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Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound

Sosuke Ito, Andreas Dechant

Published Jan 1, 2020• 95 citations
OpenAlex

Abstract

We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular, for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that the minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.

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We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular, for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that the minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.
The importance of Fisher information is increasing in nonequilibrium thermodynamics, as it has played a fundamental role in trade-off relations such as thermodynamic uncertainty relations and speed limits. In this work, we investigate temporal Fisher information, which measures the temporal information content encoded in probability distributions, for both classical and quantum systems. We establish that temporal Fisher information is bounded from above by physical costs, such as entropy production in classical Langevin and Markov processes, and the quantum entropy production and the expectation of the effective interaction Hamiltonian in open quantum systems. Conversely, temporal Fisher information is bounded from below by statistical distances (e.g., the Bhattacharyya arccos distance), leading to classical and quantum speed limits that constrain the minimal time required for state transformations. We perform numerical simulations on two quantum dot models to validate the obtained bounds. Our work provides a unified perspective on speed limits from the point of view of temporal Fisher information in both classical and quantum dynamics.
The unavoidable interaction between a quantum system and the external noisy environment can be mimicked by a sequence of stochastic measurements whose outcomes are neglected. Here we investigate how this stochasticity is reflected in the survival probability to find the system in a given Hilbert subspace at the end of the dynamical evolution. In particular, we analytically study the distinguishability of two different stochastic measurement sequences in terms of a Fisher information measure depending on the variation of a function, instead of a finite set of parameters. We find a characterization of Zeno phenomena as the physical result of the random observation of the quantum system, linked to the sensitivity of the survival probability to an arbitrary small perturbation of the measurement stochasticity. Finally, the implications on the Cram\'er-Rao bound and the Zeno time are discussed, together with a numerical example.
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).
Variance and Fisher information are ingredients of the Cramér-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence between the candidates of the two concepts. We emphasize that Fisher information is obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold.
Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.
We describe a compact and reliable method to calculate the Fisher information for the estimation of a dynamical parameter in a continuously measured linear Gaussian quantum system. Unlike previous methods in the literature, which involve the numerical integration of a stochastic master equation for the corresponding density operator in a Hilbert space of infinite dimension, the formulas here derived depend only on the evolution of first and second moments of the quantum states and thus can be easily evaluated without the need of any approximation. We also present some basic but physically meaningful examples where this result is exploited, calculating analytical and numerical bounds on the estimation of the squeezing parameter for a quantum parametric amplifier and of a constant force acting on a mechanical oscillator in a standard optomechanical scenario.