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arXiv:0706.2019 · Physical Review Letters

Operational Interpretation for Global Multipartite Entanglement

Sergio Boixo, Alex Monràs

Published Mar 14, 2008• 84 citations
arXivHTMLPDF

Abstract

We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is directly related to the Meyer-Wallach multipartite entanglement measure. Using channels that depolarize across different partitions, we obtain related multipartite entanglement measures. We show that this measure is the sum of expectation values of local observables on two copies of the state.

DOI
10.1103/physrevlett.100.100503

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We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is directly related to the Meyer-Wallach multipartite entanglement measure. Using channels that depolarize across different partitions, we obtain related multipartite entanglement measures. We show that this measure is the sum of expectation values of local observables on two copies of the state.
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Recently, Meyer and Wallach [D.A. Meyer and N.R. Wallach (2002), J. of Math. Phys., 43, pp. 4273] proposed a measure of multi-qubit entanglement that is a function on pure states. We find that this function can be interpreted as a physical quantity related to the average purity of the constituent qubits and show how it can be observed in an efficient manner without the need for full quantum state tomography. A possible realization is described for measuring the entanglement of a chain of atomic qubits trapped in a 3D optical lattice.
Recently, Meyer and Wallach [Meyer and Wallach (2002), J. of Math. Phys., 43, pp. 4273] proposed a measure of multi-qubit entanglement that is a function on pure states. We find that this function can be interpreted as a physical quantity related to the average purity of the constituent qubits and show how it can be observed in an efficient manner without the need for full quantum state tomography. A possible realization is described for measuring the entanglement of a chain of atomic qubits trapped in a 3D optical lattice.
Abstract The study of multipartite entanglement is much less developed than the bipartite scenario. Recently, a string of results have proposed using tools from topological data analysis to attach topological quantities to multipartite states. However, these quantities are not directly connected to concrete information processing tasks making their interpretations vague. We take the first steps in connecting these abstract topological quantities to operational interpretations of entanglement in two scenarios. The first is we provide a bound on the average distillable entanglement, which we develop as a generalization of the Meyer–Wallach entanglement measure studied by Scott (2004 Phys. Rev. A 69 052330), via the integrated Euler characteristic (IEC) defined by Hamilton and Leditzky. This allows us to connect the distance of an error correcting code to the IEC. The second is we provide a characterization of a class of graph states containing the GHZ state via the birth and death times of the connected components and one-dimensional cycles of the entanglement complex. In other words, the entanglement distance behavior of the first Betti number β 1 ( ε ) allows us to determine if a state is locally equivalent to a GHZ state, potentially providing new verification schemes for highly entangled states.
We introduce a quantum Fisher information based measurement-induced nonlocality (QFI-MIN), which quantifies the maximal statistical distinguishability induced by locally invariant unitary dynamics. The proposed measure inherits desirable properties including positivity, local unitary invariance, monotonicity under local operations, and immunity to the local ancilla problem. Analytical expressions are obtained for pure states, arbitrary two-qubit states, and two-qubit X states, revealing a direct connection with entanglement for pure systems. We further establish clear operational interpretations of QFI-MIN in quantum parameter estimation, local channel discrimination, and correlation-assisted communication. Its behavior under amplitude damping, depolarizing, and generalized amplitude damping channels demonstrates robustness against environmental noise. The proposed framework provides a physically consistent and operationally meaningful quantifier of quantum correlations, linking nonlocality, quantum metrology, and quantum information processing
We investigate the dynamics of global entanglement, the Meyer-Wallach measure, under decoherence, analytically. We study two important classes of multipartite entangled states, the Greenberger-Horne-Zeilinger and the W\mathit{W}W state. We obtain exact results for various models of system-environment interactions (decoherence). Our results shows distinctly different scaling behavior for these initially entangled states indicating a relative robustness of the W\mathit{W}W state, consistent with previous studies.