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.
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.
We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su(2;C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external time-dependent magnetic field that specifies the su(2;C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.
We develop an information-geometric framework for describing quantum state-update processes associated with measurement and statistical distinguishability. The approach is based on the quantum relative entropy and the quantum Fisher information metric, which together induce a natural Riemannian geometry on the manifold of quantum states. Using the second-order expansion of relative entropy, we show how the Fisher metric governs the local structure of distinguishability between nearby states and defines a corresponding thermodynamic length. This geometric structure provides an effective description of finite quantum state transitions in terms of fluctuation geometry and information-space distance. The formalism is applied to thermal two-level systems and harmonic oscillator states, illustrating how the Fisher metric encodes susceptibilities, fluctuations, and geometric transition costs. We also discuss the relation between thermodynamic length, dissipation bounds, and optimal paths in state space. Within this framework, wave function collapse is interpreted not as a microscopic dynamical mechanism, but as an effective state-update process that admits a geometric characterization in the manifold of density operators. The resulting perspective unifies concepts from quantum information theory, thermodynamics, and differential geometry within a common operational framework based on statistical distinguishability. Possible connections with quantum speed limits, entanglement geometry, and holographic relations between relative entropy and gravitational dynamics are briefly discussed.
We present an information geometric analysis of both entropic speeds and entropy production rates arising from geodesic evolution on manifolds parametrized by pure quantum states. In particular, we employ pure states that emerge as outputs of suitably chosen [Formula: see text] time-dependent Hamiltonian operators that characterize analog quantum search algorithms of specific types. The [Formula: see text] Hamiltonian models under consideration are specified by external time-dependent magnetic fields within which spin-[Formula: see text] test particles are immersed. The positive definite Riemannian metrization of the parameter manifold is furnished by the Fisher information function. The Fisher information function is evaluated along parametrized squared probability amplitudes obtained from the temporal evolution of these spin-[Formula: see text] test particles. A minimum action approach is then utilized to induce the transfer of the quantum system from its initial state to its final state on the parameter manifold over a finite temporal interval. We demonstrate in an explicit manner that the minimal (that is, optimum) path corresponds to the shortest (that is, geodesic) path between the initial and final states. Furthermore, we show that the minimal path serves also to minimize the total entropy production occurring during the transfer of states. Finally, upon evaluating the entropic speed as well as the total entropy production along optimal transfer paths within several scenarios of physical interest in analog quantum searching algorithms, we demonstrate in a transparent quantitative manner a correspondence between a faster transfer and a higher rate of entropy production. We therefore conclude that higher entropic speed is associated with lower entropic efficiency within the context of quantum state transfer.
We present an information geometric characterization of quantum driving schemes specified by su(2;C) time-dependent Hamiltonians in terms of both complexity and efficiency concepts. Specifically, starting from pure output quantum states describing the evolution of a spin-1/2 particle in an external time-dependent magnetic field, we consider the probability paths emerging from the parametrized squared probability amplitudes of quantum origin. The information manifold of such paths is equipped with a Riemannian metrization specified by the Fisher information evaluated along the parametrized squared probability amplitudes. By employing a minimum action principle, the optimum path connecting initial and final states on the manifold in finite time is the geodesic path between the two states. In particular, the total entropy production that occurs during the transfer is minimized along these optimum paths. For each optimum path that emerges from the given quantum driving scheme, we evaluate the so-called information geometric complexity (IGC) and our newly proposed measure of entropic efficiency constructed in terms of the constant entropy production rates that specify the entropy minimizing paths being compared. From our analytical estimates of complexity and efficiency, we provide a relative ranking among the driving schemes being investigated. Moreover, we determine that the efficiency and the temporal rate of change of the IGC are monotonic decreasing and increasing functions, respectively, of the constant entropic speed along these optimum paths. Then, after discussing the connection between thermodynamic length and IGC in the physical scenarios being analyzed, we briefly examine the link between IGC and entropy production rate. Finally, we conclude by commenting on the fact that an higher entropic speed in quantum transfer processes seems to necessarily go along with a lower entropic efficiency together with a higher information geometric complexity.
Title: The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law Description This note addresses a persistent conceptual gap in “information physics” programs that derive familiar dynamical laws by extremizing Fisher-information-based functionals: why Fisher information, rather than some other statistical functional? The proposed answer is meta-explanatory. The observer–world interface is modeled as probabilistic, and “objectivity” is formulated as Observer Equivalence (OE): invariance under reparameterization and monotonicity under coarse-graining (information-loss channels). Under these constraints, Chentsov-type characterization results single out the Fisher information metric as the unique local Riemannian geometry on the space of probability distributions (up to an overall scale). Fisher information is thus treated not as an axiom but as an invariance residue forced by OE at the probabilistic interface. To connect geometry to dynamics, the note introduces an explicit postulate—Informational Inertia—motivated as an information-geometric analogue of least-action/geodesic principles: in the absence of forcing, interface states evolve along paths of minimal statistical resistance (geodesics of the OE-fixed geometry), with physical constraints supplying the remaining “potential” structure. A single exemplar is presented: revisiting the Madelung/Reginatto (Hall–Reginatto) route to the Schrödinger equation, where the “quantum potential” arises as the geometric term associated with the Fisher penalty for reshaping the probability density. The overall scale (and thus ℏ) is explicitly acknowledged as empirical input: OE fixes the form of the geometry, not its scale. Keywords observer equivalence; Fisher information; information geometry; Chentsov theorem; coarse-graining; monotone metrics; Schrödinger equation; Madelung transformation; Reginatto; foundations of physics How to cite Ullman, G. “The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law.” Zenodo. https://doi.org/10.5281/zenodo.18495542 Notes on AI assistance This manuscript was prepared with assistance from large language models for drafting, editing, and LaTeX formatting. The author takes full responsibility for the technical content, claims, and references.