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Journal of Physics Conference Series

Convex combination of quantum states and characterization of correlation

Guangrong Liu, Yang Liu, Xiangrong Li

Published Feb 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/2964/1/012084

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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 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.
One of the most important questions in quantum information theory is the so-called separability problem. It involves characterizing the set of separable (or, equivalently entangled) states among mixed states of a multipartite quantum system. In this thesis we study the generalization of this problem to types of quantum correlations that are defined in a manner analogous to entanglement. We start with the subset of set of pure states of a given quantum system and call states belonging to the convex hull of this subset "non-correlated" states. Consequently, the states laying outside the convex hull are referred to as "correlated". In this work we focus on cases when there exist a symmetry group that preserves the class of "non-correlated" pure states. The presence of symmetries allows to obtain a unified treatment of many types of seemingly unrelated types of correlations. We apply our general results to particular types of correlations: (i) entanglement of distinguishable particles, (ii) particle entanglement of bosons, (iii) "entanglement" of fermions, (iv) non-convex-Gaussian correlations in fermionic systems, (v) genuine multiparty entanglement, and finally (vi) refined notions of bipartite entanglement based on the concept of the Schmidt number. We investigate the natural problems and questions concerning the correlations defined above: (I) We provide explicit polynomial characterization of various types of correlations for pure states. (II) We examine cases in which it is possible to give a complete analytical characterization of correlated mixed states. (III) We derive a variety of polynomial criteria for detection of correlations in mixed states.(IV) We use the above criteria and the technique of measure concentration to study typical properties of correlations on sets of isospectral density matrices.
Abstract This chapter gives a detailed overview of entanglement for pure and mixed states. We begin with the Schmidt-decomposition theorem, which allows us to associate the subsystem mixedness of pure state with entanglement. We discuss the purification of mixed quantum states, the convexity of the set of separable states, and entanglement as a form of correlation. As an example we examine the Bloch decomposition for two qubits. For the detection of entanglement the Peres-Horodecki criterion is introduced, before we analyse the relation between entanglement and non-locality via the CHSH-Operator Criterion and the family of Werner states. We further consider Tsirelson’s bound and hidden non-locality. We then discuss important mathematical tools for entanglement detection, including the positive-map theorem, the Størmer-Woronowicz theorem, and the reduction criterion. In this context we examine the family of isotropic states. Finally, the Weyl states provide an example for which we study the geometry of two-qubit entanglement
A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.
Quantum correlations in composite and separable quantum systems are characterized by non-vanishing quantum discord. We demonstrate the necessary and sufficient conditions for existence of hermitian witness operators for quantum discord, bypassing the problem of non-convexity of the set of all classical states. We propose an example of a linear witness which can detect quantum correlations in large classes of separable bipartite systems. Decomposition of the witness in terms of a small number of locally measurable operators as shown here, paves the way for experimental detection of quantum correlations beyond entanglement.
Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.