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arXiv:1410.8752 · Journal of Physics Conference Series

Entanglement and separability in the noncommutative phase-space scenario

Alex E. Bernardini, Catarina Bastos, Orfeu Bertolami A, Nuno Costa Dias, João Nuno Prata

Published Jul 3, 2015• 3 citations
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Abstract

Quantumness and separability criteria for continuous variable systems are discussed for the case of a noncommutative (NC) phase-space. In particular, the quantum nature and the entanglement configuration of NC two-mode Gaussian states are examined. Two families of covariance matrices describing standard quantum mechanics (QM) separable states are deformed into a NC QM configuration and then investigated through the positive partial transpose criterium for identifying quantum entanglement. It is shown that the entanglement of Gaussian states may be exclusively induced by switching on the NC deformation. Extensions of some preliminary results are presented.

DOI
10.1088/1742-6596/626/1/012046

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Quantumness and separability criteria for continuous variable systems are discussed for the case of a noncommutative (NC) phase-space. In particular, the quantum nature and the entanglement configuration of NC two-mode Gaussian states are examined. Two families of covariance matrices describing standard quantum mechanics (QM) separable states are deformed into a NC QM configuration and then investigated through the positive partial transpose criterium for identifying quantum entanglement. It is shown that the entanglement of Gaussian states may be exclusively induced by switching on the NC deformation. Extensions of some preliminary results are presented.
Most quantum gravity theories propose that the fundamental concept of space-time is mostly compatible with quantum theory in noncommutative (NC) space. In the present paper, we revisit the notion of entanglement induced by NC deformations of phase space. The positive partial transpose (PPT) criterion for separability of bipartite Gaussian states is extended to a general class of Bopp's shift. In particular, we have considered both the position-position and momentum-momentum noncommutativity, with deformation parameters θθθ and ηηη, respectively. It turns out that θθθ and ηηη induce the entanglement. We have directly applied the formalism for an anisotropic two-dimensional harmonic oscillator. Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space. It turns out that the bipartite Gaussian state is almost always entangled in deformed space. To implement the theoretical idea, we provide an outline for a gedankenexperiment to identify the signature of phase-space noncommutativity, i.e., quantum gravity. In particular, the gedankenexperiment is devised to test the separability of supposedly separable Gaussian states in the usual commutative space, through the covariance matrix, which is constructed via measured output photocurrents after interaction of input Gaussian states and reference states. If the experiment shows that the supposedly separable states are actually entangled, then the entanglement is created through the intermediate background noncommutative space, which is a signature of the quantum nature of gravity.
We explore the effect of two-dimensional position-space noncommutativity on the bipartite entanglement of continuous-variable systems. We first extend the standard symplectic framework for studying entanglement of Gaussian states of commutative systems to the case of noncommutative systems residing in two dimensions. Using the positive partial transpose criterion for separability of bipartite states, we derive a condition on the separability of a noncommutative system that is dependent on the noncommutative parameter $\ensuremath{\theta}$. We then consider the specific example of a bipartite Gaussian state and show the quantitative reduction in entanglement originating from noncommutative dynamics. We show that such a reduction in entanglement for a noncommutative system arising from the modification of the variances of the phase-space variables (uncertainty relations) is clearly manifested between two particles that are separated by small distances.
In this paper, we revisit the notion of quantum entanglement induced by the deformation of phase-space through noncommutative space (NC) parameters. The geometric structure of the state space for Gaussian states in NC-space is illustrated through information geometry approach. We parametrize the phase-space distributions by their covariances and utilize the Fisher-Rao metric to construct the statistical manifold associated with quantum states. We describe the notion of the Robertson-Scrödinger uncertainty principle (RSUP) and positive partial transpose (PPT) conditions for allowed quantum states and separable states, respectively, for NC-space. RSUP and PPT provide the restrictions on all allowed states and separable states, respectively. This enables us to estimate the relative volumes of set of separable states and entangled states. Numerical estimations are provided for a toy model of a bipartite Gaussian state. We restrict our study to such bipartite Gaussian states, for which the entanglement is induced by the noncommutative phase-space parameters.
In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter. We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing. We explore the experimental viability of our result through the computation of extractable work for the current situation.
We consider two classes of non-Gaussian entangled states generated from the product of number states by the action of the beam splitter or the two-mode squeezer. It is shown that, for many of these states, the covariance matrix is compatible with the covariance matrix of separable Gaussian states and their separability cannot be verified by the measurements of the first and second moments of canonical variables. We identify a couple of continuous-variable entanglement criteria with higher-order moments to verify these non-Gaussian entanglements.