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Chinese Physics Letters

Relative Entropy of Entanglement of One Class of Two-Qubit System

Lin-Mei Liang, Ping‐Xing Chen, Cheng-Zu Li, Ming-Qiu Huang

Published Feb 8, 2001• 12 citations
Publisher

Abstract

The relative entropy of entanglement of a mixed state σ for a bipartite quantum system can be defined as the minimum of the quantum relative entropy over the set of completely disentangled states. Vedral et al. [Phys. Rev. A 57(1998)1619] have recently proposed a numerical method to obtain the relative entropy of entanglement E re for two-qubit systems. This letter shows that the convex programming method can be applied to calculate E re of two-qubit systems analytically, and discusses the conditions under which the method can be adopted.

DOI
10.1088/0256-307x/18/3/305

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The relative entropy of entanglement of a mixed state σ for a bipartite quantum system can be defined as the minimum of the quantum relative entropy over the set of completely disentangled states. Vedral et al. [Phys. Rev. A 57(1998)1619] have recently proposed a numerical method to obtain the relative entropy of entanglement E re for two-qubit systems. This letter shows that the convex programming method can be applied to calculate E re of two-qubit systems analytically, and discusses the conditions under which the method can be adopted.
The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find an explicit expression for all the entangled states for which this state is a CSS. Our formula holds for multipartite states in all dimensions. For the bipartite case of two qubits, our formula reduce to the one given by Miranowicz and Ishizaka [Phys. Rev. A 78, 032310 (2008)].
Abstract It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.
It is known that relative entropy of entanglement for an entangled state $\ensuremath{\rho}$ is defined via its closest separable (or positive partial transpose) state $\ensuremath{\sigma}$. Recently, it has been shown how to find $\ensuremath{\rho}$ provided that $\ensuremath{\sigma}$ is given in a two-qubit system. In this article we study the reverse process, that is, how to find $\ensuremath{\sigma}$ provided that $\ensuremath{\rho}$ is given. It is shown that if $\ensuremath{\rho}$ is of a Bell-diagonal, generalized Vedral-Plenio, or generalized Horodecki state, one can find $\ensuremath{\sigma}$ from a geometrical point of view. This is possible due to the following two facts: (i) the Bloch vectors of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ are identical to each other; (ii) the correlation vector of $\ensuremath{\sigma}$ can be computed from a crossing point between a minimal geometrical object, in which all separable states reside in the presence of Bloch vectors, and a straight line, which connects the point corresponding to the correlation vector of $\ensuremath{\rho}$ and the nearest vertex of the maximal tetrahedron, where all two-qubit states reside. It is shown, however, that these properties are not maintained for the arbitrary two-qubit states.
Any arbitrary state in bipartite quantum system can be transformed into a two-parameter class of states by means of local quantum operation and classical communication.In this paper,the quantum relative entropy of entanglement for this kind of states is calculated,the nearest separable states are found and the analytic expression of the entanglement is obtained.The comparation with two other measures of entanglement,the negativity and the entanglement of formation,is also made.The sequence of three measures from high to low is the negativity,the entanglement of formation and the relative entropy of entanglement.