Fisher Information and Quantum Origins of the Adiabatic Law

Carlos Alberto Gomez-Uribe

Angelo Plastino, F. Pennini

Dennis G. Collins

The paper observes that consistent lateral motion of probability within a probability density does not affect the Shannon entropy, i.e. that such movement of probability is an adiabatic or entropy‐preserving operation. Next it is observed that if such motion removes “kinks” in a probability density, it decreases the Fisher information. Then it is found that a minimum of Fisher information due to this procedure must result if all kinks are removed by pushing probability laterally toward the origin on the plus side. Finally an algorithm is developed which will effect this operation, and calculate the resulting minimum laterally‐adiabatically reduced Fisher information. Examples are worked out to illustrate the algorithm. Applications may include many processes which occur adiabatically, i.e. without entropy increase. Variational problems are also worked out.

Angelo Plastino, Victoria Vampa

Beginning with Mandelbrot’s insight that Fisher information may admit a thermodynamic interpretation, a growing body of work has connected this information-theoretic measure to fluctuation–dissipation relations, thermodynamic geometry, and phase transitions. Yet, these connections have largely remained at the level of formal analogies. In this work, we provide what is, to our knowledge, the first explicit realization of the epistemic-to-physical transition of Fisher information within a finite interacting quantum system. Specifically, we analyze a model of N fermions occupying two degenerate levels and coupled by a spin-flip interaction of strength V, treated in the grand canonical ensemble at inverse temperature β. We compute the Fisher information FN(V) associated with the sensitivity of the thermal state to changes in V, and show that it becomes an observer-independent, experimentally meaningful quantity: it encodes fluctuations, tracks entropy variations, and reveals structural transitions induced by interactions. Our findings thus demonstrate that Fisher information, originally conceived as an inferential and epistemic measure, can operate as a bona fide thermodynamic observable in quantum many-body physics, bridging the gap between information-theoretic foundations and measurable physical law.

Angelo Plastino

Carlo Cafaro, Paul M. Alsing

The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.