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Essay on the Foundations of Classical and Quantum Information Theories Essay on the Foundations of Classical and Quantum Information Theories André Tanus Cesário de Souza August 2011 Essay on the Foundations of Classical and Quantum Information Theories André Tanus Cesário de Souza Orientador: Prof. Dr. Reinaldo Oliveira Vianna Versão Final - Dissertação apresentada à UNIVERSI- DADE FEDERAL DE MINAS GERAIS - UFMG, como requisito parcial para a obtenção do grau de MESTRE EM FÍSICA. Belo Horizonte Brasil Agosto de 2011 Dedicate To my mother Roane Tanus (in memoriam). to my friend Mário Mazzoni, to Reinaldo O. Vianna, and also to Karol Zyczkowski˙ and Ingemar Bengtsson. Acknowledgements Agradeço ao grande professor e físico, meu mestre e amigo Mário Mazzoni. Sem a tua amizade e o teu apoio nesse último semestre, seria impossível nave- gar por esse mar quase infinito de tristezas e dificuldades... muito obrigado pelas lindas poesias e conversas maravilhosas. Agradeço também pelos ensi- namentos, tanto no subespaço da técnica, (quase todo o meu conhecimento em física eu aprendi direta ou indiretamente contigo), quanto em todos os outros subespaços da vida. Ao meu orientador e amigo Reinaldo O. Vianna, pelas infinitas discussões sobre o que há de mais bonito e fundamental na Mecânica Quântica. Seus conselhos sempre foram como uma luz-guia enquanto eu sutilmente divergia. Agradeço aos professores Aldo Delgado Hidalgo e Carlos Saavedra pelo excelente mês no Chile, na Universidade de Concepción. Um agradecimento especial ao Aldo, pelas discussões e ensinamentos. Agradeço à Telma e Wallon pela imensa e impagável ajuda no Chile. Agradeço os meus amigos Thiago e Wanderson pela maravilhosa viagem Chile afora, por mais de dois mil kilômetros. Sem vocês essa viagem seria impossível! Agradeço à grande professora Maria Carolina Nemes, (uma das forças da natureza). A imensidão do teu conhecimento ilumina a todos. Agradeço também aos grandes físicos C. H. Monken e Sebastião de Pádua. Espero aprender muita física com vocês ainda. Agradeço o professor Jafferson Kamphorst, e ainda os professores Marcos Pimenta e João Antônio Plascak pelo exemplo e dedicação, e também os amigos do CETEC-MG: o Reinaldo Trindade, Elenice Cavichioli, Adriano Menezes e o professor J. Roberto T. Branco - pela oportunidade de estudo na Paraíba. Grande parte do capítulo de formalismo matemático foi inspirado nos dois ótimos livros do professor Marcelo Terra Cunha, portanto o meu obrigado a ele.1 Agradeço à minha família: meu Pai, a Raquel e Gustavo (pelos desenhos em Corel) – um beijo para Helena (minha afilhada e paixão da minha vida). Um agradecimento especial à minha avó querida, sempre minha companheira e amiga. Ao meu avô, que sempre foi um exemplo para mim. Ao meu tio Renê -uma das pessoas mais cultas que eu já conheci. Agradeço ao tio Libério que, apesar de não termos contato hoje em dia, grande parte do meu desejo de 1I’d like to express my gratitude to Scott Aaronson for the idea followed in the Section 2.6. I’d like also to thank Peter E. Blöchl for the idea followed in Section 4.6.2. This text can be understood as a kind of non-geometrical introduction to the Zyczkowski˙ and Bengtsson textbook. A special thanks to the authors mentioned in the bibliography page. aprender física descende daquelas nossas conversas (juntamente com o meu pai) sobre o universo quando eu era criança. E outro agradecimento especial ao amor da minha vida Ida B. Rodríguez, (e ao Thiaguinho) pela imensa compreensão, amor e carinho. Você é a luz que ilumina o meu universo. Um agradecimento especial ao grupo InfoQuant: Meu amigo Debarba, (que me emprestou uma fonte de computador para que eu pudesse latecar), meu amigo Fernando, (um grande músico e parceiro), meu amigo e colab- orador Thiago, (aprendi muita física contigo) e ao amigo Júnior. Agradeço o Leozão por ter me ensinado muita física. Também agradeço ao grupo Enlight pela enorme colaboração e ajuda dos meus amigos Mateus Araújo e do Marco Túlio. Agradeço aos outros amigos do Enlight: Pierre, o Breno, o Marco Aurélio e o Wanderson, que muito me ensinaram. Agradeço também aos amigos e colegas do Depto. de Física da UFMG2: a Amanda e Tauanne pelo carinho, a Lígia pelos sorrisos, a Lídia, Regiane e Jenaína pelas ótimas conversas. Agradeço aos meus amigos: Geraldo Parreiras, a Luciana Cam- braia, o Daniel Parreiras, a Ingrid e a Juliana, e o Régis. Agradeço também o meu amigo Tiago Campolina, o Geraldão, a minha amiga Monique, o Daniel Massote, a Fabíola Rocha e os meus grandes amigos Bolivar e Capitão Clyffe. Agradeço também Juan Loredo (meu amigo peruano). Um Agradecimento mais que especial à Nina pelos conselhos e pela amizade: a tua amizade foi e ainda é muito importante para mim, principalmente nos tempos de piração. Aos meus segundo-irmãos Euler P. G. de Mello e Igor Laguna (o meu grande companheiro dos botecos da vida) e Felipe de Lena, que me ajudou na correção do texto. Agradeço à Marluce e Ieda. Um agradecimento especial à Shirley e Maria Clarice. Agradeço ao presidente Luiz Inácio Lula da Silva e à minha querida presidenta Dilma Vana Rousseff pelo melhor governo da História do Brasil. Agradeço também aos Beatles e ao Chico Buarque. E um agradecimento especial ao grande amor da minha vida: Clube Atlético Mineiro. – The author does not consider this work as completed due to problems after its presentation and lack of review. Suggestions for improvement and error notes can be sent to [email protected] – This text was written by a student for students. – Este trabalho teve o apoio financeiro direto e indireto da CAPES, da FAPEMIG e do CNPq. 2Enumerar pessoas é algo terrível, por isso mantive o critério de ordenamento o mais entrópico possível, com o vínculo da ordem em que os nomes apareciam em minha mente. 5 Abstract This work is an essay that assesses some important tools in Classical and Quan- tum information theory. The concept of qubit (quantum two level systems) is introduced in a simple manner by using a few statistical optics concepts (Stokes vectors and coherence matrices). Unlike most texts of Quantum infor- mation in which the qubits of spin are often addressed, in this text the concept of polarization qubit is developed. Shannon and Von Neumann entropy func- tions are introduced formally and, therefore, some of the particularities of classical and quantum information theory are highlighted and confronted. The most important properties of these entropies are demonstrated with detail. The concept of majorization is introduced for both cases: the quantum and the classical ones. The Jaynes problem is discussed in the light of classical theory of information (The Jaynes principle), in the quantum case, this problem -which is still open- is approached in a simple and intuitive manner. Keywords qubit, polarization qubit, Jones vector, Jones matrix, Stokes parameters, coherence matrix, Poincaré–Bloch sphere, convex set, Shannon entropy, conditional entropy, joint entropy, mutual entropy, Von Neumann entropy, relative entropy, entropy properties, entropy inequalities, Kullback-Leibler distance or divergence, majoration, bistochastic matrices, Schrödinger mixture theorem, Jaynes principle, maxent problem, classical information, quantum information. 6 Resumo Este trabalho é um ensaio sobre algumas ferramentas importantes para a Informação Clássica e Quântica. O conceito de qubit (sistema de dois níveis quântico) é introduzido de forma simples via óptica estatística (vetores de Stokes e matrizes de coerência). Contrariando a abordagem da maioria dos textos de informação Quântica, nos quais os qubits de spin são frequentemente abordados, nesse texto o conceito de qubit é o de polarização. As noções de entropia de Shannon e de Von Neumann são introduzidas formalmente e, com isso, algumas das particularidades das teorias de informação clássica e quântica são evidenciadas e confrontadas. As propriedades mais importantes dessas entropias são detalhadamente demonstradas. O conceito de majoração é introduzido e desenvolvido em ambos os casos: clássico e quântico. O problema de Jaynes clássico é discutido sob a luz da Teoria de Informação Clássica, (Princípio de Inferência de Jaynes) e, no caso quântico, esse problema -que ainda permanece em aberto- é abordado de forma simples e intuitiva. Palavras-chave qubit, qubit de polarização, vetor de Jones, matriz de Jones, parâmetros de Stokes, matriz de coerência, esfera de Poincaré–Bloch, conjuntos convexos, entropia de Shan- non, entropia condicional, entropia conjunta, informação mútua, entropia de Von Neumann, entropia relativa, propriedades da entropia, desigualdades para entropia, distância ou divergência de Kullback-Leibler, majoração, matrizes biestocásticas, Teo- rema da mistura de Schrödinger, princípio de Jaynes, problema maxent, infomação clássica, informação quântica. Contents Contents 8 List of Figures 11 List of Tables 12 1 Introduction 13 2 What is a Qubit? 16 2.1 Two Level Systems.......................... 16 2.2 The Polarization Qubit........................ 17 2.3 Jones Matrices............................. 18 2.4 Coherence Matrix Formalism.................... 19 2.5 The Stokes Parameters and Poincaré-Bloch Spheres....... 19 2.5.1 The Stokes Parameters.................... 19 2.5.2 The Poincaré–Bloch Spheres................ 21 2.5.3 Experimental Determination of One Qubit........ 24 2.6 What is the Essential Difference Between one Qubit and a Probabilistic Classical Bit?...................... 25 2.6.1 “Negative Probabilities” - Interference.......... 25 3 Mathematical Background–The Space of Density Matrices 29 3.1 Some Important Facts About Metric Spaces............ 29 3.1.1 The Distance Function.................... 29 3.1.2 Norms............................. 30 3.1.3 Vector Spaces and Inner Products............. 31 3.2 The Algebra of the Classical and Quantum Observables.... 31 3.2.1 Some C∗-Algebra Properties................ 32 3.2.2 The Algebra of the Classical Observables......... 32 3.2.3 States of a Commutative Algebra.............. 33 3.2.4 The Algebra of the Quantum Observables........ 34 3.2.5 States of the Non-Commutative Algebras......... 34 3.3 Convex Sets.............................. 35 3.4 The Density Matrix Formalism................... 37 3.4.1 The Postulates of Quantum Mechanics.........
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