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arXiv:1607.00164 · Quantum Information Processing

Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange’s identity and wedge product

Vineeth S. Bhaskara, Prasanta K. Panigrahi

Published Mar 20, 2017• 53 citations
arXivHTMLPDF
DOI
10.1007/s11128-017-1568-0

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Concurrence, introduced by 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 vanishing for all separable states. We present an extension of concurrence to general continuous variable pure states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara et al. [Quantum Inf. Process. 16, 118 (2017)] for pure discrete variable systems in arbitrary dimensions. A family of faithful entanglement measures, of which concurrence is a member, is constructed that admit necessary and sufficient conditions for separability across arbitrary bipartitions, which is shown as a particular invariance with connections to the partial transpose, uncovering an inherent geometry of entanglement. This framework may be useful for the further extensions to mixed states and entanglement in quantum field theories.
Concurrence provided an important measure of entanglement for general pair of qubits, which is strictly positive for entangled states and vanishing all for separable states. We present an extension of concurrence to multiparticle pure states using the Lagrange’s identity and wedge product representation, which coincides with ‘I-concurrence’ by extending the spin-flip operator to a so-called universal inverter superoperator. This framework may be useful for the extension to mixed and continuous variable systems.
Abstract The wedge product of post-measurement vectors of a two-qubit state gives rise to a parallelogram, whose ‘area’ has been shown to be equivalent to the generalized I-concurrence measure of entanglement. In multi-qudit systems, the wedge product of post-measurement vectors naturally leads to a higher dimensional parallelepiped which yields a modified faithful entanglement measure. Our new measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We present a complete analysis of the bipartite qutrit case considering all possible geometric structures where three entanglement classes of pure bipartite qutrit states can be identified with different geometries of post-measurement vectors: three planar vectors; three mutually orthogonal vectors; and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for the characterization and quantification of entanglement in higher dimensional systems.