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.
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.
We investigate two special classes of two-mode Gaussian states of light that are important from both the experimental and theoretical points of view: the mode-mixed thermal states and the squeezed thermal ones. Aiming to a parallel study, we write the Uhlmann fidelity between pairs of states belonging to each class in terms of their defining parameters. The quantum Fisher information matrices on the corresponding four-dimensional manifolds are diagonal and allow insightful parameter estimation. The scalar curvatures of the Bures metric on both Riemannian manifolds of special two-mode Gaussian states are evaluated and discussed. They are functions of two variables, namely, the mean numbers of photons in the incident thermal modes. Our comparative analysis opens the door to further investigation of the interplay between geometry and statistics for Gaussian states produced in simple optical devices.
The Hilbert-Schmidt measure is a useful geometric tool used to investigate the properties on the manifold of mixed Gaussian states in multimode continuous variable quantum systems. In this work, we study the relationship between the scalar curvature and quantum entanglement across causally disconnected regions (open charts) in de Sitter spacetime. We first consider a two-mode squeezed state shared between Alice and Bob, where the Gaussian channel models the effects of spacetime curvature. We employ logarithmic negativity as a witness for quantum entanglement. We study the impact of the mass, parameter p and squeezing parameter on the scalar curvature and entanglement. We find an interesting interplay between scalar curvature and entanglement for certain values of the mass and parameter p . By varying p , we thus explore how information geometry and entanglement are influenced by both ultraviolet and infrared modes. Moreover, the relationship between scalar curvature and entanglement is discussed under the influence of various parameters characterizing the geometric system under consideration.
This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.
The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.
Abstract In this paper, we investigate the creation of quantum correlations such as entanglement, the Gaussian quantum discord and quantum steering in a quantum system under consideration via the linear beam splitter with squeezed thermal and a single-mode Gaussian states at the inputs. This quantum system generates bright entangled output two-mode light. Specifically, the quantum entanglement can be enhanced through increasing the purity of the Gaussian state and the squeezing parameter at the input states. The quantum correlations reveal similar characteristics for each case. That means when we increase the non-classicality, purity and squeezing parameter, the quantum correlations show enhanced profile. On the other hand, upon increasing the mean thermal photon number at the input, the quantum correlations indicate inverse relation.
Different types of geometric and entropic quantum correlation quantifiers are studied for a system composed of two resonant bosonic modes embedded in a thermal bath. The description of the evolution of the correlation measures is formulated in the framework of the theory of open systems, based on completely positive quantum dynamical semigroups, by using both a geometric and entropic quantification of total nonclassical correlations of Gaussian states. We consider the special case when the initial squeezed thermal state of the system preserves its form in time. We show that time evolution of the measures strongly depends on the parameters characterising the initial state of the system (squeezing parameter and average thermal photon numbers of the two modes) and of the thermal environment (temperature of the thermal bath and dissipation rate). In the limit of large times all the considered measures asymptotically tend to zero value, corresponding to an asymptotic bimodal uncorrelated product state. We make a comparison between the behaviour of the evolution in time of the Gaussian geometric quantum correlations and Gaussian entropic quantum correlations.