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arXiv:1701.03782 · Quantum Information and Computation

Candidates for universal measures of multipartite entanglement

Samuel R. Hedemann

Published May 1, 2018• 5 citations
arXivHTMLPDF

Abstract

We propose and examine several candidates for universal multipartite entanglement measures. The most promising candidate for applications needing entanglement in the full Hilbert space is the ent-concurrence, which detects all entanglement correlations while distinguishing between different types of distinctly multipartite entanglement, and simplifies to the concurrence for two-qubit mixed states. For applications where subsystems need internal entanglement, we develop the absolute ent-concurrence which detects the entanglement in the reduced states as well as the full state.

DOI
10.26421/qic18.5-6-3

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We propose and examine several candidates for universal multipartite entanglement measures. The most promising candidate for applications needing entanglement in the full Hilbert space is the ent-concurrence, which detects all entanglement correlations while distinguishing between different types of distinctly multipartite entanglement, and simplifies to the concurrence for two-qubit mixed states. For applications where subsystems need internal entanglement, we develop the absolute ent-concurrence which detects the entanglement in the reduced states as well as the full state.
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The problems of genuine multipartite entanglement detection and classification are challenging. We show that a multipartite quantum state is genuine multipartite entangled if the multipartite concurrence is larger than certain quantities given by the number and the dimension of the subsystems. This result also provides a classification of various genuine multipartite entanglements. We present a lower bound of the multipartite concurrence in terms of bipartite concurrences. While various operational approaches are available for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and classify genuine multipartite entanglement. As applications, genuine multipartite entanglement of tripartite systems is analyzed in detail.
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 can not 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 towards this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. Kokenyesi, 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 turns also out that, physically, this structure is related to the LOCC convertibility: if a class can completely be mapped into another one by the use of LOCC, then it 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.
We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons, and fermions. In each case properties of an entanglement measure are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entanglement measure reduces to the well-known many-particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a nontrivial lower bound of thus obtained generalized concurrence.