Information geometry and entanglement under phase-space deformation through nonsymplectic congruence transformation

Abstract: The Fisher-Rao (FR) information matrix is a central object in multiparameter quantum estimation theory. The geometry of a quantum state can be envisaged through the Riemannian manifold generated by the FR-metric corresponding to the quantum state. Interestingly, any congruence transformation $GL(2n,\mathbb{R})$ in phase-space leaves the FR-distance for Gaussian states invariant. In the present paper, we investigate whether this isometry affects the entanglement in the bipartite system. It turns out that, even a simple choice of a congruence transformation, induces the entanglement in a bipartite Gaussian system. To make our study relevant to physical systems, we choose Bopp's shift in phase-space as an example of $GL(2n,\mathbb{R})$, so that the results can be interpreted in terms of noncommutative (NC) phase-space deformation. We explicitly provide a quantitative measure of the dependence of separability on NC parameters. General expressions for the metric structure are also provided. The crucial point in the present piece of study is the identification of the connection of phase-space deformation-induced entanglement through a general class of congruence transformation and the identification of a trade-off relationship between the initial correlations and deformation parameters.

An Examination of Information Geometry and Entanglement in Phase-Space Deformation

The paper titled "Information Geometry and Entanglement under Phase-Space Deformation through Nonsymplectic Congruence Transformation" delves into the subtleties of quantum information geometry and the impact of phase-space deformation on Gaussian entanglement. The authors utilize Gaussian states due to their prevalence in continuous variable quantum systems and their well-defined behavior in phase-space representation.

Summary of Contributions

1. Fisher-Rao Information Matrix:
   The Fisher-Rao (FR) information matrix forms the backbone of the study, offering a metric to analyze quantum state geometries. In this context, Gaussian states play a critical role due to their intrinsic simplicity, characterized by a Gaussian-type function in phase-space and covariance, effectively described through linear symplectic transformations.

2. Congruence Transformations in Phase-Space:
   The authors explore phase-space transformations through generalized congruence operations in the group ( GL(2n,\mathbb{R}) ). Such transformations maintain the FR-distance invariant for Gaussian states, presenting interesting implications for quantum entanglement. Specifically, even simple congruence transformations can induce entanglement in originally separable bipartite Gaussian systems.

3. Noncommutative Space:
   Utilizing Bopp’s shift as a vehicle for noncommutative (NC) phase-space deformation, the authors highlight the physical interpretation of such transformations, connecting them to phenomena observable in quantum mechanics at small scales. The study suggests that NC parameters significantly contribute to separability changes, impacting the Gaussian state's entanglement properties.

4. Measurement and Analysis:
   Quantitative measures are provided to assess these changes, leveraging symmetric pure Gaussian states and employing positive partial transpose (PPT) separability criteria. The authors systematically detail the notion of curvature in state manifolds and contrast separable state measures against entangled counterparts by observing variations in symplectic eigenvalues.

Implications and Future Directions

The work addresses the theoretical and practical implications of phase-space deformations. In scenarios where quantum gravity is relevant, Gaussian states modified by NC parameters could provide insight into the dynamics at high-energy scales. Furthermore, the paper's results can inform experimental setups, especially in contexts like the quantum Hall effect, where noncommutative properties of electron guiding centers mimic the studied deformations.

Future research could expand on the implications of different congruence transformations, exploring broader classes of quantum states beyond bipartite Gaussian arrangements. By doing so, the foundations laid in this paper can leverage algorithms in quantum information processing to implement such transformations practically, and perhaps, extend applicability across various dimensions of quantum physics and computational models.

In conclusion, the intricate relationship between phase-space geometry and quantum entanglement that this paper highlights promisingly points to new avenues in understanding quantum state behavior under deformation and transformation.