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 dá cuenta de que tan cuántica es una correlación. Así, poder distinguir las correlaciones cuánticas distintas al entrelazamiento proporciona una mejor división entre los mundos cuántico y clásico, especialmente cuando se consideran los estados mixtos.
Desde su introducción en la teoría cuántica de la información, el quantum discord cau- tivó a gran parte de la comunidad científica motivando una avalancha de publicaciones cen- tradas tanto en su interpretación física [37—40] como en su interpretación operacional [41—44], al igual que en su utilidad como recurso necesario para implementar distintos protocolos de procesamiento, almacenamiento y transferencia de información, como la transmisión local de información [45], quantum state merging [42], teleportación [17] y preparación remota de esta- dos [46].
Sin embargo, debido a la optimización involucrada en la definición del quantum discord, obtener una expresión analítica es una tarea complicada que en general requiere de cálculo numérico para ser realizada. En [47] D. Girolami, y G. Adesso traducen el problema de la optimización a encontrar las soluciones de dos ecuaciones trascendentales, formulando un algoritmo numérico que permite calcular discord para estados generales de dos qubits. Sólo se cononcen expresiones analíticas cerradas del quantum discord para sistemas de dos qubits con maximally mixed marginals [48] y para una subfamilia de los estados [49].
Por otro lado, Dakic y Vedral han interpretado el discord desde un punto de vista geométrico definiéndolo como la medida de la distancia que hay entre el estado mixto estudiado y su estado clásico más cercano [50], entregando una expresión analítica cerrada para calcular el discord entre dos qubits. Generalizaciones de ésta expresión en el caso de dos qudits ( ) ⊗ también han sido estudiadas [51]. Recientemente, Passante y colaboradores han descrito e implementado experimentalmente un algoritmo eficiente que permite cuantificar el discord geométrico [52].
1 Un ejemplo de esto son los estados (mixtos) de Werner ya que para cierto intervalo de valores son estados separables pero con discord distinto de cero [9].
iv Existen otras medidas que, siguiendo el espíritu del quantum discord, intentan también cuantificar las correlaciones cuánticas. Alguna de ellas son el quantum work deficit [53, 54], el measurement induced disturbance [55] y la disonancia [56]. Esta última es particularmente interesante ya que, de acuerdo a su definición, contiene todas aquellas correlaciones cuánticas que no son descritas por el entrelazamiento.
Es de particular interés el enfoque en que el discord, al igual que el entrelazamiento, es considerado un recurso para realizar ciertos protocolos de información cuántica [46]. En es- pecial, para aquellos protocolos en los que el entrelazamiento no está presente o no sea nece- sario [57, 58]. Esto ha motivado un gran interés en el estudio de la dinámica del discord bajo mecanismos de decoherencia [59—62]. En este sentido, se ha encontrado que el discord no es tan frágil como el entrelazamiento [63], característica importante que lo eleva por sobre los otros tipos de correlaciones cuánticas, transformándolo así en el candidato ideal para ser utilizado en computación cuántica [64].
Como objeto principal de esta tesis se plantea entonces estudiar, clasificar y cuantificar el entrelazamiento, el quantum discord y la disonancia requerida para realizar con éxito la discriminación asistida de estados no ortogonales. Además, interesa conocer la dependencia que existe entre los estados de los sistemas utilizados en tal protocolo y la cantidad de correlaciones necesarias, es decir, caracterizarlos en términos de los parámetros que definen a los estados utilizados.
Esta tesis se separa en tres partes. La primera parte consta del primer y segundo capítulo en donde se presentan aquellos conceptos, definiciones y herramientas matemáticas involucradas en la definición de entrelazamiento, quantum discord y disonancia. El primer capítulo incluye solo aquellas asociadas a la teoría clásica de la información, mientras que en el segundo se muestran sus contrapartes cuánticas. El material contenido en estos capítulos fué extraído en su totalidad de [65], libro en el cual se encuentran todas las demostraciones de los teoremas incluidos aqui.
La segunda parte de la tesis corresponde al tercer capítulo, y trata sobre las correlaciones cuánticas. En la primera parte de éste se expone brevemente el concepto de entrelazamiento, incluyendo el criterio de separabilidad de Peres [66] y algunas medidas cuantitativas de entre- lazamiento como la concurrencia [2] y la negatividad [67]. La segunda parte, basada y extraída en su mayoría desde el review de Modi et al. [68], trata el quantum discord y la disonancia, incluyendo sus definiciones y su relación con el entrelazamiento. Se finalizaelcapítulopre- sentando algunas desigualdades y criterios importantes que muestran los límites generales del
v discord y aquellas condiciones necesarias y suficientes para encontrar los estados cuyo discord es cero (estados clásicos).
El cuarto capítulo, tercera y última parte de la tesis, es el único que contiene material original. En él se presentan los resultados obtenidos a partir del trabajo de investigación prop- uesto en esta tesis, los cuales tienen relación con el estudio de las correlaciones como recursos necesarios para realizar la discriminación asistida de estados no ortogonales. Como resultado principal se encontró que para realizar el protocolo de manera óptima son necesarios tanto el entrelazamiento como el discord. Sin embargo, en el caso particular en que las probabilidades de preparación de los estados a discriminar son iguales, basta con el quantum discord para realizar de manera óptima el reconocimiento de estados.
Finalmente están las conclusiones, donde se resumen y discuten los resultados mostrados en el capítulo cuatro.
vi Introduction.
One of the key features of non-clasicality in a quantum systems is the existence of correlations which don’t have classical counterparts. Such correlations, quantum correlations, are central in the search for understanding and harnessing the power of quantum mechanics applied to information theory, giving rise to one of the most studied topics in the area and whose objective is to develop different methods to quantify such correlations.
Entanglement [1, 2] is perhaps the kind of quantum correlations more known and studied and since it was first described by Einstein, Podolsky and Rosen [3] has attracted the attention and interest of scientists being studied both theoretically [4—9] and experimentally [10—15], becoming considered a key ingredient in quantum information theory. It is a phenomenon without classical counterpart arising from the direct or indirect interaction between two or more quantum systems in which the states of the systems involved are correlated so that a measurement process performed on one affects the other, even if individual systems are spatially separated [16].
As a resource, entanglement allows innumerable tasks that are impossible classically. For example, the use of entangled states is central in the process of deterministic teleportation of unknown pure states [17]. It also is a key piece in the entanglement swapping protocol [18,19], state discrimination [20—25], cloning of non-orthogonal states [26], quantum dense and super- dense [27], quantum cryptography [28, 29], remote state preparation [30, 31] and mapping of non-orthogonal states [32], among others.
However, for about a decade ago the view that entanglement is responsible for quantum benefits changed dramatically. First, in 1998 Knill and Laflamme [33] showed that even when no entanglement is present, is possible to achieve efficiencies greater than those achieved clas- sically using mixed states.
Later, in 2001, Henderson and Vedral [34] on one side, and Ollivier and Zurek [35, 36] in
vii the other, realize that unlike what occurs with the pure state, when study and analyze various information measures in quantum theory with mixed states not all the correlations that are present are contained within the entanglement. This new type of correlation is called quantum discord.
Quantum discord includes entanglement but is not limited to it2.Thishasallowedto interpret it as a measure that accounts of how quantum is a correlation. Thus, be able to distinguish quantum correlations other than entanglement provides a better division between the quantum and classical worlds, especially when considering mixed states.
Since its introduction in quantum information theory, the quantum discord has captured the attention of most of the scientific community, motivating an avalanche of publications focusing both in its physical interpretation [37—40] as in its operational interpretation [41—44], as well as in its usefulness as a resource necessary to implement different protocols of processing, storage and transmission of information such as local information transmission [45], quantum state merging [42], teleportation [17] and remote state preparation [46].
However, due to the optimization involved in defining the quantum discord, obtaining an analytical expression is a complicated task which generally requires numerical calculation to be performed. In [47] D. Girolami, and G. Adesso translate the optimization problem to find the solution of two transcendental equations, formulating a numerical algorithm for calculating general discord for two-qubit states. Only for two-qubit systems with maximally mixed marginals [48] and for a subfamily of states [49], closed analytical expressions of quantum discord are known.
Moreover, Vedral and Dakic have interpreted the discord from a geometrical point of view, defining it as the measure of the distance between the studied mixed state and its closest classical state [50], providing a closed analytic expression for calculating the discord between two qubits. Generalizations of this expression in the case of two qudits ( )havealso ⊗ been studied [51]. Recently, Passante and colleagues have described and experimentally im- plemented an efficient algorithm that quantifies the geometric discord [52].
There are other measures that, following the spirit of quantum discord, also try to quantify the quantum correlations. Some of them are the work quantum deficit [53, 54], the mea- surement induced disturbance [55] and dissonance [56]. The latter is particularly interesting because, according to its definition, contains all the non-quantum correlations described by entanglement.
2 An example of this are the (mixed) Werner states which for certain range of values are separable but with nonzero discord [9].
viii Of particular interest is the approach in that the discord, like entanglement, is considered a resource for some quantum information protocols [46]. Especially for those protocols that entanglement is not present or is not necessary [57, 58]. This has led to a great interest in the study of the dynamics of discord under decoherence mechanisms [59—62]. In this sense, it has been found that the discord is not as fragile as entanglement [63], an important feature that rises it above the other types of quantum correlations, thus transforming it into the ideal candidate for use in quantum computing [64].
Then, the main object of this thesis is study, categorize and quantify entanglement, quan- tum discord and dissonance required to successfully carry out the assisted discrimination of non-orthogonal states. Also to know the dependency between the states of the systems used in this protocol and the amount of necessary correlations, i.e., to characterize them in terms of the parameters that define the states used.
This thesis is separated into three parts. The first one contains the first and second chap- ter. Both presents the concepts, definitions and mathematical tools involved in the definition of entanglement, quantum discord and dissonance but the first chapter includes only those associated with the classical theory of information while the second its quantum counterparts. The material in these chapters was taken entirely from [65], book which contains all the proofs of the theorems included here.
The second part of the thesis is the third chapter. In it the quantum correlations are discussed. In the firstpartofitisbriefly exposed the concept of entanglement, including Peres separability criterion [66] and some quantitative measures of entanglement such as concurrence [2] and negativity [67]. The second part, based and mainly extracted from the review of Modi et al. [68], is about the quantum discord and dissonance, including their definitions and their relation to entanglement. The chapter ends by presenting some important inequalities and criteria that show the general bounds of quantum discord and those necessary and sufficient conditions for finding the states whose discord is zero (classical states).
The fourth chapter, third and last part of the thesis, is the only one containing original material. It presents the results obtained from the research work proposed in this thesis, which are related to the study of correlations as resources needed to perform assisted discrimination of non-orthogonal states. As a main result it was found that to perform optimally the protocol are required both entanglement and discord. However, in the particular case in which the probabilities of preparing the states to be discriminate are identical, only the quantum discord is required to perform optimally the recognition of the states.
Finally in the summary are discussed the results shown in chapter four.
ix x Chapter 1 Classical Information and Shannon Entropy.
In physics the usual notion of bit refers to the physical representation that it has. In information theory instead, the bit is a measure of how much we can learn1 fromtheresultsofarandom experiment.
All physical systems can be used to register bits of information and, depending on the nature of the system, the information could be classical, quantum, or a hybrid of both. For example, an atom can register both quantum and classical information while location of a billiard ball registers classical information only.
In this chapter we provide an intuitive understanding of information measures in terms of the parties who have access to the classical systems. We introduce the entropy as the expected surprise of a random variable and then we used this notion to develop other measures of information that prove to be useful for increasing our understanding about the nature of information.
1.1 Entropy of a Random Variable.
Consider a random variable ,andbe each of its possible realizations. Let () denote the probability density function of so that () is the probability that realization occurs.
We define the information content () of a particular realization as the measure of the surprise that one has upon learning the outcome of a random experiment:
() log ( ()) (1.1) ≡− 1 Perhaps the word "surprise" better captures the notion of information as it applies in the context of information theory.
1 2 CHAPTER 1. CLASSICAL INFORMATION AND SHANNON ENTROPY.
The logarithm is base two and this choice implies that we measure surprise or information in bits.
Figure 1.1 plots the information content for values in the unit interval. This measure of surprise is higher for lower probability events that surprise us, and it is lower for higher probability events that do not surprise us. Inspection of the figure reveals that the information content is positive for any realization .
10
8
6
)
x
(
i 4
2
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p Figure 1.1: The information content or "surprise" in (1.1) as a function of a probability ranging from 0 to 1.Aneventhasalowersurpriseifitismorelikelytooccurandithasa higher surprise if it less likely to occur.
The information content is additive, due to the choice of the logarithm function. Given two independent random experiments involving random variable with respective realizations 1 and 2,wehavethat
(12)= log ( (12)) = log ( (1) (2)) = (1)+ (2) (1.2) − −
Although the information content is a useful measure of surprise for particular realizations of random variable it does not capture a general notion of the amount of surprise that a given random variable possesses. The entropy ()
() ()log( ()) (1.3) ≡− X called Shannon Entropy, captures this general notion of the surprise of a random variable , it is, the expected information content of random variable . For realizations with zero probability we adopt the convention2 that 0 (0) = 0. · 2 The fact that lim 0 ( log )=0intuitively justifies this convention. → 1.1. ENTROPY OF A RANDOM VARIABLE. 3
For example, suppose that Alice generates a random experiment that selects a realization accordingtothedensity() of random variable and Bob has not yet learned the outcome of the experiment. Then, the Shannon entropy () quantifies Bob’s uncertainty about before learning it. His expected information gain is () bits upon learning the outcome of the random experiment.
1.1.1 The Binary Entropy Function.
A special case of the entropy occurs when the random variable is a Bernoulli random variable with probability density (0) = and (1) = 1 . This Bernoulli random variable could − correspond to the outcome of a random coin flip. The entropy in this case is known as the binary entropy function:
() log (1 )log(1 ) (1.4) ≡− − − − and it quantifiesthenumberofbitsthatwelearnfromtheoutcomeofthecoinflip. If the 1 coin is unbiased ( = 2 ), then we learn a maximum of one bit (()=1). If the coin is deterministic ( =0or =1), then we do not learn anything from the outcome (()=0). Figure 1.2 reveals that the binary entropy function () is a concave function of the parameter 1 and has its peak at = 2 .
1.0
0.8
0.6 H(p) 0.4
0.2
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p Figure 1.2: The binary entropy function () displayed as a function of the parameter .
1.1.2 Mathematical Properties of Entropy.
Five important mathematical properties of the entropy () are: 4 CHAPTER 1. CLASSICAL INFORMATION AND SHANNON ENTROPY.
Property 1. (Positivity) The entropy () is non-negative for any probability density
(): () 0 (1.5) ≥
Property 2. (Concavity) The entropy () is concave in the probability density (), i.e., consider two random variables 1 and 2 with two respective probability density functions
1 () and 2 () whose realizations belong to the same set. Consider a Bernoulli random variable with probabilities and 1 corresponding to its two respective realizations. Then, − concavity of entropy is the following inequality:
() (1)+(1 ) (2) (1.6) ≥ −
Property 3. (Invariance under permutations) The entropy is invariant under permu- tations of the realizations of random variable .
Property 4. (Minimum value) The entropy vanishes for a deterministic variable.
Property 5. (Maximum value) The maximum value of the entropy () for a random variable with different realizations is log :
() log (1.7) ≤
1.2 Classical Conditional Entropy.
Suppose now that Alice possesses random variable and Bob possesses some other random variable .Randomvariables and share correlations if they are not statistically indepen- dent, and Bob then possesses "side information" about in the form of . This conditional information content is denoted by ( ) and is defined in terms of entropy as: |
( ) log ( ) (1.8) | ≡− | | ¡ ¢ Then, the entropy ( = ) of random variable conditional on a particular realization | of random variable is the expected conditional information content, where the expectation 1.2. CLASSICAL CONDITIONAL ENTROPY. 5 is with respect to :
( = )= ( )log ( ) (1.9) | | | − | | X ¡ ¢
The relevant entropy that applies to the scenario where Bob possesses side information is the conditional entropy ( ). It is the expected conditional information content where the | expectation is with respect to both and :
( )= () ( = ) (1.10) | | X = () ( )log ( ) (1.11) | | − | | X X ¡ ¢ = ( )log ( ) (1.12) | − | X ¡ ¢ which can be interpreted as follows: suppose that Alice possesses random variable and Bob possesses random variable . The conditional entropy ( ) istheamountofuncertainty | that Bob has about given that he already possesses .
The above interpretation immediately suggests that having access to a side variable should only decrease our uncertainty about another variable. We state this idea as the following theorem:
Theorem 1 (Conditioning does not increase entropy) The entropy () is greater than or equal to the conditional entropy ( ): | () ( ) (1.13) ≥ |
As well as entropy, conditional entropy is non-negative. This is because ( ) is the | expectation of the entropy ( = ) with respect to the density (), which means that | we always learn some number of bits of information upon learning the outcome of a random experiment involving even if we have access to some side information . 6 CHAPTER 1. CLASSICAL INFORMATION AND SHANNON ENTROPY. 1.3 Classical Joint Entropy.
The natural entropic quantity that describes the uncertainty when neither nor is known, is the joint entropy ( ). It is merely the entropy of the joint random variable ( ):
( )= ( )log( ( )) (1.14) − X
Property 6. (Chaining rule for entropy) Consider 1 random variables. Then:
(1)= (1)+ (1 2)+ + ( 11) (1.15) | ··· | − If we have only two random variables and , the relation between joint entropy ( ), conditional entropy ( ), and marginal entropy () is: | ( )= ()+ ( )= ( )+ ( ) (1.16) | |
Property 7. (Subadditivity) Entropy is subadditive:
(1) () (1.17) ≤ =1 X
Property 8. (Additivity for independent random variables) For independent random variables 1:
(1)= () (1.18) =1 X
1.4 Classical Mutual Information.
An entropic measure of the common or mutual information that two parties possess is the mu- tual information, and it quantifies the dependence or correlations of the two random variables and .
Suppose that Alice possesses random variable and Bob possesses random variable . 1.4. CLASSICAL MUTUAL INFORMATION. 7
The mutual information is the marginal entropy () less the conditional entropy ( ): | ( : ) () ( ) (1.19) ≡ − |
The mutual information measures how much knowing one random variable reduces the uncertainty about the other random variable. In this sense, it is the common information between the two random variables. Bob possesses and thus has an uncertainty ( ) | about Alice’s variable . Knowledge of gives an information gain of ( ) bits about | andthenreducestheoveralluncertainty () about ,theuncertaintywerehenottohave any side information at all about .
Property 9. (Symmetric) The mutual information is symmetric in its inputs:
( : )= ( : ) (1.20) implying additionally that ( : )= ( ) ( ) (1.21) − |
In terms of the respective joint and marginal probability density functions ( ) and
() and (), the mutual information ( : ) can be wrote as:
( ) ( : )= ( )log (1.22) () () X µ ¶ The above expression leads to two insights regarding the mutual information ( : ):
3 (i) If two random variables and are statistically independent , ( )= () (), then they possess zero bits of mutual information.
(ii) If two random variables and are perfectly correlated in the sense that = ,then they possess () bits of mutual information.
Theorem 2 The mutual information ( : ) is non-negative for any random variables and : ( : ) 0 (1.23) ≥
3 That is, knowledge of does not give any information about . 8 CHAPTER 1. CLASSICAL INFORMATION AND SHANNON ENTROPY. 1.5 Classical Relative Entropy.
This is another important entropic quantity that quantifies how "far" one probability density function 1 () is from another probability density function 2 ().
We define the relative entropy ( ) as follows: 1 || 2 () ( ) ()log 1 (1.24) 1 2 1 () || ≡ 2 X µ ¶
The above definition implies that the relative entropy is not a distance measure in the strict mathematical sense because it is not symmetric under interchange of the densities 1 () and
2 ().
It is interesting to note that the mutual information ( : ) is equivalent to the relative entropy ( ( ) () ()). In this sense, the mutual information quantifies how || far the two random variables and are from being independent because it calculates the distance of the joint density ( ) from the product of the marginals () (). Chapter 2
Quantum Information and von Neumann Entropy.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure is the quantum analog of the Shannon entropy, called von Neumman entropy.
In some sense, von Newmman entropy is an generalization of Shannon entropy because it captures both classical and quantum uncertainty in a quantum state. The von Neumann entropy gives meaning to the information qubit which is different from that of the physical qubit. The information qubit is the fundamental quantum informational unit of measure and determines how much quantum information is in a quantum system while the physical qubit is the description of a quantum state in an electron or a photon.
The definitions here are analogous to the classical definitions of entropy. However, there are at least two fundamental differences. The first one is that the conditional quantum entropy can be negative1 for certain quantum states. In fact, pure quantum states that are entangled have stronger correlations than classical states are examples of states that have negative conditional entropy. The second one is related to quantum version of mutual information. A simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information, compared with the largest classical mutual information, one bit, for the case of two maximally correlated classical bits.
1 In the classical world, this negativity simply does not occur, though it takes a special meaning in quantum information theory.
9 10 CHAPTER 2. QUANTUM INFORMATION AND VON NEUMANN ENTROPY. 2.1 Quantum Entropy.
We might expect a measure of the entropy of a quantum system to be vastly different from the classical measure of entropy because a quantum system possesses not only classical uncertainty but also quantum uncertainty that arises from the uncertainty principle. But recall that the density operator captures both types of uncertainty and allows us to determine probabilities for the outcomes of any measurement on system . Thus, a quantum measure of uncertainty should be a direct function of the density operator, just as the classical measure of uncertainty is a direct function of a probability density function.
Definition 1 (Quantum Entropy) Suppose that Alice prepares some quantum system in astate. Then the entropy () of the state is as follows:
() log (2.1) ≡− © ª The entropy of a quantum system is also known as the von Neumann entropy or the quantum entropy. We can denote it by () or () to show the explicit dependence on the density operator .Fromitsdefinition is clear that the von Neumann entropy has a special relation to the eigenvalues of the density operator: the von Neumann entropy of a density operator is the Shannon entropy of its eigenvalues.
The quantum entropy admits an intuitive interpretation. Suppose that Alice generates a random quantum state in her lab according to some probability density () of a random variable . Suppose¯ ® further that Bob has not yet received the state from Alice and does not know which one she¯ sent. The expected density operator from Bob’s point of view is then
= () (2.2) X ¯ ® ¯ The interpretation of the entropy () is that it¯ quantifies¯ Bob’s uncertainty about the state Alice sent. His expected information gain is () qubits upon receiving and measuring the state that Alice sends.
2.1.1 Mathematical Properties of Quantum Entropy.
Since the von Neumann entropy of a density operator is the Shannon entropy of its eigenvalues, quantum entropy posses similar properties to its classical version: positivity, minimum value, maximum value, invariance but now under unitaries, and concavity. 2.1. QUANTUM ENTROPY. 11
Property 10. (Positivity) The von Neumann entropy () is non-negative for any density operator : () 0 (2.3) ≥
Property 11. (Minimum Value) The minimum value of the von Neumann entropy is zero, and it occurs when the density operator is a pure state. Why should the entropy of a pure quantum state vanish? It seems that there is quantum uncertainty inherent in the state itself and that a measure of quantum uncertainty should capture this fact. This last observation only makes sense if we do not know anything about the state that is prepared. But if we know exactly how it was prepared, we can perform a special quantum measurement to verify that the quantum state was prepared, and we do not learn anything from this measurement because the outcome of it is always certain. For example,supposethatAlicealwayspreparesthestate andBobknowsthatshedoesso. | i He can then perform a measurement of the following form to verify that {| ih | − | ih |} she prepared this state. He always receives the first outcome from the measurement and never gains any information from it. Thus, it make sense to say that the entropy of a pure state vanishes.
Property 12. (Maximum Value) The maximum value of the von Neumann entropy is log where is the dimension of the system, and it occurs for the maximally mixed state.
Property 13. (Concavity) The entropy is concave in the density operator:
() () () (2.4) ≥ X where () . ≡ The physicalP interpretation of concavity is as before for classical entropy: entropy can never decrease under a mixing operation. This inequality is a fundamental property of the entropy.
Property 14. (Unitary Invariance) The entropy of a density operator is invariant under unitary operations on it:
()= † (2.5) ¡ ¢ 12 CHAPTER 2. QUANTUM INFORMATION AND VON NEUMANN ENTROPY.
2.1.2 Alternative Expression for von Neumann Entropy.
There is an interesting alternative characterization of the von Neumann entropy of a state as the minimum Shannon entropy of a rank-one POVM performed on it. That is:
()=min Λ log2 ( Λ ) (2.6) Λ − { } { } { } X where the minimum is restricted to be over rank-one POVMs (those with Λ = for some vectors such that 1and = ). In this sense, there ≤ ¯ ® ¯ is some optimal measurement to perform on such that its entropy is equivalent¯ to the¯ von ¯ ® ©¯ ® ¯ª P ¯ ® ¯ Neumann entropy,¯ and this optimal¯ measurement¯ is the "right¯ question¯ to ask".
2.2 Joint Quantum Entropy.
The joint quantum entropy () of the density operator for a bipartite system follows naturally from the definition of quantum entropy:
() log (2.7) ≡− © ª
2.2.1 Marginal Entropies of a Pure Bipartite State.
Theorem below states the most fundamental differences between classical and quantum infor- mation: the marginal entropies of a pure bipartite state are equal, while the entropy of the overall state remains zero.
Theorem 3 The marginal entropies () and () of a pure bipartite state are | i equal:
() = () (2.8) while the joint entropy () vanishes:
() =0 (2.9) 2.3. QUANTUM CONDITIONAL ENTROPY. 13
2.2.2 Additivity.
Additivity is a property that we would like to hold for any measure of information.
The quantum entropy is additive for tensor product states:
( )= ()+ () (2.10) ⊗ This property can be verified by diagonalizing both density operators and resorting to the additivity of the joint Shannon entropies of the eigenvalues.
2.2.3 Joint Entropy of a Classical-Quantum State.
A classical-quantum state () (2.11) ≡ | i h | ⊗ X is a bipartite state in which a classical system and a quantum system are classically correlated, and its joint quantum entropy takes a special form that is similar to entropies in the classical world:
Theorem 4 The joint entropy () of a classical-quantum state is as follows:
() = ()+ () () (2.12) X where () is the entropy of a random variable with distribution ().
2.3 Quantum Conditional Entropy.
The most useful definition of conditional quantum entropy in quantum information theory is inspired from the relation between joint entropy and marginal entropy:
Definition 2 (Conditional Quantum Entropy) The conditional quantum entropy ( ) | of a bipartite quantum state is the difference of the joint quantum entropy () and the marginal (): ( ) = () () (2.13) | − 14 CHAPTER 2. QUANTUM INFORMATION AND VON NEUMANN ENTROPY.
The above definition is the most natural one, because it is straightforward to compute for any bipartite state and because it obeys many relations that the classical conditional entropy obeys.
Theorem 5 (Conditioning does not increase entropy) Consider a bipartite quantum state . Then the following inequality applies to the marginal entropy () and the condi- tional quantum entropy ( ) : | () ( ) (2.14) ≥ | The above relation implies that conditioning cannot increase entropy, even if the condition- ing system is quantum.
However, conditional quantum entropy may seem a bit difficult to define because there is no formal notion of conditional probability in the quantum theory. Lets consider an arbi- trary bipartite state and suppose that Alice performs a complete von Neumann measure- ment Π of her system in the basis . This procedure leads to an ensemble ≡ {| ih |} {| i} () ,where | i h | ⊗ n o 1 (2.15) ≡ () | i h | ⊗ | i h | ⊗ n³ ´ ³ ´o () (2.16) ≡ | i h | ⊗ n³ ´ o One could then think of the density operators as being conditional on the outcome of the measurement, and these density operators describe the state of Bob given knowledge of the outcome of the measurement. With this in mind, we could potentially redefine a conditional entropy as follows:
( )Π () () (2.17) | ≡ X in analogy with the definition of the classical entropy in (1.10). This approach might seem to lead to a useful definition of conditional quantum entropy, but the problem with it is that the entropy depends on the measurement chosen. This dependence on measurement is a fundamental aspect of the quantum theory since this problem does not occur in the classical world because the probabilities for the outcomes of measurements do not themselves depend on the measurement selected.
We could then attempt to remove the dependence of the above definition on a particular measurement by defining the conditional quantum entropy to be the minimization of ( ) | Π 2.3. QUANTUM CONDITIONAL ENTROPY. 15 over all possible measurements. The intuition here is perhaps that entropy should be the minimal amount of conditional uncertainty in a system after employing the best possible measurement on the other.
Theaboveideaisusefulandwewillcomebacktoitwhenwedefine the quantum discord.
Quantum Conditional Entropy of a Classical-Quantum State.
A classical-quantum state is an example of a state where conditional quantum entropy behaves as in the classical world. Suppose that two parties share a classical-quantum state of the form in (2.11), where the system is classical and the system is quantum, and the correlations between them are entirely classical, determined by the probability distribution
(). The conditional quantum entropy ( ) for this state is: | ( ) = () () (2.18) | − = ()+ () () () (2.19) − X = () () (2.20) X The last form for conditional entropy is completely analogous with the classical formula in (1.10) and holds whenever the conditioning system is classical.
Negative Quantum Conditional Entropy.
One of the properties of the conditional quantum entropy in Definition 2 that seems coun- terintuitive at first sight is that it can be negative. This negativity holds for an ebit Φ+ | i shared between Alice and Bob. The marginal state on Bob’s system is the maximally mixed state . Thus, the marginal entropy () is equal to one, but the joint entropy vanishes, so the conditional quantum entropy is ( )= 1. | −
This is the second of the fundamental differences between the classical world and the quantum world, and it can be understood as follows: we can sometimes be more certain about the joint state of a quantum system than we can be about any one of its individual parts. 16 CHAPTER 2. QUANTUM INFORMATION AND VON NEUMANN ENTROPY. 2.4 Quantum Mutual Information.
The standard informational measure of correlations in the classical world is the mutual informa- tion, and such a quantity plays a central role in measuring classical and quantum correlations inthequantumworldaswell.
Definition 3 (Quantum Mutual Information) The quantum mutual information of a bi- partite state is as follows:
( : ) () + () () (2.21) ≡ −
The following relations hold for quantum mutual information, in analogy with the classical case:
( : ) = () ( ) (2.22) − | = () ( ) (2.23) − |
Theorem 6 (Positivity of Quantum Mutual Information) The quantum mutual infor- mation ( : ) of any bipartite quantum state is positive:
( : ) 0 (2.24) ≥
Theorem 7 (Bound on Quantum Mutual Information) The following bound applies to the quantum mutual information:
( : ) 2min log log (2.25) ≤ { } where is the dimension of system and is the dimension of system .
2.4.1 Holevo Information.
Suppose that Alice prepares some classical ensemble () and then hands this ≡ ensemble to Bob without telling him the classical index .© The expectedª density operator of this ensemble is = () (2.26) X 2.4. QUANTUM MUTUAL INFORMATION. 17 which characterizes the state from Bob’s perspective because he does not have knowledge of the classical index .
Bob’s task is to determine the classical index by performing some measurement on his system . The accessible information
()=max ( : ) (2.27) Λ { } quantifies his information gain after performing some optimal measurement Λ on system { } ,where is a random variable corresponding to the outcome of the measurement.
In general, the accessible information of the ensemble is quantity is difficult to compute, but another quantity, called the Holevo information, provides a useful upper bound. The Holevo information () oftheensembleis
() () (2.28) ≡ − ¡ ¢ X ¡ ¢ and it characterizes the correlations between the classical variable and the quantum system .
Theorem 8 (Quantum Mutual Information of Classical-Quantum States) Consider the following classical-quantum state representing the ensemble :
() (2.29) ≡ | i h | ⊗ X
The Holevo information () is equivalent to the mutual information ( : ):
()= ( : ) (2.30)
In this sense, the quantum mutual information of a classical-quantum state is most similar to the classical mutual information of Shannon.
Theorem 9 The following bound applies to the Holevo information:
( : ) log (2.31) ≤ where is the dimension of the random variable and the quantum mutual information is with respect to the classical-quantum state. 18 CHAPTER 2. QUANTUM INFORMATION AND VON NEUMANN ENTROPY. 2.5 Quantum Relative Entropy.
The quantum relative entropy ( ) between two states and is as follows: || ( ) (log log ) (2.32) || ≡ { − } Similar to the classical case, we can intuitively think of it as a distance measure between quantum states. But, in a mathematical sense, it is not strictly a distance measure because it is not symmetric and does not obey a triangle inequality. Nevertheless, the quantum relative entropy is always non-negative.
Theorem 10 (Positivity of Quantum Relative Entropy) The relative entropy ( ) || is positive for any two density operators and :
( ) 0 (2.33) || ≥
Corollary 1 (Subadditivity of Quantum Entropy) The von Neumann entropy is subad- ditive for a bipartite state :
() + () () (2.34) ≥
Property 15. The following identity holds:
= ( : ) (2.35) || ⊗ ¡ ¢
Property 16. The following identity holds:
= ( ) (2.36) || ⊗ − | ¡ ¢
Property 17. The relative entropy is invariant under unitary operations:
( )= † † (2.37) || || ¡ ¢ 2.5. QUANTUM RELATIVE ENTROPY. 19
Property 18. (Additivity of Quantum Relative Entropy) The quantum relative entropy is additive for tensor product states:
( 1 2)= ( 1)+ ( 2) (2.38) 1 ⊗ 2|| ⊗ 1|| 2|| In general it follows, ⊗ ⊗ = ( ) (2.39) || || ¡ ¢
Property 19. (Quantum Relative Entropy of Classical-Quantum States) Quantum relative entropy between classical-quantum states and is as follows:
= () ( ) (2.40) || || ¡ ¢ X where
() (2.41) ≡ | i h | ⊗ X () (2.42) ≡ | i h | ⊗ X 20 CHAPTER 2. QUANTUM INFORMATION AND VON NEUMANN ENTROPY. Chapter 3 Classical and Quantum Correlations.
Non-classical correlations in quantum systems (or simply, quantum correlations) can be seen as a signature that subsystems are genuinely quantum. They have come to be recognized as a novel resource that may be used to perform tasks that are either impossible or very inefficient in the classical realm, providing the seed for the development of modern quantum information science.
The notion of entanglement has been related to non-classical correlations. However, entan- glement is no the only type of correlations that can be found in multipartite quantum systems. Recently, quantum discord has proved to be other kind of quantum correlation based on the effects of measurements made on any of the parties of the system. Since the measurements do not alter the correlations present in the classical states, quantum discord has been interpreted as a measure of the quantumness of the correlations.
3.1 Entanglement.
The concept of entanglement has played a crucial role in the development of quantum physics. In the early days entanglement was mainly perceived as the qualitative feature of quantum theory that most strikingly distinguishes it from our classical intuition. In Wooter’s words [69]: entanglement is the quantum mechanical property that Schrödinger singled out many decades ago as “the characteristic trait of quantum mechanics”.
Entanglement has been studied extensively in connection with Bell’s inequality [5] allowing that the non-local characteristics being accessible to experimental verifications [5,70, 71].
Using the concept of entanglement is possible to classify the states of a quantum system in separable and in entangled states. If the state of a pair of quantum system is pure, it is called
21 22 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS. entangled if it is unfactorizable. Now, if the state is a mixed state, it is entangled if it cannot be represented as a mixture of factorizable pure states.
3.1.1 PPT Criterion and Negativity.
In his work [66], A. Peres showed that if the state was separable, i.e., if it can be written into a sum of direct products:
= 0 00 (3.1) ⊗ X where the positive weights satisfy =1and where 0 and 00 are density matrices for the two subsystems, then after partial transpose on one of the subsystems of a compound bipartite P system it is still a legitimate state. In other words, a necessary condition for separability is that a matrix, obtained by partial transposition of , has only non-negative eigenvalues.
The derivation of this separability condition is best done by writing the density matrix elements of (3.1) explicitly, with all their indices [16]:
= (0 ) (00) (3.2) X where Latin indices refer to the first subsystem, Greek indices to the second one (the subsystems may have different dimensions).
Let us now define a new matrix
= (3.3) where the Latin indices of have been transposed, but not the Greek ones. This is not a unitary transformation but, nevertheless, the matrix is Hermitian. When Eq. (3.1) is valid, we have = (0 ) 00 (3.4) ⊗ X Since the transposed matrices (0 ) (0 )∗ are nonnegative matrices with unit trace, they can ≡ also be legitimate density matrices. It follows that none of the eigenvalues of is negative which is a necessary condition for Eq. (3.1) to hold.
For low dimensional systems Peres criterion is called Peres-Horodecki criterion [72] and it gives a necessary and sufficient condition of separability. 3.1. ENTANGLEMENT. 23
A computable measure of entanglement is the Negativity [67]. It essentially measures the
degree to which the partial transpose of the bipartite mixed state fails to be positive, and therefore it can be regarded as a quantitative version of Peres’ criterion for separability [66]. Thismeasureisdefined as 1 () 1 − (3.5) N ≡ ° °2 and is based on the trace norm ° °
( )† (3.6) 1 ≡ ° ° q which corresponds to the absolute° value° of the sum of negative eigenvalues of .Nega- tivity () vanishes for unentangled states and does not increase under LOCC [67], i.e., it N is an entanglement monotone [73], and as such it can be used to quantify the degree of the entanglement in composite systems [74].
3.1.2 Entanglement of Formation and Concurrence.
Perhaps the most basic physically motivated quantitative measures of entanglement is the entanglement of formation [75], which is intended to quantify the resources needed to create a given entangled state.
The entanglement of formation is defined as follows [75]:
Definition 4 Given a density matrix of a pair of quantum systems and ,considerall possible pure-state decompositions of , that is, all ensembles of states with probabilities | i , such that
= (3.7) | ih | X For each pure state, the entanglement is defined as the entropy of either of the two subsystems and [76,77]: ()= ( log )= ( log ) (3.8) − − Here is the partial trace of over subsystem ,and has a similar meaning. The | ih | entanglement of formation of the mixed state is then defined as the average entanglement of the pure states of the decomposition, minimized over all decompositions of :
()=min () (3.9) X 24 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS.
The minimum value specified in Eq. (3.9) can be expressed as an explicit function of [2], i.e., the entanglement of formation of a mixed state oftwoqubitsisgivenby
()=E ( ()) (3.10) where 1+√1 2 E ()= − (3.11) 2 µ ¶ with ()= log (1 )log(1 ) (3.12) − − − − The concurrence () is defined as
()=max 01 2 3 4 (3.13) { − − − } where the non-negative real numbers ’s are the square roots, in decreasing order, of the eigenvalues of the non-Hermitian matrix ˜. The density matrix ˜ =( ) ∗ ( ) is ⊗ ⊗ the spin-flipped state, with the Pauli matrix and ∗ denoting complex conjugated (when the latter is written in the standard basis).
Note that the function E () is monotonically increasing and ranges from 0 to 1 as goes from 0 to 1,sothatonecantaketheconcurrence(3.13)asameasureofentanglementinits own right.
3.2 Quantum Discord.
Another method to quantify the quantum correlations of the system is to use the fact that measurements disturb quantum systems but does not classical ones. Using this idea, Ollivier and Zurek introduced the notion of quantum discord [35].
Basically, quantum correlations are present between two systems if a disturbance is detected when a measurement is performed on one of the parties. Otherwise it implies that they are absent.
The quantum mutual information function quantifies this disturbance. It gives an indi- cation of how much information is shared between parties and .Thedifference between the mutual information function before the measurement and after measurement defines the discord. However, the quantum mutual information function before the measurement depends 3.2. QUANTUM DISCORD. 25 on the set of projectors that are applied, for example on . Therefore they have to be chosen so that they give the maximal value for the measurement-induced mutual information function.
Due to the "optimization" involved, quantum discord has to be obtained numerically. How- ever, for certain classes of mixed states it is possible to calculate it analytically. For example, S. Luo evaluated analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord [48]. In [49] M. Ali and coworkers derived an explicit expressions for quantum discord for a seven-parameter family of so called states but, despite of this result, it is not possible to find an analytic expression for discord for the general state [78].
In general the problem of the calculation of the quantum discord can be cast into the solution of two transcendental equations as it is shown in [47].
There is in general a more complex hierarchy of quantum correlations’ quantifiers in which different types of measurement schemes are applied of which the quantum discord is a particular case [79].
3.2.1 Positive Operator Valued Measure.
A positive-operator-valued measure (POVM), denoted as , is a set of positive operators { } called POVM elements that sum to identity, reflecting positivity and normalization condition for probabilities.
As positive operators, each can be diagonalized and the number of its nonzero eigenval- ues gives the rank of the POVM element. Rank-one POVMs are of special interest and they are defined to be POVMs with only rank-one elements. These elements are proportional to projectors, but these projectors need not be orthogonal.
(1) (0) ThesetofPOVMsisconvex,i.e.if and are elements of a POVM, then the convex (1) (0) combination of elements +(1 ) defines a valid POVM . This structure ≡ − reflects an experimentalist’s freedom to randomly choose one of many measuring apparatuses.
A POVM is called extremal if it cannot be represented as a convex combination of other
POVMs. A rank-one POVM is extremal if and only if its elements are linearly independent [80].
Every POVM element can be written as = † where is called measurement operator. This decomposition is not unique and therefore knowledge of POVM elements is not 26 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS. sufficient to describe post-measurement states. The full physical evolution is codified by the measurement operators. The post-measurement state, ignoring the measurement outcome, is given by the map 0 = ()= †. E P
3.2.2 Entropic Definition of Quantum Discord.
If we measure the lack of information by entropy, the total correlations is captured by the mutual information ( : ) ()+ () () (3.14) ≡ − where () is the Shannon entropy ()= log if is a classical variable − with values occurring with probability ,or () is the von Neumann entropy ()= P ( log ) if is a quantum state of system (all logarithms are base two). For clas- − sical variables, Bayes’ rule defines a conditional probability as = .Thisimpliesan | equivalent form for the classical mutual information
( )= () ( ) (3.15) | − | where the conditional entropy ( )= ( ) is the average of entropies ( )= | | | log . The classical correlations can therefore be interpreted as information gain − | | P about one subsystem as a result of a measurement on the other. P
In the quantum case, there are many different measurements that can be performed on a system and in general they disturb the quantum state. A measurement on subsystem is described by a POVM with elements = †,where is the measurement operator and is the classical outcome. Under the measurement, if we don’t know the result, the initial state is then transformed to
0 = † (3.16) → X where party observes outcome a with probability = () and has the conditional () state = . |
With this in mind, we can define a classical-quantum version of the conditional entropy,
( ) , and introduce classical correlations of the state in analogy |{ } ≡ | with Eq. (3.15), [34]: P ¡ ¢ ( ) () ( ) (3.17) |{ } ≡ − |{ } 3.2. QUANTUM DISCORD. 27
Now, to quantify the classical correlations of the state independently of a measurement,
( ) has to be maximized over all measurements: |{ }
( ) max ( ) (3.18) | ≡ |{ } { }
When the measurement is carried out by a set of rank-one orthogonal projections Π ,the { } state on the right hand side of Eq. (3.16) has the form
= Π (3.19) | ⊗ X which involves only fully-distinguishable states for and some indistinguishable states for . Such states are often called classical-quantum (CQ) states1. Note that for a CQ state there exists a von Neumann measurement of which does not perturb the state.
Thus, the quantum discord of a state under a measurement is defined as a { } difference between total correlations measured by Eq. (3.14) and the classical correlations Eq. (3.17), [35]:
( ) ( : ) ( ) | ≡ − | =min + () () (3.20) | − { } X ¡ ¢ The minimization here is equivalent to maximization in Eq. (3.18).
Eq.(3.20) is just a difference between two classically-equivalent versions of conditional en- tropy
( )=min ( ) ( ) (3.21) | |{ } − | { } where ( )= () () is the usual conditional entropy, [81]. This equivalence holds | − for rank-one POVM measurements which in classical theory correspond to questions about a value of a classical random variable. It turns out that rank-one POVM measurements minimize the discord.
Quantum discord has the following properties: Property 20. It is not symmetric, i.e. in general
( ) = ( ) (3.22) | 6 | 1 Or quantum-classical (QC) when one exchanges the roles of and 28 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS. which may be expected because conditional entropy is not symmetric. This can be interpreted in terms of the probability of confusing certain quantum states.
Property 21. Discord is nonnegative,
0 (3.23) ≥ which is a direct consequence of the concavity of conditional entropy [82].
Property 22. Discord is invariant under local unitary transformations, i.e. it is the same for state and state ( ) ( )†. This follows from the fact that discord is ⊗ ⊗ defined via entropies, and the value obtained for measurement on the state can also { } be achieved with measurement † on the transformed state. Note that discord is not contractiven undero general local operations, and therefore should not be regarded as a strict measure of correlations satisfying postulates of [83]. However, ( ) is contractive under general local operations. | Property 23. Discord ( ) vanishes if and only if the state is classical-quantum, [35,84]. |
Property 24. Discord is bounded from above as ( ) (), while ( ) | ≤ | ≤ min () () [85]. { }
3.2.3 Dissonance.
Modi et al. [56] studied the problem of the separation of total correlations in a given quantum state using the concept of relative entropy as a distance measure of correlations. This allowed to put all correlations on an equal footing and unified the approach to various correlations.
Their work is based on the idea that a distance from a given state to the closest state without the desired property (e.g., entanglementordiscord)isameasureofthatproperty. Using the relative entropy ( ) ( log ) () as a measure of that distances, || ≡− − the resulting measures are the relative entropy of entanglement [86—88]
=min ( ) (3.24) ∈S || the relative entropy of discord
=min ( ) (3.25) ∈C || 3.2. QUANTUM DISCORD. 29 and the relative entropy of dissonance
=min ( ) (3.26) ∈C || The state in these expressions belongs to the set of entangled states , is in the set of E separable states and is in the set of classical states . S C
Quantum dissonance is thus defined as nonclassical correlations which exclude entan- glement.
An advantage of using distance-like measures is that everything can be defined for multi- partite states. It also turns out that and are optimized by an orthogonal projective measurement [56].
Diagram in Figure 3.1 shows the relations between these measures, where the state ∈ E (the set of entangled states), (the set of separable states), (the set of classical ∈ S ∈ C states), and (the set of product states). An arrow from to , , indicates that ∈ P → is the closest state to as measured by the relative entropy ( ). and are the total || mutual information, and and are classical correlations. The quantities labeled as and has no physical interpretation yet but they play a role in forming relations such as:
+ = + and + = + (3.27)
It is shown in [56] that all relative entropies, except for entanglement, reduce to the differ- ences in entropies between the state and its closest classical state, i.e.:
= () and = ( ) () (3.28) − − ¡ ¢ where ()=min where forms the eigenbasis of . Thismeansthatmostofthequantitiesaregivenbytheentropiccost(di| i ³ ¯ ED ¯ ¯ ED ¯´ n¯ Eo fference of entropies) P ¯ ¯ ¯ ¯ ¯ of performing operations bringing¯ the¯ initial¯ state¯ to the¯ closest state without the desired property. 30 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS.
Figure 3.1: Relative entropy of discord and dissonance. This figure is reproduced from [56].
3.2.4 Geometric Measure of Quantum Discord.
Dakic et al [50] introduced a measure of quantum discord based on the Hilbert-Schmidt dis- tance: 2 2 min =min ( ) (3.29) ≡ ∈C k − k ∈C − called geometric quantum discord. In the above expression£ is the¤ set of classical-quantum C states given by Eq. (3.19). Like the relative entropy of discord, the geometric measure gives the Hilbert-Schmidt distance to the state after the (optimal) measurement [51]:
2 =min 0 (3.30) Π k − k { } where 0 = ΠΠ. P Recently Bellomo et al. [89] study a unified version of geometric discord in a manner similar to the study of Modi et al. [56]. They found that the closest product state to a given quantum state is not the product of the marginal states, which makes computing the total correlations with a geometric measure nontrivial. They also found that unlike for the relative entropy measures, geometric measures of correlations are not additive. They give an additivity expression for correlations as function of the original state for -states, given by
11 0014 0 0 = ⎛ 22 23 ⎞ (3.31) 0 0 ⎜ 32 33 ⎟ ⎜ ⎟ ⎜ 41 0044 ⎟ ⎜ ⎟ ⎝ ⎠ where =1and 2, 2 must be satisfied for to be a density 22 33 ≥ | 23| 11 22 ≥ | 14| matrix. P 3.3. QUANTUM DISCORD AND GENERALIZED MEASUREMENTS. 31
The advantage of the geometric measure is that the minimization present in the definition (3.29) can be performed explicitly. For a general two-qubit states written in the representation
1 3 3 = (3.32) 4 =0 =0 ⊗ X X where = 1 is the th Pauli operator and the real are experimentally- { } accessible averages = ( ), the geometric discord of a quantum state equals: ⊗ 3 3 1 2 = max (3.33) 4 − =0 =0 X X where max is the largest eigenvalue of the matrix = + ˆˆ , built from the local Bloch vector =(102030) and correlation matrix ˆ having as entries for =1 2 3 [50].
For an explicit form of max see [90].
Hassan et al. [91] and [92] claim similar results for more general bipartite states. Shi et al. [93] give an analytic formula for symmetric geometric discord for two-qubit systems. Geometric discord can be established directly from experimental data measured on up to six copies of a quantum state [94]. The idea is to rephrase the discord in terms of functions of powers of density operators and use known circuits for their implementation [95].
Girolami and Adesso [90] introduce a remarkably tight lower bound on geometric discord
of two qubits: 1 = 2() 6(2) 2()2 (3.34) 12 − − ∙ q ¸ where is defined below Eq. (3.33). A similar bound exists for systems in 2 dimensions. × The value of (numerically) upper bounds the negativity of two-qubit states squared, i.e. 2 , with equalities for pure states [96]. In terms of quantum discord, the geometric ≤ ≤ 1 2 discord of two qubits admits the bound [51]. Girolami and Adesso [47] give another ≥ 2 lower bound on geometric discord, in terms of the correlation tensor of a general bipartite state.
3.3 Quantum Discord and Generalized Measurements.
From the definition of quantum discord we realize that there is involved an optimization over measurements. Naturally, one would like to know whether projective measurements or POVMs are optimal. 32 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS.
In [84] A. Datta showed that quantum discord is optimized by rank-one POVM. Using a
fine-graining = measurement to minimize the classical-quantum version of condi- tional entropy ( ) he found that ( ) ( ). Similar line of reasoning |{P } |{ } ≥ |{ } shows that the optimal rank-one POVM has to be extremal [97].
However, Galve et al. [98] showed that orthogonal projective measurements give a pretty- tight upper bound on discord, and there is only a tiny set of states for which numerics shows the difference. They also show that for rank-two states (with only two nonzero eigenvalues) orthogonal projective measurements are optimal.
3.4 Relation between Entanglement and Discord: Koashi- Winter Identity.
For some tasks quantum discord has been related to various measures of entanglement. This relation is often derived from the Koashi-Winter relation [99] and the purification process (any mixed state comes from a partial trace of a pure state).
3.4.1 Purification.
Any system in a mixed state can be seen as part of a larger pure state, and constructing a pure state from a given mixed state is called purification. This important feature of quantum mechanics can be used to distinguish quantum mechanics from other theories [100].
A pure state can be constructed (purified) from a mixed state that is in the spectral decomposition of = by means of using the Schmidt decomposition | ih | P = √ (3.35) | i | i ⊗ | i X where are orthonormal in the space of . {| i}
3.4.2 Koashi-Winter Relation.
Recall that quantum discord (3.21) is the difference between the two definitions, ( ) and | ( ) of conditional entropies (which classically are equal but not quantum). |{ } 3.4. RELATION BETWEEN ENTANGLEMENT AND DISCORD. 33
While the classical conditional entropy is always a positive quantity, its quantum version ( ) can become negative2. On the other hand, the second definition of quantum condi- | tional entropy suffers from classicalization, that is there must be a measurement on the state in order to determine its outcome [34]. This quantity is always positive and it is related to entanglement of formation due to the monogamy relation [99]:
( : )+ ( )= () (3.36) | for any tripartite pure state . | i
The Koashi-Winter relation (3.36) and the relation between concurrence and [2] give an explicit algorithm for calculating the quantum discord of rank-two states of 2 dimensional × systems [98,104—106]. It reads:
( )= ( : ) ( ) (3.37) | − |
2 where system purifies . For rank-two states = =1 the purifica- 2 | ih | tion reads = √ where is any orthonormal basis of and | i =1 | i| i {| i} P accordingly is a qubit. Therefore, = ( ) is a state of two qubits and P | ih | Wootters’ formula (3.13) can be applied for calculation of discord.
3.4.3 Conservation Law for Correlations.
Fanchini et al. [107] derived the following relation
( : )+ ( : )= ( )+ ( ) (3.38) | | called a quantum conservation law. It says that "Given an arbitrary tripartite pure system, the sum of all possible bipartite entanglement shared with a particular subsystem, as given by the , cannot be increased without increasing, by the same amount, the sum of all discord shared with this same subsystem."
Similarly, the difference in discord as measured by a single party can be understood as the difference in entropies of the unmeasured parties:
( ) ( )= () () (3.39) | − | − 2 This quantity has proven to be very useful, for instance, the negativity is an entanglement witness [101,102], and yet for a long time it lacked an operational interpretation. The key breakthrough came in the form of a task known as quantum state merging [103]. 34 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS.
In another paper, same authors [105] give the discord chain rule, which expresses entangle- ment of formation in terms of different discords:
( : )= ( )+ ( ) ( ) (3.40) | | − |
3.5 General Bound for Quantum Discord.
A very general bound relating discord to the von Neumann entropy of the measured subsystem was proved by A. Datta [84] and Xi et al. [108]. They found the following inequality:
( ) () (3.41) | ≤ Determining which states saturate this bound is more demanding, and was done in [109].
A further work [110] presents bounds for discord that apply to arbitrary finite dimensional
. These bounds are much weaker than the previous type, but are experimentally accessible and can be measured by joint measurements on two-fold copies of an unknown state.
3.6 Classical States and Nullity Conditions for Quan- tum Discord.
Vanishing discord corresponds to a key notion of classicality, for which maximal information about a subsystem can be obtained by some specific local measurement without altering cor- relations with the rest of the system. Therefore several nullity conditions have been proposed.
A theorem which characterizes the zero-discord states is the following.
Theorem 11 A state satisfies ( )=0if and only if there exists a complete set of | rank-one orthogonal projectors Π on , satisfying Π =1and ΠΠ0 = 0 Π,suchthat P = Π (3.42) | ⊗ X
The set of states classical with respect to is denoted as . C 3.6. CLASSICAL STATES AND NULLITY CONDITIONS FOR QUANTUM DISCORD.35
The theorem above gives a physical interpretation for zero-discord states: for any state in
there exists a basis for for which the locally-accessible information is maximal and, from C the perspective of an external observer, this information can be obtained without disturbance to the combined system.
A necessary but not sufficient condition for vanishing discord can be inferred from of Eq. (3.42) and is due to [111]:
( )=0= [ 1 ]=0 (3.43) | ⇒ ⊗
On the other hand, a simple necessary and sufficient nullity condition for a state to have discord zero with respect to one party, first presented by Chen and coworkers [112] and latter by Huang et al. [113], is: ( )=0if and only if there exists a complete-orthonormal basis | for party which simultaneously diagonalizes all the operators 0 ,that {| i} | 0 ≡ h | | i is if and only if the operators commute. To check for classicality for a bipartite state | 0 4 therefore, it is necessary to verify a number () of commutation relations. The condition canbeappliedforanyfinite number of parties and dimensionality. This condition is applied to two-qudit circulant states in [114].
Finally, an alternative nullity condition is proposed by [50], and makes use of the singular- value decomposition. The idea is as follows: Given a state , of arbitrary finite dimensions, one first obtains the (real-valued) correlation matrix =() by making the expansion
= ,where ( )defines a basis of Hermitian operators for party ⊗ { } { } (). By the singular-value decomposition, can be diagonalized as = ,where P matrices and are orthogonal, and the diagonal entries of are the nonnegative singular values of .Then = where = and = . ⊗ The existence of the block diagonalization of Eq. (3.42) is equivalent to the simultaneous P P P diagonalizability of the operators . This gives the nullity condition ( )=0if and { } | only if the operators commute. The number of commutation relations to check is given by (12) () ( () 1), a number which has been substantially reduced by the × × − singular-value decomposition. If () is greater than the dimension of ,then cannot be classical with respect to . 36 CHAPTER 3. CLASSICAL AND QUANTUM CORRELATIONS. Chapter 4 Dissonance is Required for Assisted Optimal State Discrimination.
The roles of quantum correlations, entanglement, discord, and dissonance needed for perform- ing unambiguous quantum state discrimination assisted by an auxiliary system are studied. In general, this procedure for conclusive recognition between two non-orthogonal states relies on the availability of entanglement and discord. However, we find that there exist special cases for which the procedure can be successfully achieved without entanglement. In particular, we show that the optimal case for discriminating between two non-orthogonal states prepared with equal a prior probabilities does not require entanglement but quantum dissonance only.
Unambiguous discrimination among linearly independent non-orthogonal quantum states is a problem of fundamental interest [22—24, 115—117]. Two non-orthogonal states require a three-dimensional Hilbert space for implementing an optimal procedure of unambiguous state discrimination [16, 118]. When the states are codified strictly in a two-dimensional Hilbert 1 space, like a spin- 2 particle, the process for unambiguous discrimination has to be assisted by an ancillary system in order to increase the dimension of the Hilbert space [16,118].
Naively thinking, entanglement would be the main ingredient for performing the assisted state discrimination protocol [24]. However, we know that entanglement is not the only cor- relation present in quantum states so, it is natural to ask about what kind of correlations, entanglement, quantum discord, or dissonance, are behind a successful discrimination out- come.
Consider that a qubit is randomly prepared in one of the two non-orthogonal states + or with a prior probabilities + and =1 +, respectively [24, 119]. Let us assume¯ ® − − − ¯ that¯ the® system can be coupled to an auxiliary qubit by a joint unitary transformation ¯ 37 38 CHAPTER 4. CORRELATIONS FOR STATE DISCRIMINATION such that
2 = 1 + + 0 + + 0 1 (4.1a) + | i − | | | i| i | i| i ¯ ® q 2 ¯ = 1 0 + 0 1 (4.1b) − | i − | −| |−i| i − | i| i ¯ ® q where is a known initial¯ state and 0 1 is an orthonormal basis of the auxiliary | i {| i | i} system. We have also considered the orthonormal basis 0 1 of the principal system and 0 1 {| i | i} the orthonormal states = | i±| i [24]. |±i √2 The a prior fixed overlap + = = does not change due to the joint unitary − | | transformation; thus from (4.1) we¯ see® that + and probability amplitudes satisfy the ¯ − constraint = +∗ , i.e., = + and + = ,with the phases of ,and − | −| | | | | − − ± ± + 1.Thus,the+ amplitude defines the joint unitary transformation which allows | | ≤ | | ≤ to couple in average the quantum system of interest with the auxiliary one.
We must consider that in principle the relative phase could be managed by properly choosing the axes on the Bloch sphere. As we will see, one convenient choice would be =0. In this manner after applying the unitary we have the mixed states
+ = † + + + + † (4.2) | | − − | i − h | | i h | ¯ ® ¯ ¯ ® ¯ This expression reveals in principle¯ the presence¯ of quantum¯ correlations¯ between the system and the ancilla.
The process of discriminating unambiguously the prepared initial states + or ,is − achieved by performing a von Neumann measurement on the basis 0 1 of the ancillary {| i | i}¯ ® ¯ ® system. The recognition is successful when the ancilla is projected onto the¯ state 0 ¯ ,since | i in this case the system of interest collapses to the orthogonal states + or , depending | i |−i in which state, + or it was initially prepared. Otherwise, the process fails when the − projection is onto 1 . In this case, the initial information disappears since the principal ¯ |® i ¯ ® system collapses¯ into 0 ¯, whatever be the prepared state. | i
The probability of success depends on the + parameter and is given by: | | 2 2 ( + )=1 | | 2 + + (4.3) | | − − + − | | | |
Notice that ( + ) is different from zero for any value of + . This means that this process | | | | always allows discriminating probabilistically and unambiguously the prepared state. 39
The optimal success probability is attained for + = 4 − ,or + = (+ ), | | + | | | | | | ≥ − or + =1( +), and can be expressed as: q | | − ≥ p
1 2√+ 0 ˜ − max = − 2 | | ≤ | | ≤ (4.4) ( 1 max + ˜ 1 − | | { −} ≤ | | ≤ ¡ ¢ + where ˜ =min − . It is worth emphasizing that for [0 ˜ [ both states could + − | | ∈ | | be recognized andnq theq probabilityo is linear in .For [ ˜ 1[, only one state can be | | | | ∈ | | discriminated, say + ( )if+ (+ ) and the probability is quadratic in . − ≥ − ≤ − | | In addition, we note¯ that,® ¯ as® is known, the probability is 1 for discriminating two orthogonal states ( =0), whereas¯ it¯ is 0 when the two states are different only by a phase factor ( =1). | | We now answer our main question about what kind of correlation allows performing the procedure of conclusive non-orthogonal state discrimination when it is assisted by an auxiliary system. We can ask first how much entanglement between the systems is required. The amount of entanglement contained in state (4.2) is given by the concurrence [2]
2 2 2 + = 2 1 + + + + 1 | | 2 | | | | ⎧ Ã − | | | | s − + + −! ⎨ q | | | | ¡ ¢ 1 2 2 ⎩ 2 2 8 1 + 1 | | 2 + cos (4.5) − | | − | | s − + − 2 q | | ) We see that the concurrence depends on the phase of the overlap , and it has its minimum value when is zero. The maximal concurrence holds for = , which corresponds just to ± the average concurrence of the decomposition (4.2). This is illustrated in Figure 4.1, where concurrence is shown as a function of + for different values of , ,and+. The solid line | | | | is the probability ( + ) of Eq. (4.3). | |
It is clear from Figure 4.1 that we can not relate the probability of success to a given amount of entanglement, given that, for different values of attaining different values of entanglement, we have the same probability of success. Even more, for =0there are some values of + | | forwhichtheentanglementiszero.
In particular we note that there is one zero of entanglement around the maximal prob- ability of discrimination for + = .For+ = , still we have successful discrimination − 6 − 40 CHAPTER 4. CORRELATIONS FOR STATE DISCRIMINATION
(a) (b) 0.6 0.6
0.4 0.4 , C , C s s
P 0.2 P 0.2
0 0 0.4 0.6 0.8 1 0.6 0.8 1 |α | |α | + +
0.8 (c) 0.8 (d) 0.6 0.6
0.4 0.4 , C , C s s P P 0.2 0.2
0 0 0.2 0.4 0.6 0.8 1 0.6 0.8 1 |α | |α | + +
Figure 4.1: Concurrence + as a function of + for different values of : 0 (dashed), | | | | 2 (dotted), (dash-dot). The solid line is the probability ( + ). The a priori probability ¡ ¢ | | + =12 for (a) and (b) and + =25 for (c) and (d). The corresponding overlaps are (a) 13,(b)1√3, (c) 15,and(d)1√5. | | with zero entanglement but with non-optimal probability. These facts are indicating that the discrimination processes do not necessarily require entanglement.
We now consider the optimal process of state discrimination (the maximum value of ), for which the concurrence is obtained by evaluating (4.5) in + = 4 − ,when [0 ˜], | | + | | | | ∈ or in + = ( + =1)when [˜ 1] and + (+ q). | | | | | | | | ∈ ≥ − ≤ − p
The concurrence is symmetric under the exchange of + and and it takes 4 − √ + | | − µ ¶ themaximalvaluefor = and the minimal one for =0( [0 ˜[), whereas and | | ∈ | | ( ) do not depend on the phase ( [˜ 1]). 1 | | ∈ ¡ ¢ These features are illustrated in Figures 4.2(a) and 4.2(b) which show the concurrence as functions of for different values of and +, i.e., in the interval [0 ˜] 4 − √ | | + | | | | ∈ and ( ) in the interval [˜ 1]. µ ¶ 1 | | ∈
In the optimal success probability the concurrence can be zero only when + = and − =0as is illustrated by the solid line in Figure 4.2(b). In this case the concurrence is given by: = 2 (1 ) sin (4.6) √ | | − | | 2 | | ¯ ¯ ³ ´ p ¯ ¯ ¯ ¯ ¯ ¯ 41
0.7 0.7
0.6 (a) 0.6 (b)
0.5 0.5
0.4 0.4 C C 0.3 0.3
0.2 0.2
0.1 0.1
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 |α | |α |
0.7 0.7 0.6 (c) 0.6 (d) 0.5 0.5
0.4 0.4 D D 0.3 0.3 0.2 0.2
0.1 0.1
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 | α | |α |
Figure 4.2: Concurrence (a), (b) and discord (c), (d) as functions of for the case of optimal success probability of discriminating. We consider different values of |: |0 (solid), 3 (dashed), 2 (dotted), and (dot-dash) in the interval [0 ˜] and solid line in the interval | | ∈ | | ∈ [˜ 1].In(a),(c)+ =14 and in (b), (d) + =12.
It is clear from this expression that for =0there is an optimal discrimination process without assistance of entanglement with =1 . − | |
In the general case + , expression (4.5), can exhibit other zeros only when = | | 0. Specifically we get that¡ concurrence¢ (4.5) is zero when + is a root of the 4th degree 2 | | polynomial in + ,givenby: | | 2 2 8 6 2 2 + + + − + − =0 (4.7) | | − | | | | | | − | | µ + ¶ µ + ¶ A simple analytical solution of this equation is found for equal a prior probabilities, namely:
+ = 0 1 (4.8a) | |=0 | | ≤ | | ≤ p 2 1 1 4 1 + = v ± − | | 0 (4.8b) | |=0 u q 2 ≤ | | ≤ 2 u t Equation (4.8a) coincides with the case of optimal success probability. In the solution (4.8b) 1 the process of discrimination happens with constant probability = 2 , as can be seen by replacing (4.8b) in (4.3).
Figure 4.3 shows the solutions + of Eq. (4.7) as a function of for which the concurrence | | | | is equal to zero. The solutions given in (4.8) are shown in Figure 4.3(a), where the solid line corresponds to the (4.8a) solution, dashed line to (4.8b) solution with plus sign, and dotted line to (4.8b) solution with minus sign. The case = + is illustrated in Figures 4.3(b), − 6 42 CHAPTER 4. CORRELATIONS FOR STATE DISCRIMINATION
4.3(c), and 4.3(d) where also three solutions appear in one interval and one solution in another interval.
1 1 0.8 (a) 0.8 (b)
| 0.6 | 0.6 + + α α | 0.4 | 0.4 0.2 0.2 0 0 0 0.5 1 0 0.5 1 |α| |α|
1 1 0.8 (c) 0.8 (d)
| 0.6 | 0.6 + + α α | 0.4 | 0.4 0.2 0.2 0 0 0 0.5 1 0 0.5 1 |α| |α|
Figure 4.3: + solutionsasfunctionsof for which the concurrence is zero.In (a) + =12, | | | | (b) + =4991000,(c)+ =13,(d)+ =35.
From the previous analysis we learned that the assisted state discrimination process can be performed in the absence of entanglement. In those cases it is important to know which correlation is behind the state recognition. In this respect, recent progress in the understanding of correlations other than entanglement, such as quantum discord, dissonance, or classical ones, can shed light to answer this question. In the absence of entanglement in a mixed state, quantum dissonance is present if discord is different from zero [56]. If discord is zero, then only classical correlations could be present [56]. As is well known, quantum discord for a bipartite mixed is given by [35]
= () sup () () (4.9) − ˆ − ( ) { } X where ( )= ( )+ ( ) ( ) is the quantum mutual information and () is the − von Neumann entropy. The second term on the right-hand side of this expression corresponds to the classical correlations. The supreme is taken over all the measurement sets ˆ applied ˆ on system , is the probability for outcomes ,and is the partial projectionn o of ˆ [(1 )] defined as = ⊗ [35]. In this way, discord can be calculated numerically. However, there are some cases where the optimization problem was solved analytically [48]. 43
0.20
0.15
D 0.10
0.05
0.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |α | +
Figure 4.4: Quantum dissonance as a function of + for + = and =0,forsolutionsin (4.8). Solid line corresponds to (4.8a), dashed line| to| (4.8b) with− minus sign and dotted line to (4.8b) with plus sign.
In our study, we avoid the optimization problem by using the Koashi and Winter identity [99] since the rank-two state (4.2) can be written as a tripartite pure state:
Ψ = √+ ( + ) 0 + √ ( ) 1 (4.10) | i | i| i | i − | −i| i | i where we introduced a fictitious qubit that, once traced, led to the mixed state (4.2).
FollowingtheKoashiandWinter[99]recipe,wehavethat =min ˆ ( ) = { } ( ),where ( ) is the entanglement of formation between the system{ } of interest and the P fictitious qubit . Then, for calculating the optimization it is easier to calculate ()= 2 1+√1 log (1 )log (1 ) with = − and being the concurrence of the − 2 − − 2 − 2 reduced density matrix. Thus, the quantum discord is given by:
( )= ( ) ( )+ ( ) (4.11) − This expression can be calculated more easily than (4.9).
1 In Figure 4.2(c), the discord is shown for the optimal probability of success when + = 4 , and can be compared qualitatively with the corresponding concurrence in Figure 4.2(a). We realize that discord also depends on the phase when [0 ˜[ whereas for [˜ 1] it 1 | | ∈ | | ∈ does not depend on . For the case + = 2 , a similar dependence on is shown for discord in Figure 4.2(d) as compared with concurrence in Figure 4.2(b). 44 CHAPTER 4. CORRELATIONS FOR STATE DISCRIMINATION
In general, we cannot say which correlation is responsible for the state discrimination process. However, we can say that in the optimal case with equal a prior probabilities and =0, the process is assisted exclusively by dissonance. Similarly, for the roots (4.8b), the non-optimal case, the quantum dissonance can be calculated by using Eq. (4.11).
Figure 4.4 shows the quantum dissonance as a function of + for + = and for solutions | | − in (4.8) with =0. Noticethatinallofthesethreecasesthequantumdissonanceisresponsible for successfully completing the procedure.
One can show that there are always solutions of Eq. (4.7), some of them illustrated in Figure 4.3, for which the process of state discrimination is assisted only by dissonance and not by entanglement. Summary.
In summary, we have shown that the protocol for unambiguous discrimination of two non- orthogonal quantum states, assisted by an auxiliary system, in general requires quantum cor- relations in order to be implemented. The particular case with optimal probability of success requires both entanglement and discord except the case with equal a prior probabilities, which is performed with zero entanglement and nonzero discord; i.e., only quantum dissonance is needed in this important case.
We also found other non-optimal state discrimination procedures with different a prior probabilities which are assisted by quantum dissonance only, since entanglement is absent and the success probability is different from zero. In other words, here we have found that an assisted unambiguous state discrimination protocol always can be implemented successfully aided only by quantum dissonance.
Finally, we would like to emphasize that the optimal assisted state discrimination protocol with equal a prior probabilities does not make use of an entangled state but of a nonclassical separable state in general. However, there are two cases for which a classical state appears: (i) two orthogonal states for which the probability of discrimination is one and (ii) two parallel states for which the probability of discrimination is zero.
45 46 Conclusiones.
En resumen, hemos demostrado que el protocolo para la discriminación asistida de dos estados cuánticos no ortogonales en general requiere correlaciones cuánticas para ser realizada con éxito. Para tener probabilidades de discriminación óptimas se requiere entrelazamiento y quantum discord, mientras que el caso en que las probabilidades aprioride los estados son iguales no se requiere de entrelazamiento pero si quantum discord. Es decir, sólo disonancia es necesaria en este importante caso.
También encontramos que para probabilidades aprioridiferentes es posible realizar la discriminación asistida solo por la disonancia, solo que en estos caso las probabilidades de éxito no son óptimas. En otras palabras, un protocolo de discriminación asistida siempre se puede implementar con éxito teniendo como recurso solo el quantum discord.
Por último, se reafirma que el protocolo óptimo de discriminación asistida con probabili- dades apriorino hace uso de un estado entrelazado, sino de un estado separable en general. Sin embargo, hay dos casos en los que aparece un estado clásico. (i) Cuando los estados son ortogonales, caso en el cual la probabilidad de discriminación es uno. (ii) Cuando los estados son paralelos, caso en el cual la probabilidad de discriminación es cero.
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