Information-Geometric Aspects of Quantum Statistical-Mechanical Systems

Takemi Nakamura

Amari, Shun-ichi, Nagaoka, Hiroshi

Mathieu Molitor

Let N be the space of Gaussian distribution functions over ℝ, regarded as a 2-dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper, we show that the tangent bundle of N, endowed with its natural Kähler structure, is the Siegel-Jacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, and relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler structure of the complex projective space. This paper is a continuation of our previous work [M. Molitor, “Remarks on the statistical origin of the geometrical formulation of quantum mechanics,” Int. J. Geom. Methods Mod. Phys. 9(3), 1220001, 9 (2012); M. Molitor, “Information geometry and the hydrodynamical formulation of quantum mechanics,” e-print arXiv (2012); M. Molitor, “Exponential families, Kähler geometry and quantum mechanics,” J. Geom. Phys. 70, 54–80 (2013)], where we studied the quantum formalism from a geometric and information-theoretical point of view.

Dėnes Petz

Florio M. Ciaglia, Fabio Di Cosmo, Domenico Felice +3 more

The purpose of this paper is to exploit the geometric structure of quantum mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon’s entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.

Jincheng Zhang

Quantum Entanglement Geometric Framework explores the intersection of quantum mechanics and geometry through the application of entanglement. This framework posits that entanglement profoundly influences geometric properties, offering a novel approach to understanding quantum systems and their computational processes. The core mechanism involves constructing a quantum geometric model using entanglement, simulating geometric aspects and investigating the impact of entanglement on these properties. This research aims to identify potential applications of this framework in fields such as quantum computing and information processing. The paper presents a theoretical foundation and preliminary results demonstrating the potential of this approach.