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Entanglement Measure and Geometric Approach to a Seperability Problem

C. Bangarusai, Supervisor Prof. Prasanta K. Panigrahi

Published May 1, 2016• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a ``universal inverter,'' which acts on quantum systems of arbitrary dimension, and we introduce the corresponding generalized concurrence for joint pure states of ${D}_{1}\ifmmode\times\else\texttimes\fi{}{D}_{2}$ bipartite quantum systems. We call this generalized concurrence the I concurrence to emphasize its relation to the universal inverter. The universal inverter, which is a positive, but not completely positive superoperator, is closely related to the completely positive universal-NOT superoperator, the quantum analogue of a classical NOT gate. We present a physical realization of the universal-NOT superoperator.