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International Conference on Applied Mathematics, Modelling and Intelligent Computing (CAMMIC)

Characterization of quantum correlation in two-body quantum system by probability distribution

Guangrong Liu, Yang Liu, Xiangrong Li

Published Jul 21, 2024• 0 citations
Publisher

Abstract

The characterization of the correlation of quantum States plays a key role in the study of quantum information theory. Classical correlation is a quantum state that remains unchanged under local orthogonal projection, and it is not affected by decoherence. The entangled quantum state must be quantum correlation, and there is also quantum correlation in the separable quantum state. Relevance is a common phenomenon in nature. From the perspective of information theory, classical relevance can be described under Shannon's information theory. Therefore, it is necessary to explore the relationship among classical correlation, quantum correlation and quantum nonlocality. For the two-body quantum system, this paper gives the corresponding definitions of classical correlation state, left classical associated state and right classical associated state from the perspective of probability density, especially the definition of classical independent associated state. Different from the definition of local orthogonal projection measurement, it is easier to construct classical correlation state and quantum correlation state from the perspective of probability distribution, and further analyze the characteristics of eigenvalues and their corresponding eigenvectors. The selection of standard orthogonal basis, the corresponding relationship between separable ground state and classical correlation, entangled ground state and quantum correlation in two-body quantum system are studied. The corresponding characteristics of left classical correlation state and right classical correlation state are explored, and the relationship between separability and correlation, the relationship between convex combination and correlation of quantum States, and the left classical correlation state and right classical correlation state may also be quantum correlation States are illustrated by examples, and the definition of independent classical correlation state and its relationship with edge distribution are further obtained.

DOI
10.1117/12.3036546

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The characterization of the correlation of quantum States plays a key role in the study of quantum information theory. Classical correlation is a quantum state that remains unchanged under local orthogonal projection, and it is not affected by decoherence. The entangled quantum state must be quantum correlation, and there is also quantum correlation in the separable quantum state. Relevance is a common phenomenon in nature. From the perspective of information theory, classical relevance can be described under Shannon's information theory. Therefore, it is necessary to explore the relationship among classical correlation, quantum correlation and quantum nonlocality. For the two-body quantum system, this paper gives the corresponding definitions of classical correlation state, left classical associated state and right classical associated state from the perspective of probability density, especially the definition of classical independent associated state. Different from the definition of local orthogonal projection measurement, it is easier to construct classical correlation state and quantum correlation state from the perspective of probability distribution, and further analyze the characteristics of eigenvalues and their corresponding eigenvectors. The selection of standard orthogonal basis, the corresponding relationship between separable ground state and classical correlation, entangled ground state and quantum correlation in two-body quantum system are studied. The corresponding characteristics of left classical correlation state and right classical correlation state are explored, and the relationship between separability and correlation, the relationship between convex combination and correlation of quantum States, and the left classical correlation state and right classical correlation state may also be quantum correlation States are illustrated by examples, and the definition of independent classical correlation state and its relationship with edge distribution are further obtained.
Abstract In the study of optical quantum information, quantum teleportation can be realized by preparing multiphoton entangled states. Entanglement is a special quantum correlation, and there is also a quantum correlation in separable quantum States, which comes from the measurement of quantum states. Convex combinations of arbitrary quantum states are still quantum states. From the spectral decomposition of a quantum state, an ensemble formed by a complete eigenvector system can be determined, which corresponds to an orthogonal projection operator family with rank 1, and the convex combination of the operator family will correspond to a quantum state. The characterization of quantum correlation is an important issue in the study of quantum information theory. A Quantum state that remains unchanged under local orthogonal projection is a classical correlation state, which is not affected by decoherence. Based on the standard orthogonal basis, this paper considers the correlation of the quantum states formed by the convex combination of the operator family. According to the separability and entanglement of the ground state and the mutual dissimilarity of the corresponding non-zero eigenvalues, it is concluded that the convex combination of the operator family formed by the separable standard orthogonal basis in the composite system must be a classical correlation state. When the entangled pure state is an eigenvector, it is a quantum correlation state if the corresponding eigenvalue is a non-zero single value. In the case of multiple roots, we can illustrate that both are possible. For the correlation of unitary evolution of quantum states under different standard orthogonal bases, the evolution law is revealed by considering important quantum gates.
The aim of this paper is to distinguish classical correlations from quantum ones. First, a new characterization of a classical correlated (CC) state is established. Corresponding results for the left and right classical correlations are also obtained. Second, a sufficient and necessary condition for a convex combination of two CC states to be CC is given. It is proved that the set CC of all CC states on is a perfect, nowhere dense and compact subset of the metric space of all states (density operators) on . Based on our new characterization of CC states, a quantity Q (ρ) is associated with a state ρ. It is shown that a state ρ is CC if and only if Q (ρ) = 0. In particular, we prove that the Werner state W λ in a two-qubit system with a single real parameter λ is CC if and only if λ = 0.25.
A bipartite state is called classical (with regard to correlations) if it is left undisturbed by a certain local von Neumann measurement, and is called separable if it can be represented as a convex combination of product states. Due to the perfect distinguishability of orthogonal vectors, a classical state can essentially be identified with a convenient bivariate probability distribution, and moreover, it is separable, but not vice versa. The notion of separability plays a key role in quantum information theory because entanglement is defined via separability. However, the definition of separability is ad hoc and formal. In this paper, we present an intrinsic characterization of separable states via classical states from the measurement perspective: Separable states are precisely those states that are reductions of classical states in higher dimensions with the natural partitions. Consequently, entangled states are precisely those states that cannot be represented as such reductions of classical states. This observation highlights the hidden mutually exclusive and complementary relations between classicality and entanglement.
Quantum information theory (QIT) is an emerging field of physics which aims to develop new methods of dealing with information by harnessing the power of quantum mechanics. Besides its potential to revolutionize the techniques of information processing and communication, it also provides novel approaches to better comprehend the foundations of quantum mechanics. Among many important problems in QIT, manipulation and dynamical characterization of correlations present in quantum systems stand out due to their relevance for the practical applications of the theory. This thesis intends to explore such correlations of quantum and classical nature from various perspectives. In particular, our discussions involve the investigation of local transformations among a class of entangled states and the examination of correlation measures in some physical models. We first examine the classification of the flip (0-1) and exchange symmetric (FES) states under local quantum operations. We study the optimal local one-shot conversions of FES states to determine the entanglement transformations that relate multiqubit FES states with the maximum possible probability of success. Next, we investigate the exchange symmetry properties of certain symmetric states when the qubits evolve according to a dephasing model which is also invariant under swap operation. We find that there exist states which do not preserve the exchange symmetry with unit probability during the time evolution, leading to the spontaneous breaking of exchange symmetry. Later, we turn our attention to the dynamics of quantum and classical correlations for qubit-qutrit systems in independent and global dephasing environments. In these cases, we demonstrate several interesting phenomena such as the transition from classical to quantum decoherence. Lastly, we investigate the thermal quantum and total correlations in the one-dimensional anisotropic XY model in transverse field. We discuss the ability of different measures to estimate the critical point of the quantum phase transition at finite temperature. We also consider the relation between correlations and the factorized ground state in this model. Furthermore, we study the effect of temperature on long-range correlations.
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.
Quantum correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement---as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2\ifmmode\times\else\texttimes\fi{}d$-dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-\`a-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].