Information geometry approach to quantum stochastic thermodynamics

TL;DR

This paper develops a comprehensive information-geometric framework for quantum stochastic thermodynamics by decomposing the quantum Fisher information (QFI) into incoherent and coherent parts, linking the incoherent component to entropic acceleration and thermodynamic currents under GKSL dynamics. It extends classical results that relate Fisher information and entropy rates to the quantum domain, derives a tightened bound on the entropy-rate change that includes a non-negative contribution from coherent dynamics, and demonstrates that the quantum Mpemba effect is captured within this geometric perspective. The approach leverages the Morozova–Čencov–Petz family of QFIs and provides general insights into the geometry of quantum state evolution, including geodesic paths and completion ratios. Overall, the work offers a unifying mathematical framework for understanding how information geometry governs quantum thermodynamic processes and non-equilibrium phenomena, with potential implications for optimizing quantum protocols and interpreting quantum relaxation behavior.

Abstract

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 non-uniqueness 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 generalised 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 generalise a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a non-negative 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.

Figures (4)

• Figure 1: a) Time-evolution of the total SLD QFI of the reference state $\hat{\rho}$ (blue) and the rotated state $\hat{\rho}^\prime$ (red). b) The rotated state evolves classically (red), therefore the quantum Fisher information reduces to the incoherent contribution. The reference state, however, is coherent in the energy eigenbasis and thus has both incoherent (dashed blue) and coherent (dash-dotted) contributions. We find that the incoherent contribution decays exponentially faster than the coherent contribution.
• Figure 2: a) Statistical distance (SLD) traced by the path of the reference state $\hat{\rho}$ (blue solid) and the rotated state $\hat{\rho}^\prime$ (red solid) in state space. The semi-transparent lines indicate the distance traced by the respective paths in the infinite time limit, indicating that sufficient thermalisation is reached within the chosen time period. The statistical distance for the rotated state is shorter than for the reference state. However, the SLD geodesic distance between the reference state and the thermal state (blue dash-dotted) is shorter than that between the rotated state and the thermal state (red dash-dotted), which coincides with the statistical distance. b) Furthermore, we find that the rotated state (red) completes the path at a faster rate as shown by the higher ratio of completion at all times.
• Figure 3: The path-dependent time-integrated ratio between the heat current and the standard deviation of the Hamiltonian with respect to the time-evolving state is upper bounded by twice the statistical distance associated with the path of the time-evolving density matrix in state space, shown for a) the reference state $\hat{\rho}$ and b) the rotated state $\hat{\rho}^\prime$. The path traced by the rotated state coincides with that of the geodesic (even though it is not traced at constant speed). Therefore, the bound set by the statistical distance coincides with the generally tightest one provided by the length of the geodesic. The bound is saturated because the state evolves purely incoherently.
• Figure 4: a) Geometric uncertainty $\delta$ about the path of the rotated state $\hat{\rho}^\prime$ (red). We compute the geometric uncertainty of the time-evolving reference state $\hat{\rho}$ using the SLD, WY and HM metric (blue). The SLD yields the smallest geometric uncertainty. Deviations from the geodesic connecting the reference state and rotated state to the thermal state, respectively, are accumulated at short times. b) Trade-off between the time-averaged QFI and the geometric uncertainty.