Difficulties in analytic computation for relative entropy of entanglement

Abstract

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.