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Correlaciones En Mecánica Cuántica: Entrelazamiento Y Quantum Discord Como Recursos Para Realizar Procesos En Información Cuántica

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Correlaciones En Mecánica Cuántica: Entrelazamiento Y Quantum Discord Como Recursos Para Realizar Procesos En Información Cuántica UNIVERSIDAD DE CONCEPCIÓN FACULTAD DE CIENCIAS FÍSICAS Y MATEMÁTICAS DEPARTAMENTO DE FÍSICA Correlaciones en Mecánica Cuántica: Entrelazamiento y Quantum Discord como Recursos para Realizar Procesos en Información Cuántica Tesis para optar al grado académico de Doctor en Ciencias Físicas por Marcelo Javier Alid Vaccarezza Concepción, Chile Septiembre 2012 UNIVERSIDAD DE CONCEPCIÓN FACULTAD DE CIENCIAS FÍSICAS Y MATEMÁTICAS DEPARTAMENTO DE FÍSICA Correlaciones en Mecánica Cuántica: Entrelazamiento y Quantum Discord como Recursos para Realizar Procesos en Información Cuántica Tesis para optar al grado académico de Doctor en Ciencias Físicas por Marcelo Javier Alid Vaccarezza Director de Tesis : Dr. Luis Roa Oppliger Comisión : Dra. M. Loreto Ladrón de Guevara Dr. Gustavo Lima Concepción, Chile Septiembre 2012 Resumen. En la teoría cuántica de la información las correlaciones cuánticas son esenciales. Por ejemplo, el entrelazamiento, un fenómeno sin contraparte clásica, es fundamental tanto desde el punto de vista teórico como para el desarrollo tecnológico futuro que esté basado en la computación cuántica. Además del entrelazamiento existen otros tipos de correlaciones, presentes sólo entre sis- temas cuánticos, que también han despertado el interés entre los investigadores. El quantum discord y la disonancia son algunos de ellos. En esta tesis se estudia, clasifica y cuantifica el entrelazamiento, el quantum discord y la disonancia necesarios para llevar a cabo con éxito los protocolos de discriminación asistida de estados no ortogonales. Además, se estudia la dependencia que existe entre éstas correlaciones y los estados de los sistemas utilizados para tales procesos, logrando caracterizar la cantidad de entrelazamiento y quantum discord en términos de los parámetros que definen a los estados utilizados. Abstract. In quantum information theory quantum correlations are essential. For example, entanglement, a phenomenon without classical counterpart, is crucial from theoretical perspective as well as for technological development based on quantum computation. Besides Entanglement, other types of correlations present only between quantum systems have also attracted interest among researchers. The quantum discord and dissonance are some of them. In this thesis we study, classify and quantify the quantum correlations such as entangle- ment and quantum discord necessary to successfully perform various quantum information protocols as assisted optimal state discrimination. In addition, we study the dependency be- tween the states of the systems used for such processes and the amount of entanglement and quantum discord needed, i.e., we characterize the different quantum correlations in terms of the parameters that define the states used. Dedicado a Ligia, Emilia y OdY. Agradecimientos. Son muchas a las personas que quisiera agradecer, partiendo por mi esposa Ligia. Has sido y serás siempre mi pilar principal. Sin tu apoyo y empuje de seguro hace tres años atrás no me hubiese decidido a dar este paso. El sacrificio y esfuerzo de todo este tiempo valió la pena. A mi pequeña hija, Emilia, le agradezco por iluminar mi vida. Con tu llegada me regalaste la motivación que me faltaba para terminar esta etapa y para comenzar lo que se viene por delante. OdY, siempre fiel y leal. Gracias por esa cuota de locura que día a día me ayudó a dejar a un lado las preocupaciones y el cansancio. A mis padres y hermano les agradezco por estar siempre detrás, alentándome y deseándome lo mejor. A mis suegros por su hospitalidad y por hacerme sentir como en casa. Gracias también a mis amigos Patricio Mella, Cristian Salas, Cristian Jara, Alejandra Maldonado, Pablo Solano y Esteban Sepúlveda. Vuestra amistad ha sido fundamental tanto personal como profesionalmente. A mi profesor, Luis Roa, le agradezco por la confianza y la oportunidad. A los profesores Gustavo Lima y M. Loreto Ladrón de Guevara les agradezco por haberse interesado en mi trabajo. Sole, a ti también gracias por el tiempo dedicado y por las gestiones realizadas para que los trámites no fuesen tan lentos. Finalmente, agradezco a las instituciones que me apoyaron económicamente durante el tiempo que me tomó desarrollar esta investigación. A CONICyT por financiar mis estudios atravésdelabecadedoctoradonacional.AldepartamentodeFísicadelaUniversidadde Concepción, a la Dirección de Postgrado de la Universidad de Concepción, y al Centro de Óptica y Fotónica - CEFOP, por otorgarme co-financiamiento para asistir a conferencias y para realizar pasantías de investigación en el extranjero. Contents Contents i Introducción iii Introduction vii 1 Classical Information and Shannon Entropy. 1 1.1EntropyofaRandomVariable........................... 1 1.1.1 TheBinaryEntropyFunction........................ 3 1.1.2 MathematicalPropertiesofEntropy.................... 3 1.2ClassicalConditionalEntropy............................ 4 1.3 Classical Joint Entropy............................... 6 1.4ClassicalMutualInformation............................ 6 1.5ClassicalRelativeEntropy.............................. 8 2 Quantum Information and von Neumann Entropy. 9 2.1 Quantum Entropy. ................................. 10 2.1.1 MathematicalPropertiesofQuantumEntropy............... 10 2.1.2 AlternativeExpressionforvonNeumannEntropy............. 12 2.2 Joint Quantum Entropy............................... 12 2.2.1 MarginalEntropiesofaPureBipartiteState............... 12 2.2.2 Additivity. .................................. 13 2.2.3 JointEntropyofaClassical-QuantumState................ 13 2.3QuantumConditionalEntropy........................... 13 i 2.4QuantumMutualInformation............................ 16 2.4.1 Holevo Information.............................. 16 2.5QuantumRelativeEntropy............................. 18 3 Classical and Quantum Correlations. 21 3.1Entanglement..................................... 21 3.1.1 PPTCriterionandNegativity........................ 22 3.1.2 EntanglementofFormationandConcurrence............... 23 3.2 Quantum Discord. ................................. 24 3.2.1 PositiveOperatorValuedMeasure..................... 25 3.2.2 Entropic DefinitionofQuantumDiscord.................. 26 3.2.3 Dissonance. ................................. 28 3.2.4 GeometricMeasureofQuantumDiscord.................. 30 3.3QuantumDiscordandGeneralizedMeasurements................. 31 3.4RelationbetweenEntanglementandDiscord.................... 32 3.4.1 Purification.................................. 32 3.4.2 Koashi-WinterRelation........................... 32 3.4.3 ConservationLawforCorrelations..................... 33 3.5GeneralBoundforQuantumDiscord........................ 34 3.6 Classical States and Nullity Conditions for Quantum Discord. ......... 34 4 Correlations for State Discrimination 37 Summary 45 Conclusiones 47 Bibliography 49 ii Introducción. Una de las principales características de la no-clasicalidad en un sistema cuántico es la ex- istencia de correlaciones que no tienen contraparte clásica. Este tipo de correlaciones, las correlaciones cuánticas, ocupan una posición central en la búsqueda de la comprensión y el aprovechamiento del poder de la mecánica cuántica aplicada a la teoría de la información, dando origen a uno de los tópicos más estudiados en esa área y cuyo objetivo es desarrollar diferentes métodos que permitan cuantificar dichas correlaciones. El entrelazamiento [1, 2] es quizás el tipo de correlaciones cuánticas más conocido y estu- diado y desde que fue descrito por primera vez por Einstein, Podolsky y Rosen [3] ha atraído la atención y el interés de los científicos, siendo estudiado tanto teórica [4—9] como experimen- talmente [10—15], llegando así a ser considerado un ingrediente clave en la teoría cuántica de la información. Es un fenómeno sin contraparte clásica que surge de la interacción directa o indirectaentredosomássistemascuánticosenelcuallosestadosdelossistemasinvolucrados se correlacionan de forma tal que un proceso de medición realizado sobre uno de ellos afecta a los otros, inclusive si los sistemas individuales se encuentran espacialmente separados [16]. Al ser considerado como un recurso, el entrelazamiento permite realizar innumerables tareas que clásicamente son imposibles. Por ejemplo, el uso de estados entrelazados es fundamental en el proceso determinista de teleportación de estados puros desconocidos [17]. También se constituye como pieza clave en los protocolos de entanglement swapping [18,19], discriminación de estados [20—25], clonación de estados no ortogonales [26], quantum dense coding y super dense [27] , criptografía cuántica [28, 29], preparación remota de estados [30, 31] y mapeo de estados no ortogonales [32], entre otros. Sin embargo, hace alrededor de una década atrás la visión de que el entrelazamiento es el responsable de las ventajas cuánticas cambió dramáticamente. Por un lado, en 1998 Knill yLaflamme [33] mostraron que, incluso cuando no hay entrelazamiento, es posible lograr eficiencias superiores a las logradas clásicamente usando estados mixtos. iii Posteriormente, en 2001, Henderson y Vedral [34] por un lado y Ollivier y Zurek por otro [35, 36] se dán cuenta al estudiar y analizar diferentes medidas de información en teoría cuántica que a diferencia de lo que ocurre con los estados puros, con estados mixtos no todas las correlaciones presentes quedan contenidas dentro del entrelazamiento. Este nuevo tipo de correlación es llamado quantum discord. El quantum discord incluye al entrelazamiento pero no se limita a él1. Estohapermitido interpretarlo como una medida que
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