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arXiv:quant-ph/0603035

Existence Criterion of Genuine Tripartite Entanglement

Chang-shui Yu He-shan Song

Published Mar 6, 2006• 0 citations
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Abstract

In this paper, an intuitive mathematical formulation is provided to generalize the residual entanglement for tripartite systems of qubits (Phys. Rev. A \textbf{61}, 052306 (2000)) to the tripartite systems in higher dimension. The spirit lies in the tensor treatment of tripartite pure states (Phys. Rev. A \textbf{72}, 022333 (2005)). A distinct characteristic of the present generalization is that the formulation for higher dimensional systems is invariant under permutation of the subsystems, hence is employed as a criterion to test the existence of genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the kronecker product approximate technique. As applications, we give the analytic approximation of the criterion for weakly mixed tripartite quantum states and consider the existence of genuine tripartite entanglement of some weakly mixed states.

DOI
10.48550/arxiv.quant-ph/0603035

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In this paper, we present a new approach to study genuine tripartite entanglement existing in (2×2×n)−(2\times 2\times n)-(2×2×n)−dimensional quantum pure states. By utilizing the approach, we introduce a particular quantity to measure genuine tripartite entanglement. The quantity is shown to be an entanglement monotone in 2-dimensional subsystems (semi-monotone) and reaches zero for separable states and (2×2×2)−(2\times 2\times 2)-(2×2×2)−dimensional WWW states, hence is a good criterion to characterize genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the kronecker product approximation technique. As applications, we give the analytic approximation for weakly mixed states, and study the genuine tripartite entanglement of two given weakly mixed states.
In this paper, we present a method to construct full separability criterion for tripartite systems of qubits. The spirit of our approach is that a tripartite pure state can be regarded as a three-order tensor that provides an intuitionistic mathematical formulation for the full separability of pure states. We extend the definition to mixed states and give out the corresponding full separability criterion. As applications, we discuss the separability of several bound entangled states, which shows that our criterion is feasible.
A new formulation called as entanglement measure for simplification, is presented to characterize genuine tripartite entanglement of (2×2×n)−(2\times 2\times n)-(2×2×n)−dimensional quantum pure states. The formulation shows that the genuine tripartite entanglement can be described only on the basis of the local (2×2)−(2\times 2)-(2×2)−dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang (Phys. Rev. A \textbf{71}, 030302(R) (2005)) is reconsidered by employing the entanglement measure. The results show that a discontinuity in the first derivative of the entanglement measure or in the entanglement measure itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously prior to concurrence. Hence, the given entanglement measure may become a new alternate candidate to help study the connection between quantum entanglement and quantum phase transitions.
A new simplified formula is presented to characterize genuine tripartite entanglement of (2 ⊗ 2 ⊗ n )-dimensional quantum pure states. The formula turns out equivalent to that given in ( Quant. Inf. Comp. 7 (7) 584 (2007)), hence it also shows that the genuine tripartite entanglement can be described only on the basis of the local (2 ⊗ 2)-dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang ( Phys. Rev. A 71 030302(R) (2005)) are reconsidered by employing the formula. The results show that a discontinuity in the first derivative of the formula or in the formula itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously different from the concurrence.
In this paper, we investigate the genuine entanglement in tripartite systems based on partial transposition and the norm of correlation tensors of the density matrices. We first derive an analytical sufficient criterion to detect genuine entanglement of tripartite qubit states combining with the partial transposition of the density matrices. Then, we use the norm of correlation tensors to study genuine entanglement for tripartite qudit quantum states and obtain a genuine entanglement criterion by constructing certain matrices. With detailed examples, our results are seen to be able to detect more genuine tripartite entangled states than previous studies.