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arXiv:quant-ph/0603002 · Quantum Information and Computation

Quantum entanglement measure based on wedge product

Hoshang Heydari

Published Mar 1, 2006• 6 citations
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Abstract

We construct an entanglement measure that coincides with the generalized concurrence for a general pure bipartite state based on wedge product. Moreover, we construct an entanglement measure for pure multi-qubit states, which are entanglement monotones. Furthermore, we generalize our result on a general pure multipartite state.
DOI
10.26421/qic6.2-2

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The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
Concurrence, introduced by Hill and Wootters [Hill and Wootters, Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is strictly positive for entangled states and vanishes for all separable states. We present an extension of the entanglement measure to general pure continuous variable states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara and Panigrahi [Bhaskara and Panigrahi, Quantum Inf. Process. 16, 118 (2017)] for pure discrete-variable systems in arbitrary dimensions and extending the concept to mixed continuous-variable states. A family of faithful entanglement measures is constructed that admits necessary and sufficient conditions for separability across arbitrary bipartitions presented by Vedral et al. [Vedral, Plenio, Rippin, and Knight, Phys. Rev. Lett. 78, 2275 (1997)]. The computed entanglement measure in the present approach for general Gaussian states, pair-coherent states, and non-Gaussian continuous-variable Bell states matches with known results. We also quantify entanglement of phase-randomized squeezed states and superposition of squeezed states. Our results also simplify several results in quantum entanglement theory.