Fisher Information and Thermodynamics: Fluctuations in Quantum Many-Body Systems

F. Pennini, Angelo Plastino

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.

Sergio Curilef, F. Pennini, Angelo Plastino

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. We reveal a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.

Ye-Ming Meng, Zhe-Yu Shi

Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter $θ$ and its corresponding extensive quantum operator $\hat{O}$ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter $θ$ is quantified by its quantum Fisher information $\mathcal{F}_θ$. The framework is based on an exact integral representation that relates $\mathcal{F}_θ$ to the autocorrelation function of operator $\hat{O}$. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: $Δθ\,\overline{ΔO} \ge k_\text{B}T$ with $\overline{ΔO}\equiv\partial_θ\langle\hat{O}\rangle\,Δθ$ and $T$ is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.

B. Roy Frieden

Fisher information (FI) is normally developed and applied within the context of classical mathematical statistics. Here we show that it alternatively arises as physical properties of a generally decoherent system. Two kinds of Fisher information are treated: the classical FI $I[p]$ and the FI channel capacity 〈I〉. These are shown to be linearly related to the partition function Z, the coherence function $\ensuremath{\rho}{(x,x}^{\ensuremath{'}}),$ the two-point coherence function $\ensuremath{\tau}(\ensuremath{\Delta}x),$ the entropy of fluctuation $H(\ensuremath{\Delta}x),$ and the cross entropy ${H}_{\mathrm{KL}}(\ensuremath{\rho}{/r}_{\ensuremath{\Delta}})$ between the system probability density function $\ensuremath{\rho}(x,x)$ and its shifted version $\ensuremath{\rho}(x+\ensuremath{\Delta}x,x+\ensuremath{\Delta}x).$