Unified speed limits in classical and quantum dynamics via temporal Fisher information (2504.04790v1)

Abstract: 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 variance of interaction Hamiltonians 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. Our work provides a unified perspective of speed limits from the point of view of temporal Fisher information in both classical and quantum dynamics.

Unified Speed Limits in Classical and Quantum Dynamics via Temporal Fisher Information

The paper of speed limits in physical dynamics, which imposes constraints on the minimum time required for a system to undergo a state transformation, is a topic of significant interest across both classical and quantum domains. This paper addresses this very topic by presenting a unifying framework that leverages temporal Fisher information to establish speed limits in both classical and quantum systems. It extends the relevance of Fisher information, traditionally a cornerstone in statistical inference, to the quantitative investigation of nonequilibrium thermodynamics.

Conceptual Framework

Temporal Fisher information serves as the focal point in this framework, quantifying the information about time evolution encoded in a system's probability distribution. This concept generalizes Fisher information beyond its conventional statistical context to a dynamical setting, offering a new perspective on characterizing the temporal properties of systems. The paper posits that temporal Fisher information is not only bounded by physical parameters such as entropy production or the variance of interaction Hamiltonians but also leads directly to speed limits through relations with statistical distances.

Main Contributions

1. Classical Dynamics:
   • For classical Langevin and Markov processes, temporal Fisher information is constrained by the rate of entropy production. Specifically, it is shown that in both these classical frameworks, the upper bound of the temporal Fisher information is determined by the entropy production rate divided by the square of time.
   • The resulting classical speed limit is expressed in terms of the Bhattacharyya arccos distance, a statistical measure of distance between probability distributions, thus providing a metric for the minimal required transformation time based on how statistical distance accumulates over time.
2. Quantum Dynamics:
   • In the quantum field, the analysis focuses on open quantum systems and presents a novel bounding of temporal Fisher information via the variance of interaction Hamiltonians.
   • The quantum speed limits derived provide a quantitative measure using the Bures angle, thereby relating the temporal evolution to fundamental dissipation measures in quantum mechanics.
   • The paper includes a consideration of non-hermitian dynamics, emphasizing the variance of the dissipative components.

Implications and Future Directions

The implications of unifying classical and quantum speed limits under a common framework of temporal Fisher information are profound. Practically, this unification aids in the development of more efficient control protocols in quantum technologies and offers a rigorous tool for assessing performance bounds in stochastic thermodynamics. Theoretically, it bridges concepts across statistical mechanics and information geometry, enhancing the interpretative clarity of speed limits in broader physical contexts.

Future research can explore optimizing these bounds, exploring their implications in more complex systems, and extending the framework to relativistic systems or biological processes.

In summary, the paper contributes significantly to the understanding of dynamical speed limits by offering a coherent and comprehensive framework that spans both classical and quantum domains, rooted in the informational structure of temporal evolution.