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International Journal of Quantum Information

Relative entropy of entanglement of two-qubit Ux-invariant states

Zhen Wang, Zhi‐Xi Wang

Published Dec 1, 2015• 0 citations
Publisher

Abstract

It is strictly proved that a two-qubit [Formula: see text]-invariant state reaches its relative entropy of entanglement (REE) by the separable state having the same matrix structure. We also formulate three quadratic equations for the corresponding closest separable state (CSS) of [Formula: see text]-invariant states by their symmetric property. Thus, the CSS of [Formula: see text]-invariant state can be provided. Furthermore, to illustrate our result we consider two concrete examples.

DOI
10.1142/s0219749915500586

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It is strictly proved that a two-qubit [Formula: see text]-invariant state reaches its relative entropy of entanglement (REE) by the separable state having the same matrix structure. We also formulate three quadratic equations for the corresponding closest separable state (CSS) of [Formula: see text]-invariant states by their symmetric property. Thus, the CSS of [Formula: see text]-invariant state can be provided. Furthermore, to illustrate our result we consider two concrete examples.
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)].
We calculate the relative entropy between two quantum Gaussian states with given correlation matrices by demonstrating a practical method of transforming one of the correlation matrices into an exponential quadratic operator matrix. We show that the closest Gaussian separable state achieving the Gaussian relative entropy of entanglement is at the border between separable and inseparable Gaussian state sets. For a two-mode Gaussian state, the calculation of the Gaussian relative entropy of entanglement is greatly simplified by deducing a matrix with ten undetermined parameters to one with only three. The two-mode Gaussian states are classified into four types. Numerical calculations strongly suggest that the Gaussian relative entropy of entanglement for each type is realized by a Gaussian separable state within the same type. For a symmetric Gaussian state it is strictly proven that the Gaussian relative entropy of entanglement is achieved by a symmetric Gaussian separable state.
We prove conjectures on the relative entropy of entanglement (REE) for two families of multipartite qubit states. Thus, analytical expressions of the REE for these families of states can be given. The first family of states is composed of mixtures of some permutation-invariant multiqubit states. The results generalized to multiqudit states are also shown to hold. The second family of states contains D\"ur's bound entangled states. Along the way, we discuss the relation of the REE to two other measures: robustness of entanglement and the geometric measure of entanglement, slightly extending previous results.