Difficulties in analytic computation for relative entropy of entanglement

Abstract

It is known that relative entropy of entanglement for entangled state $\rho$ is defined via its closest separable (or positive partial transpose) state $\sigma$. Recently, it has been shown how to find $\rho$ provided that $\sigma$ is given in two-qubit system. In this paper we study on the inverse process, i.e. how to find $\sigma$ provided that $\rho$ is given. It is shown that if $\rho$ is one of Bell-diagonal, generalized Vedral-Plenio and generalized Horodecki states, one can always find $\sigma$ from a geometrical point of view. This is possible due to the following two facts: (i) The Bloch vectors of $\rho$ and $\sigma$ are identical with each other (ii) The qubit-interaction vector of $\sigma$ can be computed from a crossing point between 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 qubit-interaction vector of $\rho$ and the nearest vertex of the maximal tetrahedron, where all two-qubit states reside. It is shown, however, that these nice properties are not maintained for the arbitrary two-qubit states.

22 pages, 5 figures included, Fig.1, Fig. 3(a,b,c,d), Fig. 4(a,b,c,d), Fig. 5(a,b) and Fig. 6(a,b) are not included due to file size problem. You can get them directly from http://qubit.kyungnam.ac.kr V2: will appear in PRA

References in corpus (4)

• Experimental quantum computing without entanglement
• Closed formula for the relative entropy of entanglement
• Relative entropy of entanglement for certain multipartite mixed states
• Some Open Problems in Quantum Information Theory

Cited by in corpus (14)

• Quantum discord and geometry for a class of two-qubit states
• Two-qubit mixed states more entangled than pure states: Comparison of the relative entropy of entanglement for a given nonlocality
• On the quantum discord of general X states
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• Entanglement and Discord of superposition of Greenberger-Horne-Zeilinger states
• Geometric interpretation of the geometric discord
• Building separable approximations for quantum states via neural networks
• The One-Way Information Deficit and Geometry for a Class of Two-qubit States
• One-Way Deficit of Two Qubit X States
• Universal behavior of the geometric entanglement measure of many-qubit W states
• Phase diagram for the one-way quantum deficit of two-qubit X states
• Testing the Monogamy Relations via Rank-2 Mixtures
• REE From EOF
• Relative Entropy of Entanglement for Two-Qubit State with $z$-directional Bloch Vectors