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Extending quantum detailed balance through optimal transport

Rocco Duvenhage, Samuel Skosana, Machiel Snyman

TL;DR

This work addresses how to quantify and extend quantum detailed balance to non-equilibrium settings by embedding systems into a transport framework. It introduces transport plans between systems on possibly different observable algebras and defines Wasserstein distances that measure how closely a system reflects a model with detailed balance, even across noncommutative boundaries. The paper develops a comprehensive theory—including $W$, $W_\sigma$, and $W_{\sigma\sigma}$—proving existence of optimal plans and establishing metric properties, dualities, and bounds that connect deviations from detailed balance to transport distance, with concrete classical and quantum examples. The proposed approach provides a structured, quantitative pathway to analyze non-equilibrium steady states and common structural features across complex quantum systems, offering potential applications in non-equilibrium statistical mechanics and noncommutative ergodic theory.

Abstract

We develop a general approach to setting up and studying classes of quantum dynamical systems close to and structurally similar to systems having specified properties, in particular detailed balance. This is done in terms of transport plans and Wasserstein distances between systems on possibly different observable algebras.

Extending quantum detailed balance through optimal transport

TL;DR

This work addresses how to quantify and extend quantum detailed balance to non-equilibrium settings by embedding systems into a transport framework. It introduces transport plans between systems on possibly different observable algebras and defines Wasserstein distances that measure how closely a system reflects a model with detailed balance, even across noncommutative boundaries. The paper develops a comprehensive theory—including , , and —proving existence of optimal plans and establishing metric properties, dualities, and bounds that connect deviations from detailed balance to transport distance, with concrete classical and quantum examples. The proposed approach provides a structured, quantitative pathway to analyze non-equilibrium steady states and common structural features across complex quantum systems, offering potential applications in non-equilibrium statistical mechanics and noncommutative ergodic theory.

Abstract

We develop a general approach to setting up and studying classes of quantum dynamical systems close to and structurally similar to systems having specified properties, in particular detailed balance. This is done in terms of transport plans and Wasserstein distances between systems on possibly different observable algebras.
Paper Structure (19 sections, 9 theorems, 152 equations)

This paper contains 19 sections, 9 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. A reader's guide
  3. Basic concepts
  4. Preliminary definitions, conventions and notations
  5. Transport plans between systems
  6. Conceptual exposition
  7. Extending dynamical properties by optimal transport
  8. Wasserstein distances
  9. A simple classical example
  10. The setting
  11. Extending probability transfer to or from a point
  12. An explicit example
  13. Quantum detailed balance and KMS duals
  14. A simple quantum example
  15. Wasserstein distances
  16. ...and 4 more sections

Key Result

Theorem 7.10

The functions $W$ and $W_{\sigma}$ are asymmetric pseudometrics, while $W_{\sigma\sigma}$ is a pseudometric. In addition, optimal transport plans always exist for $W$, $W_{\sigma}$ and $W_{\sigma \sigma}$.

Theorems & Definitions (41)

  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 3.1
  • Definition 5.1
  • Example 5.2
  • Definition 7.1
  • Definition 7.2
  • Definition 7.3
  • Example 7.4
  • ...and 31 more