Warped Riemannian metrics for manifolds of two-mode entangled Gaussian states under beam splitting and squeezing unitary operations

We investigate the geometrical features of the Riemannian manifolds associated with two-mode mixed thermal states and squeezed thermal states. These two-mode Gaussian states of light are significant from both experimental and theoretical perspectives. The quantum correlated states are mapped to points of a four-dimensional manifold whose variables are the mean values of the photons of the incident modes and the two remaining variables consist of the parameters specifying the splitting or the squeezing process. For uncorrelated quantum Gaussian states, the manifold becomes a two-dimensional flat manifold. We have employed the Hilbert–Schmidt distance to determine the tensor metrics and subsequently the scalar curvature which is surprisingly identical for both manifolds and is independent of the coupling parameters characterizing the [Formula: see text] beam splitter and the [Formula: see text] squeezing unitary operations. Another important result emerges from this geometrical description of quantum correlated Gaussian states. The warped Riemannian structure of the corresponding tensor metrics reflects the presence of quantum correlations between the two Gaussian modes. These correlations are geometrically encoded in the warping function, which appears in the metric tensor. This warping function offers a geometric interpretation of how quantum correlations are generated between two Gaussian modes through beam-splitting and squeezing operations.