Entropy, Information and Quantum Geometry

Nilanjana Datta

Germano Resconi, Ignazio Francesco Licata, Davide Fiscaletti

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.

Jincheng Zhang

This paper investigates a novel approach to optimizing universal quantum algorithms by leveraging the framework of information geometry. The core idea is to treat quantum algorithms as points embedded within the information geometry of a Hilbert space, specifically tailored to the underlying quantum system. We propose a methodology utilizing gradient descent, guided by the curvature of the information geometry, to identify optimal parameter settings for these algorithms. This approach contrasts with traditional, often empirically-driven, optimization techniques. We formulate the optimization problem as minimizing the quantum relative entropy between the evolved quantum state and the ideal output state, considering the algorithm's Hamiltonian and the geometric constraints imposed by information geometry. The theoretical framework presented here offers a deeper understanding of quantum algorithm optimization and potentially unlocks new avenues for enhancing their efficiency. Key mathematical concepts include the Kullback-Leibler divergence (KL divergence), the quantum relative entropy (also known as the Von Neumann entropy), and the curvature tensor of the information manifold. The primary contribution lies in the integration of these concepts to create a systematic optimization procedure.

Subhash C. Kak

Göran Lindblad

Addisu Alemsefa

This work proposes a theoretical framework in which entropy is treated as a geometric degree of freedom in an extended quantum state space. By introducing a resolution parameter defined as the inverse of von Neumann entropy, an informational metric is constructed on the space of density matrices. Within this framework, a causal structure emerges from a null condition on the metric, defining a light-cone-like boundary for the propagation of quantum correlations. Entanglement propagation is derived and shown to depend on entropy evolution, leading to a dynamical causal structure. In the near-equilibrium limit, the model reproduces a Lieb–Robinson-type bound with finite correlation speed. Connections to holography are established by interpreting entropy as a radial coordinate controlling scale, and an analogue of the Ryu–Takayanagi relation is derived, where entropy corresponds to the length of minimal curves in the informational geometry. Black hole horizons are interpreted as maximal entropy surfaces corresponding to a collapse of distinguishability in quantum state space. The framework suggests that spacetime geometry, causality, and gravitational behavior may emerge from entropy-weighted quantum information structure. This work is an independent theoretical exploration aimed at providing a unified perspective linking quantum information, geometry, and gravity.