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arXiv:0709.4642 · Physical Review A

Multipartite entanglement in four-qubit cluster-class states

Yan-Kui Bai, Z. D. Wang

Published Mar 7, 2008• 45 citations
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Abstract

Based on quantitative complementarity relations (QCRs), we analyze the multipartite correlations in four-qubit cluster-class states. It is proven analytically that the average multipartite correlation ${E}_{ms}$ is entanglement monotone. Moreover, it is also shown that the mixed three-tangle is a correlation measure compatible with the QCRs in this kind of quantum state. More arrestingly, with the aid of the QCRs, a set of hierarchy entanglement measures is obtained rigorously in the present system.

DOI
10.1103/physreva.77.032313

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Based on quantitative complementarity relations (QCRs), we analyze the multipartite correlations in four-qubit cluster-class states. It is proven analytically that the average multipartite correlation Ems{E}_{ms}Ems​ is entanglement monotone. Moreover, it is also shown that the mixed three-tangle is a correlation measure compatible with the QCRs in this kind of quantum state. More arrestingly, with the aid of the QCRs, a set of hierarchy entanglement measures is obtained rigorously in the present system.
Based on quantum complementary relations (QCRs) and a purification scenario, we analyze a class of NNN-qubit mixed states that are entangled but do not have two-, and genuine three-, four-,\dots{},NNN-qubit entanglements. It is shown that entanglement (one-tangle or negativity) in these mixed states is closely related to the QCR entanglement of their purified states. In particular, it is elaborated that when the mixed state does not have multipartite tangles (two- and higher tangles), its entanglement is actually a kind of genuine multipartite QCR entanglement between the system and its environment.
Based on the quantitative complementarity relations, we analyze thoroughly the properties of multipartite quantum correlation and entanglement in four-qubit pure states. It is found that, unlike the three-qubit case, the single residual correlation and the genuine correlations of three and four qubits are unable to quantify entanglement appropriately. More interestingly, from our qualitative and numerical analysis, it is conjectured that the sum of all residual correlations is a good quantity for characterizing the multipartite entanglement in the system.
In this work we consider the permutational properties of multipartite entanglement monotones. Based on the fact that genuine multipartite entanglement is a property of the entire multiqubit system, we argue that ideal definitions for its characterizing quantities must be permutation invariant. Using this criterion, we examine the three four-qubit entanglement monotones introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005)]. By expressing them in terms of quantities whose permutational properties can be easily derived, we find that one of these monotones is not permutation invariant. We propose a permutation-invariant entanglement monotone to replace it, and show that our new monotone properly measures the genuine four-qubit entanglement in four-qubit cluster-class states. Our results provide some useful insights in understanding multipartite entanglement.
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement measures such as one-tangle, two-tangles and three-tangles of an N-qubit pure state.
The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. K\"ok\'enyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the local operations and classical communication convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some ``unified'' notion of entanglement.
Local or nonlocal character of quantum states can be quantified and is subject to various bounds that can be formulated as complementarity relations. Here, we investigate the local vs nonlocal character of pure three-qubit states by a four-way interferometer. The complete entanglement in the system can be measured as the entanglement of a specific qubit with the subsystem consisting of the other two qubits. The quantitative complementarity relations are verified experimentally in an NMR quantum information processor.