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arXiv:1007.4544 · Journal of Mathematical Physics

An explicit expression for the relative entropy of entanglement in all dimensions

Shmuel Friedland, Gilad Gour

Published May 1, 2011• 30 citations
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Abstract

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)].

DOI
10.1063/1.3591132

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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 relative entropy of entanglement of a mixed state σ for a bipartite quantum system can be defined as the minimum of the quantum relative entropy over the set of completely disentangled states. Vedral et al. [Phys. Rev. A 57(1998)1619] have recently proposed a numerical method to obtain the relative entropy of entanglement E re for two-qubit systems. This letter shows that the convex programming method can be applied to calculate E re of two-qubit systems analytically, and discusses the conditions under which the method can be adopted.
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.
Abstract It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.
We determine the relative entropy of entanglement of a bipartite mixed state ρABρ_{AB}ρAB​ obtained by tracing out one subsystem of a tripartite Haar-random pure state ∣ψ⟩ABC|ψ\rangle_{ABC}∣ψ⟩ABC​, finding ER(ρAB)=log⁡dAdBmax⁡(dA,dB,dC)+O(1)E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)ER​(ρAB​)=logmax(dA​,dB​,dC​)dA​dB​​+O(1). Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation EF(ρAB)E_F(ρ_{AB})EF​(ρAB​) and the mutual information I(A:B)I(A:B)I(A:B). The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.