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

TL;DR

The paper investigates how noncommutative (NC) deformation of phase space alters entanglement properties of bipartite Gaussian states by employing an information-geometric approach. Gaussian states are parametrized by covariance matrices and endowed with the Fisher-Rao metric, with RSUP $\Sigma+\frac{i}{2}\Omega \ge 0$ and PPT-based separability constraints defining quantum, separable, and entangled regions in NC space. A regularized volume measure is introduced to quantify the relative abundance of quantum versus separable states, and these volumes are computed in a toy eight-dimensional NC model, showing that entangled-state volume grows with NC parameters $\theta$ and $\eta$. This framework provides a quantitative handle on NC-induced entanglement and suggests potential experimental probes in systems related to Landau levels or low-dimensional condensed-matter platforms.

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-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.

Figures (4)

• Figure 1: Measure of states for which $\Sigma>0$, with respect to scale factor $\kappa$. Here we vary $\kappa$ from $1/2$ to $4$.
• Figure 2: Measure of allowed states with respect to $\theta \in (0.1,1)$. The measure of the entangled state increases with $\theta$, as desired since the entanglement is induced with the deformation of phase-space. Here, we take the scale factor $\kappa \to 4$.
• Figure 3: The ratio of the measure of entangled states and separable states with respect to NC parameters $\theta$. Here we take scale factor $\kappa \to 4$. We vary $\theta$ from $0.1$ to $1$. The measure of the entangled state increases with the NC parameter since the entanglement is due to deformation in phase-space.
• Figure 4: The ratio of the measure of entangled states and separable states with respect to NC parameter $\eta$. Here we take scale factor $\kappa \to 4$. We vary $\eta$from $0.1$ to $1$. The measure of the entangled state increases with the NC parameter, as expected.

Theorems & Definitions (3)

• Theorem 1: Symplectic spectrum
• Theorem 2: PPT Condition
• Theorem 3: Separability Condition for Gaussian system in NC space