Difficulties in analytic computation for relative entropy of entanglement

TL;DR

This paper explores the geometric methods for finding the closest separable state to certain entangled two-qubit states, highlighting the challenges and limitations in the general case.

Contribution

It introduces a geometric approach to determine the closest separable state for specific two-qubit entangled states, extending previous work on the inverse problem.

Findings

Geometric method successfully finds closest separable states for Bell-diagonal, Vedral-Plenio, and Horodecki states.

The approach relies on Bloch vector and qubit-interaction vector properties.

The method does not generalize to arbitrary two-qubit states.

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.