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Journal of Physics A Mathematical and Theoretical

Robust multipartite thermal entanglement

Chang Chi Kwong, Ye Yeo

Published Jun 26, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/41/29/295301

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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.
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.
The entanglement of superpositions [Linden et al., Phys. Rev. Lett. 97, 100502 (2006)]is generalized to the multipartite scenario: an upper bound to the multipartite entanglement of a superposition is given in terms of the entanglement of the superposed states and the superposition coefficients. This bound is proven to be tight for a class of states composed of an arbitrary number of qubits. We also extend the result to a large family of quantifiers, which includes the negativity, the robustness of entanglement, and the best separable approximation measure.
We derive the entangled thermal mixed states by using the formalism of phase states for a finite-dimensional algebra of a multi-qubit system in contact with an independent thermal environment of absolute temperature [Formula: see text]. Thermal entangled states describing the multi-qubit system in equilibrium with the thermal bath are a special kind of mixed states that exhibit genuine multipartite correlation. We define the unitary phase operators for a multipartite system of non-interacting qubits. Entangled density matrices are derived for qubits interacting via an Hermitian Hamiltonian of Heisenberg type [Formula: see text]. By assuming that the noisy interaction of the entangled qubit ensemble with the bath is governed by a local Hamiltonian [Formula: see text], we show that the entangled phase states can be decohered. When the multi-qubits entangled system reaches the equilibrium with the thermal bath, the decohered mixed states are identified with entangled thermal states. The thermal mixed states for bipartite and multipartite systems are explicitly expressed and their bipartite entanglement properties are investigated.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric N qubit states and provide the analytical expression for their Best Separable Approximation (BSA [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled one with maximal weight of the separable one.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric NNN-qubit states and provide the analytical expression for their best separable approximation (BSA, [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled part, with maximal weight of the separable part.
In this paper, we generalize the residual entanglement to the case of multipartite states in arbitrary dimensions by making use of a different method. Through the introduction of a special entanglement measure, the residual entanglement of mixed states takes on a form that is more elegant than that in Ref. [7] used by Coffman et al. [Phys. Rev. A 61, 052306 (2000)]. The result obtained in this paper is different from the previous one given in Ref. [8] by Wong and Christensen [Phys. Rev. A 63, 044301 (2000)]. Several examples demonstrate that our present result is a good measurement of the multipartite entanglement. Furthermore, the original residual entanglement is a special case of our result.