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arXiv:quant-ph/0304019 · Journal of Physics A Mathematical and General

Optimal Lewenstein–Sanpera decomposition for some bipartite systems

Seyed Javad Akhtarshenas, M. A. Jafarizadeh

Published Feb 11, 2004• 10 citations
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Abstract

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.

DOI
10.1088/0305-4470/37/8/008

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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.
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.
Lewenstein-Sanpera Decomposition for 2⊗22\otimes 22⊗2 Systems
As it is well known, every bipartite 2⊗22\otimes 22⊗2 density matrix can be obtained from Bell decomposable states via local quantum operations and classical communications (LQCC). Using this fact, the Lewenstein-Sanpera decomposition of an arbitrary bipartite 2⊗22\otimes 22⊗2 density matrix has been obtained through LQCC action upon Lewenstein-Sanpera decomposition of Bell decomposable states of 2⊗22\otimes 22⊗2 quantum systems, where the product states introduced by Wootters in [W. K. Wootters, Phys. Rev. Lett. {\bf 80} 2245 (1998)] form the best separable approximation ensemble for Bell decomposable states. It is shown that in these systems the average concurrence of the Lewenstein-Sanpera decomposition is equal to the concurrence of these states.
We investigate optimal separable approximations (decompositions) of states ϱ of bipartite quantum systems A and B of arbitrary dimensions M × N following the lines of Lewenstein and Sanpera. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two-qubit systems ( M = N = 2) the best separable approximation has the form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove for arbitrary M and N that the best separable approximation corresponds to a mixture of separable and entangled states, both of which are unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures for constructing such decompositions.
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.
We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.