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arXiv:quant-ph/0211156

Robustness of entanglement for two qubit density matrix

Seyed Javad Akhtarshenas, M. A. Jafarizadeh

Published Nov 24, 2002• 1 citation
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Abstract

By considering the decomposition of a generic two qubit density matrix presented by Wootters [W. K. Wootters, Phys. Rev. Lett. {\bf 80} 2245 (1998)], the robustness of entanglement for any mixed state of two qubit systems is obtained algebraically. It is shown that the robustness of entanglement is proportional to concurrence and in Bell decomposable density matrices it is equal to the concurrence. We also give an analytic expression for two separable states which wipe out all entanglement of these states. Since thus obtained robustness is function of the norm of the vectors in the decomposition we give an explicit parameterization for the decomposition.

DOI
10.48550/arxiv.quant-ph/0211156

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By considering the decomposition of a generic two qubit density matrix presented by Wootters [W. K. Wootters, Phys. Rev. Lett. {\bf 80} 2245 (1998)], the robustness of entanglement for any mixed state of two qubit systems is obtained algebraically. It is shown that the robustness of entanglement is proportional to concurrence and in Bell decomposable density matrices it is equal to the concurrence. We also give an analytic expression for two separable states which wipe out all entanglement of these states. Since thus obtained robustness is function of the norm of the vectors in the decomposition we give an explicit parameterization for the decomposition.
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.
Adding the maximally mixed state with some weight to the entanglement system leads to disentanglement of the latter. For each predefined entangled state there exists a minimal value of this weight for which the system loses its entanglement properties. These values were proposed to be used as a quantitative measure of entanglement called robustness [G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999)]. Using the concurrence, we propose the derivation of this measure for the system of two-qubit. Namely, for a two-qubit pure state, an exact expression of robustness is obtained. Finally, in the same way, the robustness of special cases of mixed two-qubit states is calculated.
Recently, Mintert et al (MKB) (2005 Phys. Rev. Lett. 95 260502) proposed generalizations of the Wootters concurrence for multipartite quantum systems, which can be evaluated efficiently for arbitrary mixed states. In this paper, we study in detail the origin of the robustness of multipartite thermal entanglement in N -qubit XY models with N = 2, 4 and 6 using the MKB concurrences. For N = 2, we establish the principle for the occurrence of robust bipartite thermal entanglement (Kamta and Starace 2002 Phys. Rev. Lett. 88 107901). We show explicitly that similar robust multipartite thermal entanglement can be found in the N = 4 case, and how this can be explained employing the same principle. It is then deduced that the six-qubit XY model also possesses robust multipartite thermal entanglement like the N = 2 case. More generally, our work spells out the characteristic properties for a given realistic model to have robust multipartite thermal entanglement like that in the two-qubit XY model.
Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.
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.