We derive a parameter-free, falsifiable inequality linking entropy production, Fisher information, and spatial dispersion for systems undergoing free diffusion. The result follows from the matrix Cramér–Rao bound combined with De Bruijn’s identity and holds at all times for probability densities evolving under the diffusion equation ∂ₜρ = TΔρ. Gaussian states saturate the bound, while all other initial conditions obey it strictly. The inequality provides a direct experimental test of the information-geometric structure underlying thermodynamic response and can be validated using reconstructed density profiles in ultracold-atom experiments. This preprint is deposited on Zenodo to establish priority and enable open access prior to journal submission.
We derive a parameter-free, falsifiable inequality linking entropy production, Fisher information, and spatial dispersion for systems undergoing free diffusion. The result follows from the matrix Cramér–Rao bound combined with De Bruijn’s identity and holds at all times for probability densities evolving under the diffusion equation ∂ₜρ = TΔρ. Gaussian states saturate the bound, while all other initial conditions obey it strictly. The inequality provides a direct experimental test of the information-geometric structure underlying thermodynamic response and can be validated using reconstructed density profiles in ultracold-atom experiments. This preprint is deposited on Zenodo to establish priority and enable open access prior to journal submission.
On some interrelations of generalized q-entropies and a generalized Fisher information, including a Cramér-Rao inequality
In this communication, we describe some interrelations between generalized q-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of entropy and Fisher information. More precisely, a generalized Fisher information naturally pops up in the expression of the derivative of the Tsallis entropy. This generalized Fisher information also appears as a special case of a generalized Fisher information for estimation problems. Indeed, we derive here a new Cramér-Rao inequality for the estimation of a parameter, which involves a generalized form of Fisher information. This generalized Fisher information reduces to the standard Fisher information as a particular case. In the case of a translation parameter, the general Cramér-Rao inequality leads to an inequality for distributions which is saturated by generalized q-Gaussian distributions. These generalized q-Gaussians are important in several areas of physics and mathematics. They are known to maximize the q-entropies subject to a moment constraint. The Cramér-Rao inequality shows that the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a fixed moment. Similarly, the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a given q-entropy.
This paper provides a third independent derivation of the critical coherence threshold K_crit ≈ 0.127, complementing the dynamical derivation (GCT) and thermodynamic derivation (Landauer bound). The falsifiability component F(t) is identified with the Fisher information of the system’s self-estimating process. The Cramér–Rao bound then establishes that as F(t) → 0, coherence estimation variance diverges — the system becomes informationally blind to its own state. K_crit is the Fisher information collapse point: below it, no estimator can locate the system’s coherence state with bounded precision. The 1=1 invariant is identified as the geodesic endpoint of the Fisher information manifold — the natural destination of the geometry itself under contractive evaluation dynamics. Three independent frameworks (dynamical systems, thermodynamics, information geometry) converge on the same critical threshold and the same invariant. Part of the Spektre corpus (github.com/spektre-labs/corpus).
This work introduces a unified dynamical law of irreversibility that links three fundamental quantities in open quantum systems: coherence change, information flux, and irreversible entropy production.For finite-dimensional systems evolving under completely positive trace-preserving (CPTP) Markovian semigroups, we prove the exact identity κ˙(t)=Φinfo(t)−ΔS(t),\dot{\kappa}(t) = \Phi_{\text{info}}(t) - \Delta S(t),κ˙(t)=Φinfo(t)−ΔS(t), where κ(t)=S(ρdiag)−S(ρ)\kappa(t) = S(\rho_{\text{diag}}) - S(\rho)κ(t)=S(ρdiag)−S(ρ) is the entropy-valued coherence, Φinfo(t)=−ddtS(ρt∥ν)\Phi_{\text{info}}(t) = -\frac{d}{dt} S(\rho_t \Vert \nu)Φinfo(t)=−dtdS(ρt∥ν) is the information flux relative to a canonical maximum-entropy reference state ν\nuν, and ΔS(t)=Tr[L(ρt)(logρt−logσ)]≥0\Delta S(t) = \text{Tr}[ \mathcal{L}(\rho_t)(\log \rho_t - \log \sigma)] \ge 0ΔS(t)=Tr[L(ρt)(logρt−logσ)]≥0 is the Spohn entropy production rate. The result establishes an exact entropy-balance identity connecting quantum coherence, thermodynamic irreversibility, and information flow — all expressed in consistent entropy-per-time units.Operational protocols are provided for experimental validation using modern quantum platforms (e.g., superconducting qubits, trapped ions, or cavity QED).A residual test, R(t)=κ˙−Φinfo+ΔSR(t) = \dot{\kappa} - \Phi_{\text{info}} + \Delta SR(t)=κ˙−Φinfo+ΔS, offers a falsifiable criterion for detecting non-Markovianity or hidden memory flux. The framework unifies information geometry, resource-theoretic coherence, and nonequilibrium thermodynamics into a single experimentally testable law, illuminating how quantum structure can be sustained only when information inflow compensates entropy production. Keywords:Quantum thermodynamics, information theory, open quantum systems, coherence, entropy production, relative entropy, Lindblad dynamics, irreversibility, information geometry.
A wide class of physical problems requires the estimation of probability laws. Diffraction patterns and quantum mechanical probability laws on position are examples. Minimum Fisher information is one approach to estimating such laws; maximum entropy is another. In this paper, we show that the minimum Fisher information approach may be derived from a prior principle of maximum Cramer-Rao bound (MCRB). The MCRB-Fisher approach is then applied to some fundamental physical problems, including diffraction theory and quantum mechanics. It is found to give the correct physical solutions, that is the Helmholtz and Schrödinger wave equations respectively. By comparison, the maximum entropy approach gives incorrect solutions to these problems, except in the special (thermodynamic) case of a harmonic oscillator potential.
The inequality , with L being the grand orbital quantum number, and its conjugate relation for (⟨ r 2 ⟩, ⟨ p −2 ⟩) are shown to be fulfilled in the D -dimensional central problem. Their use has allowed us to improve the Fisher-information-based uncertainty relation ( I ρ I γ ⩾ const) and the Cramer–Rao inequalities (⟨ r 2 ⟩ I ρ ⩾ D 2 ; ⟨ p 2 ⟩ I γ ⩾ D 2 ). In addition, the kinetic energy and the radial expectation value ⟨ r 2 ⟩ are shown to be bounded from below by the Fisher information in position and momentum spaces, denoted by I ρ and I γ , respectively.