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arXiv:1005.4777 · International Journal of Quantum Information

RELATIVE ENTROPY OF ENTANGLEMENT FOR TWO-QUBIT STATE WITH z-DIRECTIONAL BLOCH VECTORS

DaeKil Park

Published Aug 1, 2010• 6 citations
arXivHTMLPDF

Abstract

So far there is no closed formula for relative entropy of entanglement of arbitrary two-qubit states. In this paper we present a method, which guarantees the derivation of the relative entropy of entanglement for most states that have z-directional Bloch vectors. It is shown that the closest separable states for those states also have z-directional Bloch vectors though there are few exceptions.

DOI
10.1142/s0219749910006629

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So far there is no closed formula for relative entropy of entanglement of arbitrary two-qubit states. In this paper we present a method, which guarantees the derivation of the relative entropy of entanglement for most states that have z-directional Bloch vectors. It is shown that the closest separable states for those states also have z-directional Bloch vectors though there are few exceptions.
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.
The long-standing problem of finding a closed formula for the relative entropy of entanglement (REE) for two qubits is addressed. A compact-form solution to the inverse problem, which characterizes an entangled state for a given closest separable state, is obtained. Analysis of the formula for a large class of entangled states strongly suggests that a compact analytical solution of the original problem, which corresponds to finding the closest separable state for a given entangled state, can be given only in some special cases. A few applications of the compact-form formula are given to show additivity of the REE, to relate the REE with the Rains upper bound for distillable entanglement, and to show that a Bell state does not have a unique closest separable state.
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)].
The separable state closest to a given entangled state in the relative entropy measure is called the closest disentangled state. We provide an analytical formula connecting the entangled states and the closest disentangled state in two qubits. Using this formula, when any disentangled state $\ensuremath{\sigma}$ located at the entangle-disentangle boundary is given, entangled states to which $\ensuremath{\sigma}$ is closest can be obtained analytically. Further, this formula naturally defines the direction normal to the boundary surface. The direction is uniquely determined by $\ensuremath{\sigma}$ in almost all cases.