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Quantum Onsager relations

Mankei Tsang

TL;DR

This work recasts near-equilibrium quantum dynamics through quantum information geometry, showing that entropy-production-like behavior emerges from a Fisher-information metric defined via Petz density maps. It derives quantum analogs of Onsager’s reciprocity and Onsager–Casimir relations for open quantum systems modeled by GKSL dynamics, using both geometric and physically interpretable currents. A generalized time-reversal framework and quantum detailed-balance conditions underpin symmetry relations for the transport tensors, tying thermodynamics to parameter-estimation theory. The results illuminate a fundamental link between statistical mechanics and quantum metrology, while remaining applicable to a broad class of divergences and density maps and highlighting limitations to near-equilibrium, Markovian settings.

Abstract

Using quantum information geometry, I derive quantum generalizations of the Onsager rate equations, which model the dynamics of an open system near a steady state. The generalized equations hold for a flexible definition of the forces as well as a large class of statistical divergence measures and quantum-Fisher-information metrics beyond the conventional definition of entropy production. I also derive quantum Onsager-Casimir relations for the transport tensors by proposing a general concept of time reversal and detailed balance for open quantum systems. The results establish a remarkable connection between statistical mechanics and parameter estimation theory.

Quantum Onsager relations

TL;DR

This work recasts near-equilibrium quantum dynamics through quantum information geometry, showing that entropy-production-like behavior emerges from a Fisher-information metric defined via Petz density maps. It derives quantum analogs of Onsager’s reciprocity and Onsager–Casimir relations for open quantum systems modeled by GKSL dynamics, using both geometric and physically interpretable currents. A generalized time-reversal framework and quantum detailed-balance conditions underpin symmetry relations for the transport tensors, tying thermodynamics to parameter-estimation theory. The results illuminate a fundamental link between statistical mechanics and quantum metrology, while remaining applicable to a broad class of divergences and density maps and highlighting limitations to near-equilibrium, Markovian settings.

Abstract

Using quantum information geometry, I derive quantum generalizations of the Onsager rate equations, which model the dynamics of an open system near a steady state. The generalized equations hold for a flexible definition of the forces as well as a large class of statistical divergence measures and quantum-Fisher-information metrics beyond the conventional definition of entropy production. I also derive quantum Onsager-Casimir relations for the transport tensors by proposing a general concept of time reversal and detailed balance for open quantum systems. The results establish a remarkable connection between statistical mechanics and parameter estimation theory.
Paper Structure (17 sections, 14 theorems, 180 equations, 1 figure, 1 table)

This paper contains 17 sections, 14 theorems, 180 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Open quantum system theory
  4. Quantum information geometry
  5. Results
  6. Quantum Onsager theory
  7. Geometric version
  8. Physical version
  9. Quantum Onsager-Casimir relations
  10. Discussion
  11. Conclusion
  12. Classical case
  13. Gaussian states and channels
  14. Some results for coupled and damped harmonic oscillators
  15. Time reversal on a larger Hilbert space
  16. ...and 2 more sections

Key Result

Theorem 1

Define a generalized force as and an averaged conjugate variable as which I call a geometric variable. The convergence rate is then given by The geometric current $\dot x(t)$ obeys the equation of motion where the transport tensor $K$ is the information loss rate given by For any time $t$, $K(t)$ is symmetric, viz., and positive-semidefinite, so $K$ can be regarded as a Riemannian metric.

Figures (1)

  • Figure 1: Top left: the curved surface represents a manifold of density operators $\{\tau(\theta):\theta \in \Phi\}$ that can be prepared for the initial state. The $\tau(\theta_1,0)$ curve represents the set of density operators for varying $\theta_1$ and fixed $\theta_2 = 0$; the $\tau(0,\theta_2)$ curve is defined similarly. Each score $X_j(0) = \mathcal{E}_\sigma^{-1}\partial_j\tau$ represents the tangent vector of a curve at $\tau(0) = \sigma$. Top right: The CPTP map $\mathcal{F}(t)$ maps the manifold of density operators to a new manifold, while the pushforward $\mathcal{F}_*(t)$ maps each tangent vector $X_j(0)$ of a curve to the tangent vector $X_j(t) = \mathcal{F}_*(t) X_j(0)$ of the corresponding curve in the new manifold. Bottom left: Given an initial state $\tau(\epsilon v)$, $v^jX_j(0)$ represents the tangent vector that points from $\sigma$ to $\tau(\epsilon v)$. Bottom right: The pushforward $\mathcal{F}_*(t)$ determines the new tangent vectors after time $t$.

Theorems & Definitions (36)

  • Example 1: label=exa_gibbs
  • Example 2: label=exa_kick
  • Example 3: continues=exa_gibbs
  • Example 4: continues=exa_kick
  • Theorem 1
  • Example 5: continues=exa_kick
  • Theorem 2
  • Example 6: continues=exa_gibbs
  • Example 7: continues=exa_kick
  • Theorem 3
  • ...and 26 more