Information geometry of entangled states induced by noncommutative deformation of phase space

TL;DR

This paper explores how noncommutative phase-space deformations influence quantum entanglement in Gaussian states using information geometry, providing geometric insights and numerical estimates of entangled versus separable states.

Contribution

It introduces a geometric framework for analyzing entanglement induced by noncommutative phase-space parameters in Gaussian states, including criteria and volume estimations.

Findings

RSUP and PPT conditions restrict allowed and separable states

Numerical estimates of entangled versus separable state volumes

Geometric structure of state space characterized by Fisher-Rao metric

Abstract

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-Scr\"{o}dinger 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.