A quantum entropy space is suggested as the fundamental arena describing the quantum effects. In the quantum regime the entropy is expressed as the superposition of many different Boltzmann entropies that span the space of the entropies before any measure. When a measure is performed the quantum entropy collapses to one component. A suggestive reading of the relational interpretation of quantum mechanics and of Bohm’s quantum potential in terms of the quantum entropy are provided. The space associated with the quantum entropy determines a distortion in the classical space of position, which appears as a Weyl-like gauge potential connected with Fisher information. This Weyl-like gauge potential produces a deformation of the moments which changes the classical action in such a way that Bohm’s quantum potential emerges as consequence of the non classical definition of entropy, in a non-Euclidean information space under the constraint of a minimum condition of Fisher information (Fisher Bohm- entropy). Finally, the possible quantum relativistic extensions of the theory and the connections with the problem of quantum gravity are investigated. The non classical thermodynamic approach to quantum phenomena changes the geometry of the particle phase space. In the light of the representation of gravity in ordinary phase space by torsion in the flat space (Teleparallel gravity), the change of geometry in the phase space introduces quantum phenomena in a natural way. This gives a new force to F. Shojai’s and A. Shojai’s theory where the geometry of space-time is highly coupled with a quantum potential whose origin is not the Schrödinger equation but the non classical entropy of a system of many particles that together change the geometry of the phase space of the positions (entanglement). In this way the non classical thermodynamic changes the classical geodetic as a consequence of the quantum phenomena and quantum and gravity are unified. Quantum affects geometry of multidimensional phase space and gravity changes in any point the torsion in the ordinary four-dimensional Lorenz space-time metric.
In classical mechanics, we have individual particle and invariant density in the phase space. In quantum mechanics, any particle is sensitive in a different way from all other particles, for its position and also to the measure process. Thus, we substitute the classical probability in the phase space with the conditional probability in the network of communicating particles. Any probability and entropy are functions of the phase position conditioned by the position of the other particles. Therefore, for different measures we have different conditional entropies. The space of the entropies is a curved and possible torque multidimensional space where the derivative is the covariant derivative on a manifold of the entropic space. At the zero quantum field, the covariant derivative commutes and Fisher matrix is part of the kinetic terms in the Lagrangian where the derivative is the covariant derivative. With Lagrange minimum condition and the entropic space it is possible to show a connection between entropy space and Bohm potential in quantum mechanics. Entropy multidimensional space includes dependence and entanglement as geometric structure of the entropy. Now we can create a non-zero quantum field approach when the covariant derivative does not commute so we have curvature and torsion. The non-zero quantum field can be the Casimir field of forces. Therefore, Casimir force as gravity in the space-time is modelled by curvature and torsion of the entropic space. Useful connection between dependence and covariant derivatives are obtained by copula (dependence measure) and quantum mechanics.
Abstract We develop a novel framework connecting quantum mechanics with gravitational geometry through a deformation of probability amplitudes. Inspired by Rohit Dhormare’s geometric formulation of probability under gravitational fields, we reinterpret the influence of curvature on probability not within classical settings, but in the domain of quantum amplitudes. We propose a quantum probabilistic manifold where space-time curvature induces topological and metric deformations on Hilbert space projections, affecting interference, measurement outcomes, and entropy. This formulation offers new pathways for understanding quantum gravity, decoherence, and information localization.
We develop an information-geometric framework for describing quantum state-update processes associated with measurement and statistical distinguishability. The approach is based on the quantum relative entropy and the quantum Fisher information metric, which together induce a natural Riemannian geometry on the manifold of quantum states. Using the second-order expansion of relative entropy, we show how the Fisher metric governs the local structure of distinguishability between nearby states and defines a corresponding thermodynamic length. This geometric structure provides an effective description of finite quantum state transitions in terms of fluctuation geometry and information-space distance. The formalism is applied to thermal two-level systems and harmonic oscillator states, illustrating how the Fisher metric encodes susceptibilities, fluctuations, and geometric transition costs. We also discuss the relation between thermodynamic length, dissipation bounds, and optimal paths in state space. Within this framework, wave function collapse is interpreted not as a microscopic dynamical mechanism, but as an effective state-update process that admits a geometric characterization in the manifold of density operators. The resulting perspective unifies concepts from quantum information theory, thermodynamics, and differential geometry within a common operational framework based on statistical distinguishability. Possible connections with quantum speed limits, entanglement geometry, and holographic relations between relative entropy and gravitational dynamics are briefly discussed.
Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean,
α-
z, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter
α, the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter
α even though the original quantities do not. Additionally, I derive and establish basic properties of
α-
z information matrices resulting from the
α-
z Rényi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their
α-
z information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.
The general framework of Entropic Dynamics (ED) is used to construct non-relativistic models of relational Quantum Mechanics from well-known inference principles-probability, entropy and information geometry. Although only partially relational-the absolute structures of simultaneity and Euclidean geometry are still retained-these models provide a useful testing ground for ideas that will prove useful in the context of more realistic relativistic theories. The fact that in ED the positions of particles have definite values, just as in classical mechanics, has allowed us to adapt to the quantum case some intuitions from Barbour and Bertotti's classical framework. Here, however, we propose a new measure of the mismatch between successive states that is adapted to the information metric and the symplectic structures of the quantum phase space. We make explicit that ED is temporally relational and we construct non-relativistic quantum models that are spatially relational with respect to rigid translations and rotations. The ED approach settles the longstanding question of what form the constraints of a classical theory should take after quantization: the quantum constraints that express relationality are to be imposed on expectation values. To highlight the potential impact of these developments, the non-relativistic quantum model is parametrized into a generally covariant form and we show that the ED approach evades the analogue of what in quantum gravity has been called the problem of time.