Optimal Lewenstein?Sanpera decomposition for some bipartite systems

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.

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