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arXiv:quant-ph/0606075 · International Journal of Quantum Information

BEST SEPARABLE APPROXIMATION WITH SEMI-DEFINITE PROGRAMMING METHOD

M. A. Jafarizadeh, Mehdi Mirzaee, M. Rezaee

Published Dec 1, 2004• 11 citations
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Abstract

The present methods for obtaining the optimal Lewenestein–Sanpera decomposition of a mixed state are difficult to handle analytically. We provide a simple analytical expression for the optimal Lewenstein–Sanpera decomposition by using semi-definite programming. In particular, we obtain the optimal Lewenstein–Sanpera decomposition for some examples such as: the Bell decomposable state, the iso-concurrence state, the generic two-qubit state in the Wootters basis, the 2⊗3 Bell decomposable state, the d⊗d Werner and isotropic states, a one parameter 3⊗3 state, and finally a multi-partite isotropic state.

DOI
10.1142/s0219749904000924

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The present methods for obtaining the optimal Lewenestein–Sanpera decomposition of a mixed state are difficult to handle analytically. We provide a simple analytical expression for the optimal Lewenstein–Sanpera decomposition by using semi-definite programming. In particular, we obtain the optimal Lewenstein–Sanpera decomposition for some examples such as: the Bell decomposable state, the iso-concurrence state, the generic two-qubit state in the Wootters basis, the 2⊗3 Bell decomposable state, the d⊗d Werner and isotropic states, a one parameter 3⊗3 state, and finally a multi-partite isotropic state.
It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of a product set) on the separable–entangled boundary, the optimal Lewenstein–Sanpera (LS) decomposition (with respect to an arbitrary separable set) can be obtained via a direct optimization procedure for a generic entangled density matrix. On the basis of this, we obtain the optimal LS decomposition for some bipartite systems such as 2 ⊗ 2 and 2 ⊗ 3 Bell decomposable (BD) states, a generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one-parameter and three-parameter local operations and classical communications (LOCC), d ⊗ d Werner and isotropic states and a one-parameter 3 ⊗ 3 state. We also obtain the optimal decomposition for multi-partite isotropic states. It is shown that in all 2 ⊗ 2 systems considered here the average concurrence of the decomposition is equal to the concurrence. We also show that for some 2 ⊗ 3 Bell decomposable states, the average concurrence of the decomposition is equal to the lower bound of the concurrence of the state presented recently in Lozinski et al (2003 Preprint quant-ph/0302144), so an exact expression for concurrence of these states is obtained. It is also shown that for a d ⊗ d isotropic state where decomposition leads to a separable and an entangled pure state, the average I-concurrence of the decomposition is equal to the I-concurrence of the state.
An analytical expression for optimal Lewenstein-Sanpera (L-S) decomposition of a generic two qubit density matrix is given. By evaluating the L-S decomposition of Bell decomposable states, the optimal decomposition for arbitrary full rank state of two qubit system is obtained via local quantum operations and classical communications (LQCC). In Bell decomposable case the separable state optimizing L-S decomposition, minimize the von Neumann relative entropy as a measure of entanglement. The L-S decomposition for a generic two-qubit density matrix is only obtained by using Wootters's basis. It is shown that the average concurrence of the decomposition is equal to the concurrence of the state. It is also shown that all the entanglement content of the state is concentrated in the Wootters's state |x_1> associated with the largest eigenvalue \lambda_1 of the Hermitian matrix \sqrt{\sqrt{rho}\tilde{rho}\sqrt{rho}} . It is shown that a given density matrix rho with corresponding set of positive numbers \lambda_i and Wootters's basis can transforms under SO(4,c) into a generic 2x2 matrix with the same set of positive numbers but with new Wootters's basis, where the local unitary transformations correspond to SO(4,r) transformations, hence, \rho can be represented as coset space SO(4,c)/SO(4,r) together with positive numbers lambda_i.
We use the language of semidefinite programming and duality to derive necessary and sufficient conditions for the optimal Lewenstein-Sanpera decomposition (LSD) of two-qubit states. We first provide a simple and natural derivation of the Wellens-Ku\ifmmode \acute{s}\else \'{s}\fi{} equations for full-rank states. Then, we obtain a set of necessary and sufficient conditions for the optimal decomposition of rank-3 states. This closes the gap between the full-rank case, where optimality conditions are given by the Wellens-Ku\ifmmode \acute{s}\else \'{s}\fi{} equations, and the rank-2 case, where the optimal decomposition is analytically known. We also give an analytic expression for the optimal LSD of a special class of rank-3 states. Finally, our formulation ensures efficient numerical procedures to return the optimal LSD for any arbitrary two-qubit state.
We propose a simple geometrical approach for finding the Lewenstein-Sanpera decomposition of Bell decomposable states of 2 otimes 2 quantum systems. We show that in these systems, the weight of the pure entangled part in the decomposition is equal to the concurrence of the state. It is also shown that the optimized separable part of L-S decomposition minimizes the von Neumann relative entropy. We also obtain the decomposition for a class of mixed states by using some LQCC actions. It is also shown that for these states the average concurrence of L-S decomposition is equal to their concurrence.
In general, the calculation of robustness of entanglement for the mixed entangled quantum states is rather difficult to handle analytically. Using the convex semi-definite programming method, we claculate exactly the robustness of entanglement of some mixed entangled quantum states such as the generic two-qubit state in the Wootters basis, 2 ⊗ 2 Bell decomposable (BD) states, iso-concurrence decomposable states, 2 ⊗ 3 Bell decomposable states, d ⊗ d Werner and isotropic states, a one parameter 3 ⊗ 3 state and a multi-partite isotropic state. The results are in agreement with those of 2 ⊗ 2 density matrices and have already been calculated in Refs. 1 and 2. Also, an analytic expression is given for separable states that wipe out all the entanglements. It is further shown that they are on the boundary of separable states, as pointed out in Ref. 3.
We investigate the problem of finding the optimal convex decomposition of a bipartite quantum state into a separable part and a positive remainder, in which the weight of the separable part is maximal. This weight is naturally identified with the degree of separability of the state. In a recent work, the problem was solved for two-qubit states using semidefinite programming. In this paper, we describe a procedure to obtain the optimal decomposition of a bipartite state of any finite dimension via a sequence of semidefinite relaxations. The sequence of decompositions thus obtained is shown to converge to the optimal one. This provides a systematic method to determine the so-called optimal Lewenstein-Sanpera decomposition of any bipartite state. Numerical results are provided to illustrate this procedure, and the special case of rank-2 states is also discussed.