems that newer insights have emerged on Bells Theorem, or at least some questions are asked, or additional remarks are made. One such thought is on how to integrate the Heisenberg relations into the Theorem, and the test results. Here is a good example of such an article: The authors state that near simultaneously measurements, implicitly relies on the Heisenberg uncertainty relations. This is indeed true, since if Alice measures the spin along the z-direction and if she finds up, then we may say that if Bob would also measure his member particle along the z-direction too, then he will certainly find down. Therefore, the full experiment will use (also) axes for Alice and Bob which do not align, but have a variety of different angles. Then, afterwards, all records are collected, and correlations are established, and then using Bells inequalities, we try to see if those inequalities are violated (in which case LHV gets a blow, and QM seems to win). The point of the authors is however, that the measurements will occur at the same time. If now a time element is introduced in the derivation of Bells theorem, a weakening of the upper bound of the Theorem is found. As the main cause of this, the authors make it clear that second-order Broglie-Bohm type of wavefunctions may work as local operators in the Labs of Alice and Bob. I personally cant really find mistakes, apart from the fact that Broglie-Bohm is actually another interpretation of QM, which might not have a place in the argument. However, I am not sure at all. By the way, the Broglie-Bohm pilot wave interpretation, is a very serious interpretation of QM, with many supporting physicists. However, the main point is that the traditional Bell inequalities (or CHSH inequality) in combination with the experimental setup, is not unchallenged (as good physics should indeed operate). -Werner states. Amazingly, as was discovered by Werner, there exist certain entangled states that likely will not violate any Bell inequality, since such states allows a local hidden variable (LHV) model. His treatment (1989) is a theoretical argument, where he first considers the act of preparing states, which are not correlated, thus not entangled, like the example in equation 3 which is a seperable product state. Next, he considers two preparing devices, which have a certain random generator, which makes it possible to generate states where the joint Expectation value . is no longer seperable or factorizable. His artice is from 1989, where at that time it was hold that systems which are not classically correlated, then they are EPR correlated. Using a certain mathematical argumentation, he makes it quite plausible to have a semi-entangled state, or Werner state, which has the look and feel of entanglement, and where a LHV can operate. He admits its indeed a model, but it has triggered several authors to explore this idea in a more general setting. The significance is ofcourse, to have non seperable systems, using a LHV. If you are interested, take a look at his original paper: -Countless other pros and contras: There are many articles, (somewhat) pro- or contra Bells Theorem. Many different arguments are used in the battle. You can found them easily, for example, if you Google with the terms criticism Bells theorem arxiv, where the arxiv will produce the free, uneditorial, scientific papers. Here is one that makes a strong point against LHV, and is very much pro QM: This article is great, since it uses a model of 2 entangled particles without a common origin . and thus this system is very problematic for any type of classical or LHV related theory. I am not suggesting that you should read the complete artice. Contrary, often only the introduction of such articles is good enough, since then the authors outline their intentions and arguments. The next article uses a truly different perspective. According to this author, we do not need non-locality and all strange observed effects, are simply due (under the hood) to the superpostion principle. Furhermore, he makes a case that QM simply does not give a complete view on reality, just like Einstein said. You can find that article here. However, the EPR experiments, and what we have seen in Quantum Teleportation, is probably hard to understand just by superposition alone, and not thinking in terms of non-locality. So, what do we have up to now Taken all together, you cannot say that there exists a full consensus among physicist on how to exactly interpret the EPR experiments, together with Bells inequalities. However, a majority of physicists still thinks in terms of non-locality, to explain the experimental results, and see sufficient backing of theoretical considerations, for their position. Sofar, what we have seen in section 1.2 (EPR entangled bi-particle experiment), and chapter 2, (Quantum Teleportation), is that something that behaves like an immediate action at a distance, seems to be at work. This does not suggest that any form of signallingcommunication exist, that surpasses the speed of light. As said in section 1.2, the no communication theorem states exactly that. However, not all folks would agree on this. By the way, the QT effect we saw in section 1.4, simply also needed a classical channel in order to transport the state of particle 1, to particle 3 at Bobs place. That is also supporting the view, that true information transfer does not go faster than c. There exists a number of interpretations of QM, like e.g. the Broglie-Bohm pilot-wave interpretation. Rather recently, also newer parallel universe models were proposed, with a radical different view on QM. For about the latter: you might find that strange, but some models are pretty strong. The most commonly used interpretation, is the one that naturally uses superpositions of states. That model works, and is used all over the World. For example, most articles have no problem at all in writing a state (Ket) as a superposition of basis states, like in a pure state, as we have seen in section 2.1. In fact, once describing QM in the framework of Hilbertspaces (which are vectorspaces), superpostion is then sort of imposed or un-avoidable. But ofcourse, the very first descriptions using wave-functions to describe particles and quantum systems in general, is very much the same type of formulation. And this vector formulation, fits the original postulates of QM, quite well. But it seems quite fair to say that it is actually just this principle of superposition . that has put us in this rather weird situation, where we still cannot fully and satisfactory understand, exactly why we see what think we see as was described in section 1.2 (or the lightgreen text above). Not all physicist like the non-locality stuff as displayed in the lightgreen text above. For quite a few, a Hidden Variable theory (or similar theory) is not dead at all. Although the experimental evidence using the Bell tests seems rather convincing, there still seems to exist quite some of counter arguments. For now, we stay on the pure QM path (superpositions, EPR non-locality, probabilities, Operators, Projectors etc..), and how most people then nowadays interpret Quantum Steering, Entanglement, and Bell non-locality. Lets go to the next section. Chapter 6. Steering, Entanglement, and Bell non-locality. 6.1 Some descriptions: Lets first try to describe steering: Quantum Steering: Quantum steering is the ability of Alice to perform a measurements on her local member of an entangled system, with different outcomes, and that leads to different states for the remote part of that entangled system (at Bobs Lab), independend of any distance between them. How did I came up with such a nice description Here you can find an article of the man who used such text for the first time (Schrodinger, 1935), as a response to Einsteins EPR paper: (in the document, of the url above, you might scroll down a bit, to view the article) If I may quote a nice paragraph from that article: (when he is dicussing two remote members of an entangled system, or entanglement in general. ) . It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenters mercy in spite of his having no access to it. This paper does not aim at a solution of the paradox, it rather adds to it, if possible. A hint as regards the presumed obstacle will be found at the end. Schrodinger already considered (or suspected) the case (as described in section 1.2), that the result that Alice measures, instantaneously steers what Bob will find. Althoug in section 1.2 we saw steering at work, I also like to try to discuss a modern test too, involving steering, and this all under the operational definitions as listed below. Many questions are left open at this point, among which are: - Can Alice steer Bob - Can Bob steer Alice - Does two-way steering exists - What is the difference when pure systems and mixed systems are considered - Does all types of entangled systems, enable steering We are not too far of from possible answers. Lets next try to describe entanglement and Bell non-locality. Entanglement: When 2 or more particles can be described as a product state (like equation 3), they are seperable. A measurement of an observable on one particle, is independent of the other particles. You can always seperate the original ket (of a certain particle) from the product state. In many cases however, two or more particles are fully intertwined (with respect to some observable), in such way, that you cannot seperate one particle from the other(s). A measurement on one particle, effects the other particle(s) too. A state as for example in equation 4, describes both particles (together in SpaceTime). They truly have a common (inseperable) state. A sort of key definition then seems to be: If you cannot write a combined system as a full product state, then its an entangled system. But then still various forms exist, like partially entangled systems (partly seperable), or maximally entangled systems (not seperable at all). Bell non-locality: This seems to apply for any situation, for which QM violates the Bell inequalities. So, it seems to be a very broad description. You might say that entangled states as in sections 1.2 and 1.4, fall under the non-locality description. How about steering Seems that this too, as a subset, is smaller than the notion of non-locality. But this is not correct. The exact difference, or applicability, between steering, entanglement, and Bell nonlocality, was not so much of a very hard issue in the minds of physicists, so it seems. We have to admit that steering, entanglement and Bell nonlocality, seemed to have much overlap in their meanings. Well, it proved to be not entirely true. Then, in 2006, the following article appeared: by Wiseman, Jones, and Doherty. They gave a pretty solid description for Steering, Entanglement, Nonlocality, in the sense of when such term applies . As the authors say themselves: they provided (sort of) operational definitions. The statements above with respect to the relative place (as subsets or supersets) of steering, entanglement, and nonlocality, were not corrects. As the article points out: Proposition 1: -We need entanglement to enable quantum steering. -But not all entangled systems provide conditions for quantum steering. The above sounds rather logical, since quantum steering, or EPR steering, is pretty much involved, and just seems to be a rather hard quality for true a non-classical phenomenon. The authors formulate it this way: Steerable states are a strict subset of the entangled states. So, if you would regard this from the perspective of Venn diagrams, then Steerable states lie within entangled states. Or, in other words: the existence of entanglement is necessary but not sufficient for steering. Thus: steering is deeper than just entanglement, although entanglement is required. Proposition 2: -Steering is a strict superset of the states that can exhibit Bell-nonlocality. This thus would imply that steering could happen in a local setting, which might be percieved as quite amazing. In other words: in a Bell local setting (thus NOT nonlocal), steering is possible too. Or, and this is important, some steerable states do not violate the Bell inequalities. As we shall see a while later, if we would only consider pure states . the original equivalence holds to a large extent. But considering mixed states too . leads to the propositions above. I recommend to read (at least) the first page of this article. True, all these sorts of scientific papers are rather spicy, but already on page one, the authors are able to explain what they want to achieve. 6.2 Entanglement Sudden Death: Maybe the following contributes to evaluating entanglement. or not. However, its an effect that has been observed (as of 2006) in certain situations. Early-stage disentanglement or ESD, is often called Entanglement Sudden Death in order to stress the rapid decay of entanglement of systems. It does not involve perse all types of quantum systems, which are entangled. Ofcourse, any sort of state will interact with the environment in time, and decoherence has traditionally been viewed as a threat, in for example, Quantum Computing. ESD however, involves the very rapid decay of the entangled pairs of particles, that is, the entanglement itself seem to dissipate very fast, maybe due to classical andor quantum noise. But the fast rate itself, which indeed has been measured for some systems, has surprised many physicists working in the Quantum field. Ofcourse, it is known that any system will at some time (one way or the other) interact with the environment. Indeed, a general phenomenon as decoherence is almost unavoidable. Its simply not possible to fully isolate a quantum system from the environment. This even holds for a system in Vacuum. Even intrinsic quantum fluctuations has been suggested as a source for ESD. However, many see as the source for the fast decay, the rather normal local noise, as e.g. background radiation. Yu and Eberly have produced quite a few articles on the subject. The sudden loss of entanglement between subsystems may be even explained in terms of how the environment seems to select a preferred basis for the system, thus in effect aborting the entanglement. Just like decoherence, ESD might also play a role in a newer interpretation of the measurement process. Whether it is noise or something else, its reported quick rate is still not fully understood. A good overview (but not very simple) can be found in the following article: To make it still more mysterious, an entanglement decay might be followed by an entanglement re-birth, in systems, observed in some experimental setups, with the purpose of studying ESD. A re-birth might happen in case of applying random noise, or when both systems are considered to be embedded in a bath of noise or other sort of thermal background. Many studies have been performed, including pure theoretical and experimental studies. A more recent article, describing the behaviour of entanglement under random noise, can be found below: As usual, I am not suggesting that you read the complete article. This time, I invite you to go to the Conclusion in the article, just to get a taste of the remarkable results. 6.3 Types of entanglement: Ofcourse, this whole text is pretty much lightweight, so if I cant find something, it does not mean a lot. So far, as I am able to observe, there is no complete method to truly systematically group entangled states into clear categories. There probably exist two main perspectives here. The perspective of formal Quantum Information Theory, in which, more than just occasionally, the physics is abstracted away. This is not a black-and-white statement ofcourse. Pure physics, that is, theoretical- and experimental research. Both sciences deliver a wealth of knowledge, and often must overlap, and often also are complementary in initiating ideas and concepts.So what types of entanglement, physicists have seen, or theoretici have conjectured How much the points below contribute to the understanding of entanglement, I do not dare to say. However, those point constitute knowledge, so at least they must have something to say to us. Anyway, lets see what this is about: 1. Pure- and mixed states can be entangled. For pure states, a general statement is, that an entangled state is one that is not a product state. Rather equivalent, is the statement: a state is called entangled if it is not separable. Mixed states can be entangled too. This is somewhat more complex, and in section 5.4 I will try do a lightweight discussion. 2. The REE distance, or strength of entanglement. Relative Entropy of Entanglement (REE) is based on the distance of the state to the closest separable state. It is not really a distance, but the relative entropy of entanglement . E R compared to the entropy of the nearest, or most similar separable state. In Physics Letters A, december 1999, Matthew J. Donalda, and Michal Horodecki, found that if two states are close to each other, then so are their entanglements per particle pair, if indeed they were going to be entangled. Over the years after, the idea was getting more and more refined, leading to the notion of REE. So, its an abstract measure of the strength of entanglement. Its an area of active research. Intuitively, its not too hard to imagine that for nonentagled states, E R 0, and for strong entangled states E R - 1. So, in general, one might say that 0 8804 E R 8804 1. You could find arguments that this is a way, to classify entangled states. 3. Bi-particle and Multi-particle entanglement. By itself, the distinction between a n2 particle system, and a n 2 system, is a way to classify or to distinguish between types of entanglement. Indeed, point 1 above, does not fully apply to multiparticle entanglement. In a n 2 system we can have fully separable states ofcourse, and also fully entangled states However, there also exists the notion of partially separable states. In ket notation, you might think of an equation like this: 936 966 1 8855 981 2,3 and suppose we cannot seperate 981 2,3 any further, then 936, which then is only separeated in the factors 966 1 and 981 2,3 , is a partially separable state . 4. Classification according to polytopes. When the number of particles (or entities) in a quantum system increases, the way entanglement might be organized, is getting very complex. While with n2 and n3 systems, its still quite manageble, with n 3, the complexity of possible entangled states, can get enourmous (exponentially with n). In 2012, an article appeared, in which the authors explicitly target multi-particle systems, which can expose a large number of different forms of entanglement. The authors showed that entanglement information of the system as a whole, can be obtained from a single member particle . The key is the following: The quantum correlation of the whole system N, affects the single- or local particle density matrices 961(1). 961(N) which relate to the reduced density matrices of the global quantum state. Thus using information from one member alone, delivers information about the entanglement of the global quantum state. From the the reduced density matrices, which thus also correspond to the density matrices 961(1). 961(N) of one member particle, the eigenvalues 955 N can be obtained. Amazingly, using the relative sizes of 955 N . a geometric polyhedron can be constructed which corresponds to an entanglement class. From this different geometric polyhedrons (visually like trapeziums) at least stronger and weaker entanglement classes can be calculated. Using a local member this way, you might say that this single member acts like a witness to the global quantum state. If you like more information, you might want to take a look at the original article of the authors Walter, Doran, Gross, and Christandl: Chapter 7. A few words on the measurement problem. This will be very short section. But I hope to say something useful on this extensive subject. Certainly, due to chapter 8, the role of the observer and the measurement problem, simply must be addressed. Its in fact a very difficult subject, and many physicists and philosophers broke their heads on this stuff. Its not really about in-accuracies in instruments and devices. One of core problems is the intrinscic probability in QM, and certain rules which have proven to be in effect, such as the Heisenberg uncertainty relations. And indeed, on top of this, is the problem of the exact role of the observer. Do not underestimate the importance of that last statement. There exists a fairly large number of (established) interpretations of QM, and the role of the observer varies rather dramatically over some interpretations. Whatever ones vision on QM is, its rather unlikely that it is possible to detach the observer completely from certain QM events and related observations, although undoubtly some people do believe so. So, I think there are at least five or six points to consider: Intrinscic probability of QM. Uncertainty relations, and non-commuting observables. Role of the observer. Disturbive (strong) measurements vs weak measurements. The quantum description of the measurement process. Decoherence and pre-selection of states nearin the measuring device. The problem is intrinsic to QM, and in many ways un-classical. 7.1 Decoherence:: For example point 6, if you are familiar with decoherence, then you know that a quantum system will always interact with the enivironment. And in particular, at or near your measurement device, decoherence takes place, and a process like pre-selection of states may take place. In a somewhat exaggerated formulation: the specific environment of your lab, may unravel your quantum system in a certain way, and dissipates certain other substates. Could that vary over different measurement devices, and with different environments What does that say, in general, about experimental results The subject is still somewhat controversial. A nice article is the following one. Its really quite large, but reading the first few pages already gives a good taste on the subject. You can also search for articles of Zurek and co-workers. 7.2 Role of the Observer: And ofcourse, the famous or infamous problem of the role of the observer. For example, are you and the measurement apparatus connected in some way Maybe that sounds somewhat hazy, but some folks even study the psychological- and physiological state of the observer, with respect to measurements. I dont dare to say anything on such studies, but you should not dismiss them. A comprehensive analysis of making an observation, and certain choices, is very complex, and is probably not fully undestood. But there exists more factual and mathematical considerations. For example, what is generally understood with the Heisenberg uncertainty relations 7.3 A few words on the Heisenberg uncertainty relations: Today, among physicists, a large degree of consensus exists on the following: The Heisenberg uncertainty relations, have nothing to do with disturbing effects of the art of making a measurement. Here too, these relations are truly intrinsic in QM, independent of any measurement. So why then a few words on the Heisenberg uncertainty relations In the early days of QM, especially Schrodingers approach, was popular, which essentially is integral and differential calculus. Diracs vector Ket (bra ket) notation emerged somewhat later. The initial way to describe a quantum system, was by using wave equations. Some time before, it was realized that even particles could exhibit a wave-like character, and that electromagnetic radiation, proved to have a particle-like character at certain observations. Since a particle is quite localized, but not exactly localized, is why physicists introduced the Wave Packet 966(r,t). Its a superpostition too, but for a free particle, the components add up in such a way, that the packet is grossly localized, like some sort of gaussian distribution, with a maximum, and it fastly diminishes at the fringes. The addition of waves to the packet, is like adding harmonics with a mode (related to the frequency) p, each with a relative amplitude g(p). The more wave components you would add, the more the particle gets localized. In the limit, the summation would become an integral like: (one dimension x only, time-independent) 966(x) 18730 (2960 8463) 8747 -8734 8734 g(p) e ipx8463 dp (equation 28) Now, the more modes you add, the narrower in position x, the gaussian bell shaped wave packet becomes. When the number of modes is really large, we get the integral over p, and a very localized x (position). This p is related to the frequency of such mode, and represents the momentum of that mode. So, with just a low number of ps, we have a very dispersed (wide) packet, in the x position. If we have a large number of ps, we have a very narrow packet, in the x position. In other words: With a large variation in p, we have a narrow x. With a large variation in x, we have a narrow p. Using Operators, or just wave functions and Fourier transforms, it can be formalized. But even in this stage, we can hopefully understand how Heisenberg came to his famous uncertainty principle with respect to momentum and position: 916 p x 916x 8805 84632 (only for x-dimension) 916 p 916 r 8805 84632 For these observables p and r, it must hold that no matter how small you try to make the variation, (the deltas), the product must always be larger than a certain number. This result is simply a consequence of the QM wave approach to the position and momentum of a particle. Using the notion of Observables, and corresponding OperatorsMatrices, it can be shown that for non commuting observables (or Operators), it is not possible to measure them simulteneously with unlimited precision. For example, if you would pinpoint the position very precisely, the momentum is very in-precise, and vice versa. The principle applies to all non-commuting Observables, like Energy E and time t, and for example also for the spin components along the x and z directions, usually notated as 963 x and 963 z . This is why the principle can be of relevance in any discussion about measurements: The principle is intrinsic to QM, however, the consequence (or collary) is that it is not possible to measure non-commuting observables simulteneously, with unlimited precision. 7.4 The strange case of section 1.2, and the role of measurements: The strange case of section 1.2, or the EPR paradox, represents one of the most fundamental, and strangest scientific debates for about 80 years (or so), since 1935. We can indeed read all about Bell, and experiments, and steeringentanglementnon-locality, and many more theoretical arguments, but the truth is. there still is no full consensus reached among physicists. Chapter 8. Some Many Worlds interpretations in QM. Some rather remarkable ideas have been published rather recently on new interpretations in QM. Although you cannot truly call them theories, its also true that you cannot call them speculations either, since the authors provide a conceptual and mathematical framework. So, in this text, I will simply call them theories. I like to spend a few words on a few of those rather remarkable theories, but I am afraid I cannot classify this in any other way than saying that it resides in my hobby sphere. So, if there was any reader at all. this stuff is one of my hobbies . so to speak. In this chapter, I like to spend a few words on rather new parallel universe theories, and, in chapter 8, a few words on recent resarch on micro blackholeswormholes, at the Planck scale. However, I will start with the Grandfather theory of Many Worlds in QM, that is, Everetts MWI theory, which was published in 1957. Much later, around 2010 and 2014, newer models were introduced. 8.1 Some important Interpretations of QM: I should start, with a sort of listing of established interpretations. There are quite a few of them, and a few represent a rather dramatic deviation from the (more or less) standard Copenhagen interpretation. Here are a few of them: The Copenhagen interpretation (or in a new jacket). QM with Hidden variables and local realism. QM with non-locality. Decoherence as a successor to the specific Copenhagen collapse. Everetts Many Worlds Interpretation (MWI). The Many Minds Interpretation. Poiriers Waveless Classical Interpretation (MIW). Quantum hydrodynamics and Trajectory analysis (e.g. Madelung, Holland). The transactional interpretation. Time symmetric theories. de Broglie - Bohm Pilot Wave Interpretation (PWI). possibly also the Two State Vector Formalism. I would not dare to say that this listing is complete. Furthermore, some of them quite overlap, but at certain issuess, there are fundamental differences (like nonlocality, and local realism).There are also quite subtle differences. For example, a Ket vector, or wavefunction, can be interpreted as just a working vehicle, but its true existence might be doubted. Some other interpretations see them as a real representation of entities and properties. There are no polls of who likes which the most, but as I observe it, it seems as if a lot of folks are attracted by some features of the Broglie - Bohm Pilot Wave Interpretation. 8.2 The Grandfather theory of Parallel universes in QM: Everetts MWI (1957). Its impossible not to spend a few words on the original Parallel universe theory in QM. It was formulated by Everett, in 1957, although Schrodinger in 1952 already hinted to such an idea, in the form of simultaneous existing outcomes. Everett published his theory as a Phd thesis, called the Relative State Formulation of Quantum Mechanics, under a certain degree of guidance by his mentor Wheeler. Slightly later, also due to the promotional work of DeWitt, it became know as the Many Worlds Interpretation or MWI. In this theory, the wavefunction is a real existing entity . and forms the basis for all entities. The key of his idea is the following. Although in the simple example below, a ket vector notation is used, its rather equivalent to a wave function setting. If you would consider the following state (rather equivalent to a wave function interpretation): and in an observation, you will find the state u 2 , then in the Copenhagen interpretation it is said that the state 966 collapsed (or was projected) to the state u 2 , with a probability that relate to a 2 . Its quite possible then, that you could have found, for example, u 3 , with a probability that relate to a 3 . However, you found one outcome, and the former state is destroyed. In Everetts theory, all 3 outcomes are realized. So, if you performed the measurement, then 3 different states forks off and undergo their further evolution. So, following the example above, the quantum superposition of the combined observer-state and observed object-state wavefunction, will resolve into three relative states, completely independed from each other. Hence the notion of branched off worlds, or Many Worlds. If you want to know more, then here is a (new version) pdf of Everetts original article: Indeed, its not a strange framework at all. Although his idea did not find any support at first, it is true that as from the late 60s, up to the 90s, it became quite popular, and many physicists considered it to be a viable interpretation. Some even considered it to be the best interpretation thus far. Note also that Everetts theory is a no-collapse formulation of QM, quite unlike the Copenhagen interpretation. -However, the popularity declined over later years, and quite a few articles expressed to have found some (supposedly) inconsistencies in MWI. As is rather usual in science, not all physicist turned around, and still some are very sceptical on those (supposedly) inconsistencies in MWI. Below I provide some links to arxiv articles, pro- and contra MWI, which will demonstrate some of those inconsistencies. -New theoretical paths in general in physics, and in QM specifically, probably did not helped MWI much. For example, around the 80s, the theoretical principle of decoherence was discovered (Zeh, Zurek), which provided a new way on how a wavefunction would interact with its environment, like a measuring device, or the environment in general. But it must be said that decoherence shows some important similarities with MWI, but then ofcourse without the branching into other Worlds. But even up to this very day, we have physicist who publish arguments in favour of MWI. Its still a valid interpretation of QM, and as usual, some folks like it, and some dont. I highly recommend to invest some time to explore some great articles in MWI. The following short article tells us about some inconsistensies in MWI, which ofcourse, you dont need to take for granted. But this is an exceptionally nice and sometimes humouristic text, and I am sure you like it. By the way, you will absolutely learn a lot of MWI, by reading this article. However, the same author, at another moment, produces an article which you can call rather pro MWI. Since this one explains MWI rather well, I like to list it here too: The next article is from Vaidman, and is rather against QM non-locality, and the article expresses lots of motivation, why MWI should the theory of choice. 8.3 Modern Parallel Universe theories in QM: beyond 2005 In 2004, a rather remarkable article appeared, from P. Holland, titled: Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation. It looks quite non-quantum to talk about precisely defined spacetime trajectories. At least, this is a common view among some people in the field. Also, the common interpretation of the wavefunction, works against such a view too. However, plenty of physicists would not not agree to such statements. Ofcourse, it also depends on which particular Interpretation of QM, you favour the most. For example, the Broglie-Bohm Pilot Wave Interpretation seems to re-introduce some classical features again. Not literally, but essentially the theory states that we have a real particle, with an accompanying pilot wave . which can be regarded as a velocity field guiding the particle. The pilot wave then (sort of) has the look and feel of the Copenhagen wavefunction. One remarkable thing is, that if you would have the initial conditions, you can further calculate future positions precisely. Hollands article is not part of my intended discussion, but ofcourse the article is great. In short, his article seem to deal about the following: Holland notes that in QM, in certain cases, certain results of hydrodynamics and fluid mechanics, can help in calculating particle positions. Such an idea was first pointed out by Madelung. In such approaches, the QM probability density then looks similar to the fluid density. Then he works out all sorts of calculations, using classical Euler-Lagrange equations to calculate the trajectories of the fluid particles as functions of their initial coordinates, as if you would regard such initial ensemble as the initial wavefunction. In 2010, Poirier published a remarkable article too, called Bohmian mechanics without pilot waves . It was first published in Chemical Physics, Volume 370, May 2010 . The amazing thing is, that Poirier has arguments to strip away the Pilot wave, from the particle, when considering Bohemian mechanics (Broglie-Bohm Pilot wave mechanics). As he says: can a trajectory ensemble itself, be possible too, in the sense that it obeys Schrodingers equation, and therby making no explicit or implicit reference to an external pilot wave Then, using kinematics and rather classical looking differential equations, and using results from (especially) Hollands work (I think), he arrives at the remarkable proposition: It seems possible to replace the wavefunction by a real-valued trajectory field, furher defined by a trajectory density weighting function. The orthodox complex valued wavefunction is unneccessary. Note this line of thinking seems like a reduction of QM, to a semi-classical statistical Mechanics. From this on, he is not so far away anymore from his final MIW interpretation, that is, the Many Interacting Worlds (MIW) interpretation. The Many Interacting Worlds (MIW) interpretation. It seems that Poirier and Schiff were first (2012) in devising a true MIW model. However, in 2014, Wiseman, Hall, and Deckert used a slightly other approach to reach (I think) a better model, since they can avoid a continuum of many interacting worlds. Generally, MIW essentially means this. The proposition is essentially, to have n Classical parallel universes, where in one such Universe, an entity (like a particle), has well defined (sharp) observables (like position), all the time. However, there exists a variation over such observables, in the various different Universes, in such a way, that an observer experiences the illusion of QM, like the implicit spreading of the wavefunction (which leads to an uncertainty in e.g. the position). So, each World it has its own private sharp values for that observable, but since there exists variation between Worlds, we percieve the illusion of a composed State vector where the same experiment might show different values all the time. In fact, in MIW, there is no state vector, no kets, no wavefunctions. This is a remarkable different view, compared to Everetts MWI. Most notably, the differences are: In Everetts MWI, the wave function is real and plays a central role. In MIW, the wave function does not exist, and in fact MIW is a sort of QM without the wavefunction. In Everetts MWI, universes branches off due to the fact that superimposed waves each interact with the environment. In Poiriers MIW, there is no branching, no wavefunctions, but the many worlds may interact. Everetts MWI uses standard Quantum Mechanics (wavefunctions, probabilities etc..), however Poirier uses established rules from many parts of physics, whereas many are simply familiar classical, like trajectories, positionimpuls, Lagrangian etc. where paths over different trajectories relate to the Many Interacting Worlds. A short survey of one of the MIW theories: I think the approach of Hall, Deckert, and Wiseman is the best one, up to this day, for constructing a MIW theory. Their original article, published in 2014, is the following: As usual, I advice to read the whole article. However, even only reading the first 2 pages (the Introduction), will tell you what Poiriers MIW theory tried to prove, and what the plan of the authors is, to provide for a better alternative derivation, in order to arrive to an (almost) equivalent MIW theory. In this paragraph, I will try to say something useful on this improved MIW theory. Essentially, it goes like this: If you would have n particles, then you might define a (single) configuration space for that set of particles, which space can be described by the vector: The number kdepends on which number of dimensions, and which attributes, you would take into consideration. The vector Q . then describes the set of particles, for example, with respect to position. The authors also see this vector as to be equal to the configuration space of the set of particles, and also called the world space for that set. For Q . at a certain t, you could define a position as Q R . Sofar, there is nothing too special here. As an nterim object its possible to define a Probability Density P( Q R ) 966( Q R ) This would then express the distribution of the positions of the particles . Next, they make the assumption that N of such worldspaces, thus Q 1 . Q N could exist, or if you like, could be postulated. If we would find that a credible assumption, we then would have N configuration spaces for those n particles, expressed as the collection Q 1 . Q N . N is not aboslutely fixed, and might even go to nfinity. The remarkable thing then is, and what is not showed here (see the article above), that a sort of local repulsive force can be derived, between those worldspaces Q i . A very important aspect of that local force is, that it is only of relevance when Q l and Q m are close, with respect 966( Q R ). If the worldspaces Q i are now viewed as seperate Universes, which might be percieved as a rather bold proposition, then at least the force between close Universes would be repulsive. The authors then apply this theory to e.g. The Ehrenfest theorem, spreading of the wave packet, tunneling, and a few other QM effects. Remarkably, they indeed seem to succeed in their examples. For example, the spreading of a particle over configuration spaces (the Universes), gives the illusion of a wave packet. However, the Universes themselves, are fully classical, and all observables have sharp values at all times. Other Multiple Worlds theories in the scope of QM: - Many Minds theory: Everetts MWI, and Poirier MIW, are not the only propositions of Many Worlds, in the scope of QM. For example, the Many Minds theory of Dieter Zeh, is one important line of thinking. This theory, actively integrates the observer (or the mind of the observer) in the process of observation, and interpretation. It maybe of interest of people working in the field of psychology, or any sort of neural science. Other Multiple Worlds theories not directly in the scope of QM: Outside the immediate scope of QM, some other Parallel Universe models were deviced. Here, you might think of Hubble Volumes, or Chaotic Inflation and some other models. Some time back, I tried to make an overview in some sort of note. I you want to try it, you can use this link. Even if you do not buy these classes of theories. they certainly gives us a different perspective on QM, and thats valuable anyway. Chapter 9. A few words on entropy. 9.1 What is entropy Entropy is a physical quantity, which is especially used in statistical mechanics, and thermodynamics. Originally, you might say that it especially useful in macroscopic systems, having many entities, like gasmolecules in a room, or cilinder etc.. In ordinary statistical mechanics, the entropy (S) provides us a pointer to the measure of the multiplicity of microstates that sits behind a particular macrostate. For example, you have a large number of gas atoms, where all of them are somehow magically placed in one single corner of a vacuum room, and you release the ban, you would observe that immediately the gas will be distributed all over that volume of that room. In this example, we can talk about on how many different ways the atoms or molecules can be arranged. Just after the ban was released, the number of arrangements increased enormously. When energy is increased into a system, or by adding new arrangements (microstates), entropy is increased. The second law of thermodynamics says that the entropy of a closed system, will never decrease. In fact, thats a remarkable law, and it even may suggest that Nature wants to promote disorder over order, in an isolated system. Ofcourse, we need to be very careful using such sort of statements. A mathematical relation has been found that relates the Entropy S, to the number of microstates W: S 8765 ln(W) (equation 29) In thermodynamics, chemistry, the usual equation is: S k B ln(W) (equation 30) where k B is Boltzmann constant. So, entropy in general seems to be a macroscopic quantity. True, but its applicability has turned out to be extremely general. For example, just think of the all of the subwaves that contribute (or sum up) to a wavefunction or wavepacket. So, entropy can be used here as well And it uses the same natural logarithm, that is, the ln function. Note that equations 29, or 30, have a certain resemblence to Shannons Law (1948) of data communication technologies, and related sciences: S B log 2 (1 SN) (equation 31) where C is the maximum attainable error-free data speed in bps that can be handled by a communication channel, B is the bandwidth of the channel in Hz, and SN is the signal-to-noise ratio of that communication channel. Also note that log N (x) is a logarithmic function, based on N, while ln is the natural logarithmic function, based on e. Although the above relation stems from 1948, modern theories consider variables, symbols, and distribitions. Indeed, entropy has a very central meaning as well in Information Sciences . 9.2 Why entropy can be expressed as a ln(W) function. Many processes in Nature can be expressed in the form of an e x function, or a ln(x) function. Both are indeed quite remarkable. -In the figure below, you see e x , which starts out rather slow, but as x increases, it almost explosively starts to climb upwards. - The inverse function of e x , is ln(x). This one is exactly the mirrored function of e x , with respect to the line yx. The function ln(x) starts out climbing extremely rapidly, but as x increases, it climbs less and less steeply, until it almost (but not quite) reaches a nearly horizontal slope. The function ln(x) is natural. For example, if you have a certain amount of radiactive material, then it will decay over time. The amount of active material thats left after some period t, can be expressed as a function of of the half-live or mean lifetime 964, that is ln(964). How can we make plausible, that ln(W) is indeed related to the entropy S of a system Suppose you add microstates to a system. Suppose you originally had 10 states. Adding for example 3 additional states, is a rather big increase in entropy. Indeed 103 is a relevant change. However, once you reach the number of 1000 states, and again add 3 microstates, then 10003 is not a large increase in entropy anymore. This behaviour reflects precisely the graph of an ln(W) function. The slope gets less and less, if W increases. 9.3 Entropy of a quantum system. The text below, perhaps, may strike you as a bit weird. By now, we know what we must understand about a pure- or mixed state. As it will turn out, the entropy of a pure state is zero. Thats remarkable, since we know that, in general, it is a superposition. First, in general, about the concept of entropy which is involved here, is the Von Neumann entropy, which is shown below. This framework was, and still is, accepted, since it works. Secondly, considering a pure state describing superposition, we talk about quantum superpostion Indeed, we can write a ket (statevector) as a sum of eigenstates (basis vectors), but we dont know really what it is in terms of classical physics. Now, you might say that this is an incorrect statement, since we can attain a probability of finding a certain eigenstate. That is still true, however, a quantum superpostion is still something very different from a statistical ensemble . which a mixed state is supposed to describe. Indeed, in case of a statistical ensemble, we have physical states . on which we can apply true classical statistics, since that statistical ensemble looks exactly like an ordinary statistical ensemble. Remember from chapter 4, that a mixed state is a statistical mixture of pure states . while superposition refers to a state carrying some other states simultaneously. It might be relatively hard to understand (or not). Lets also take a look at the statements below: -Statement: Entropy, as a quantity that in general relates to distinguishable microstates . applies well to mixed states. It simply looks like a classical statistical ensemble, on which we can apply the term entropy. -Statement: Entropy, as applied to quantum superposition, as is meant in a pure state, should indeed return 0, also since we cannot say anything definite of the state (unless we observe it). If we observe it, the former state is lost. After measurement, we simply have one eigenstate. -Statement: Since 1948, entropy started to be used in terms of mathematical communication theory, as information entropy . Folks started to think that the physical degree of distinguishable states of a system (statistical entropy), is related to its information (information entropy). For a pure state in superposition, we only know that 936 936. If we observe it (Copenhagen talk), we are left with a certain eigenstate. A mixed state is a true statistical ensemble. Hopefully, we are in the clear now, for how we must interpret entropy for pure- and mixed states. Lets see how this part of QM works. Chapter 10. EREPR models. Personally, I find these theories (or hypotheses) quite appealing. For example, do you remember the Inflationary Universe theorie(s) Those are regarded as the most plausible theories today, to explain (or describe) the origin and evolution of the Universe. At the tiniest fragment of the start of time, at the earliest phase (maybe around 10 -42 sec), a Quantum Fluctuation (according to the theory), gave rise to a pre-form of SpaceTime and a precursor of Gravity. Then, for very short period, an exponential inflation of SpaceTime took place. In case you are not familiar to Inflationary Universe theorie(s), you might want to do a websearch first on that topic. A rather bold assumption might be made: An embedded relation, even today, between SpaceTime components and for example entanglement, might be possible, according to these lines of thought. Thus the current theories try to establish plausible models for SpaceTime, Gravity, and indeed, QM effects, like entanglement. - Be warned though . that most physicists seem (or probably really are), very weary or sceptical on those models. Its true that only a rather select company of theoretical physicists are actively working on these models. - Be double warned . since the entities they try to study (from a theoretical framework), are on the smallest scale possible, that is, typically in the order of the Planck length. This scale is fully out-of-reach for direct experimental work, and the present day particle accelerators, lack many billions of orders of magnitude of Energy, to probe such small scales. Since this is really a fact, all work done up to now, is purely theoretical. - Or, be not warned at all . since science is just simply always in progress and sometimes we have an established theory, which does not hold up anymore to new experimental results, or new, sufficiently backed, theoretical considerations, especially if established theories fail in certain domains. 10.1 General overview. Lets start with a sort of overview, of which sort of ideas emerged, and when. In the second section, I will try to go somewhat deeper into the theories, but for now, having this sort of overview helps to put stuff in perspective. Entanglement seems to have (or might have) a rather large area of application. At least, that is how many theoretical physicists look at it nowadays, especially since the 2000s, and even more so since 2013. But long time ago, but after QM stood firmly in the physics books, science went on, ofcourse. All those years during the 50s, 60s etc. in the former century, enormous progress was made in astronomy, particle physics, theoretical physics etc.. Its impossible to say anything useful here, ofcourse, unless one is planning to write a book on the theme of progress in physics, during the 50s up to today. Its seems fair to say that string theory (since the 80s), AdSCFS, Quantum field theories, Quantum Gravity, SpaceTime models, Cosmological models etc. kept people busy for a long time. I cant say that those theories, ultimately, came together, but certain results from all of them, created an atmosphere (so to speak), to bring in entanglement into the picture. Holographic principle and Entanglement Entropy. Here is an article (2006) from Shinsei Ryu and Tadashi Takayanagi, where the authors link entanglement from a holographic perspective on entropy from AdSCFT: This link above, is for illustrational purposes only. You can read it ofcourse, but its rather involved. I simply only like to create a (although on a nano-scale) small historical perspective too. Thats why I listed the article above. AdSCFT is a specific SpaceTime model, and in some respects compatible with string theory, in the sense of a certain correspondence. The article seems to succeed in deriving entanglement entropy from minimal surfaces (one dimension less), in some form of AdS space of certain dimensions. Its all highly theoretical, but its getting quite concrete in using entanglement in SpaceTime models. Entanglement and the creation of SpaceTime geometries. More importantly is the following article. And this time, you are encouraged to read (or browse through) it. Possibly, its the first concrete article, postulating entanglement as the cement in the SpaceTime fabric. Its the classic article from M. van Raamsdonk (2010), and you can find it here: Ofcourse, the article is partly inspired from former work, like e.g. articles of Maldacena, but nevertheless, as far as I know, it explicitly uses quantum entanglement to build geometries of SpaceTimes. Ofcourse, a various ideas on the dicreteness of SpaceTime, spin-networks, loops, already circulated quite some time before (e.g. some ideas of Penrose and others). But finding entanglement as the fundamental sculpter of the geometry of SpaceTime, is quite new (or new). Essentially, using the methods from Ryu and Takayanagi, Raamsdonk shows that if you would slowly tear down entanglement from a certain AdS model of SpaceTime, then when entanglement is finally reduced to nothing, this SpaceTime will be no more than fully disjoint parts of SpaceTime. Hereby, making plausible that the principle of entanglement, creates SpaceTime. Remember, this was only 2010. The methods of Ryu and Takayanagi stems around 2006. Some fundamental ideas of Malcedena were from (about) 1997. For (about) 2010 to today . the ideas are really alive throughout the community of theoretical physicists, and many refinements and explorations were made. Entangled Black Holes. As a slightly other line of thought, theoretical explorations were done on the subject of entangled black holes. Although Einsteins Relativity theory allows for the principle of Wormholes (also called the Einstein-Rosen bridge, or the ER bridge), Juan Maldacena and Leonard Susskind, introduced the idea of applying entanglement on pairs of Black Holes. One of their articles is the following: In this facinating article, the authors explore the idea that an Einstein Rosen bridge between two black holes, might be very similar to the EPR-like correlations as seen in many applications and experiments in QM. In other words: entangled Black Holes. In the same article, the authors say that its tempting to suspect that any correlation through entanglement, has its roots in ER bridges (or wormholes) on a microscopic scale. Hereby, they rooted the idea of EREPR, which made quite a few folks enthousiastic for that concept. The article is quite spicy, if not at least some core ideas are introduced. Thats my challenge for the sections below. I am afraid that it all will be a bit lengthy. 10.2 AdS, Strings, Entropy, Holographic picture. 10.3 EREPR wormholes. 1057 PC Amsterdam The Netherlands KvK: 37125573 tel: Int: (0031)(0)6 2060 4148 NL: 06 2060 4148 mail: albertvanderselgmail absrantapex.org Any questions or remarks Then contact me at. albertvanderselgmail Site maintained by: Albert van der Sel last update: 16 Februari, 2017 Nederlandstalige paginas: Klik aub hier voor enkele andere Nederlandstalige paginas.