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

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, 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 the entanglement-generating congruent transformation depends upon the system and the symplectic structure of phase-space. To make our study relevant to physical systems, we choose Bopp's shift in phase space as an example of GL(2n, R), so that the results can be interpreted in terms of noncommutative (NC) phase-space deformation. We provide a quantitative estimation for the dependence of symplectic eigenvalues on the deformation parameters and explain the induced entanglement through phase-space deformation. The coexistence of FR-isometry with deformation-dependent entanglement demonstrates that statistical distinguishability and quantum correlations constitute complementary aspects of the geometry of Gaussian quantum systems. With the help of toy models of oscillators in NC-space, we illustrate our results quantitatively.