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A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature

Ainesh Bakshi, Allen Liu, Ankur Moitra, Ewin Tang

TL;DR

This work develops a quantum analogue of the classical Dobrushin framework to connect dynamics and structure in high-temperature Gibbs states. By formulating a quantum Dobrushin condition using quantum Wasserstein distance and a transport-plan viewpoint, the authors prove rapid mixing for a carefully constructed balanced Lindbladian and deduce exponential decay of conditional mutual information with distance via recovery maps. The results hold in continuous and discretized dynamics and rely on quasi-locality and detailed balance, yielding a global Markov property for high-temperature quantum Gibbs states. The framework provides a unified approach to understanding information flow, correlation decay, and Gibbs-state preparation in quantum many-body systems with potential implications for quantum simulation and quantum advantage.

Abstract

A central challenge in quantum physics is to understand the structural properties of many-body systems, both in equilibrium and out of equilibrium. For classical systems, we have a unified perspective which connects structural properties of systems at thermal equilibrium to the Markov chain dynamics that mix to them. We lack such a perspective for quantum systems: there is no framework to translate the quantitative convergence of the Markovian evolution into strong structural consequences. We develop a general framework that brings the breadth and flexibility of the classical theory to quantum Gibbs states at high temperature. At its core is a natural quantum analog of a Dobrushin condition; whenever this condition holds, a concise path-coupling argument proves rapid mixing for the corresponding Markovian evolution. The same machinery bridges dynamic and structural properties: rapid mixing yields exponential decay of conditional mutual information (CMI) without restrictions on the size of the probed subsystems, resolving a central question in the theory of open quantum systems. Our key technical insight is an optimal transport viewpoint which couples quantum dynamics to a linear differential equation, enabling precise control over how local deviations from equilibrium propagate to distant sites.

A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature

TL;DR

This work develops a quantum analogue of the classical Dobrushin framework to connect dynamics and structure in high-temperature Gibbs states. By formulating a quantum Dobrushin condition using quantum Wasserstein distance and a transport-plan viewpoint, the authors prove rapid mixing for a carefully constructed balanced Lindbladian and deduce exponential decay of conditional mutual information with distance via recovery maps. The results hold in continuous and discretized dynamics and rely on quasi-locality and detailed balance, yielding a global Markov property for high-temperature quantum Gibbs states. The framework provides a unified approach to understanding information flow, correlation decay, and Gibbs-state preparation in quantum many-body systems with potential implications for quantum simulation and quantum advantage.

Abstract

A central challenge in quantum physics is to understand the structural properties of many-body systems, both in equilibrium and out of equilibrium. For classical systems, we have a unified perspective which connects structural properties of systems at thermal equilibrium to the Markov chain dynamics that mix to them. We lack such a perspective for quantum systems: there is no framework to translate the quantitative convergence of the Markovian evolution into strong structural consequences. We develop a general framework that brings the breadth and flexibility of the classical theory to quantum Gibbs states at high temperature. At its core is a natural quantum analog of a Dobrushin condition; whenever this condition holds, a concise path-coupling argument proves rapid mixing for the corresponding Markovian evolution. The same machinery bridges dynamic and structural properties: rapid mixing yields exponential decay of conditional mutual information (CMI) without restrictions on the size of the probed subsystems, resolving a central question in the theory of open quantum systems. Our key technical insight is an optimal transport viewpoint which couples quantum dynamics to a linear differential equation, enabling precise control over how local deviations from equilibrium propagate to distant sites.

Paper Structure

This paper contains 41 sections, 42 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Technical overview
  4. Key ideas
  5. Path coupling.
  6. Quantum path coupling.
  7. A Dobrushin condition for rapid mixing.
  8. Transport plan dynamics.
  9. A detailed balanced Lindbladian.
  10. Proving that the Dobrushin condition holds.
  11. Clustering of conditional mutual information.
  12. Formal results and comparison to related work
  13. Our Hamiltonians and Lindbladians
  14. Our formal results
  15. Preliminaries and background
  16. ...and 26 more sections

Key Result

Theorem 1.1

Given a geometrically local Hamiltonian, let $\sigma$ be the corresponding Gibbs state at inverse temperature $\beta$. For any $\beta < \beta_c$, where $\beta_c$ is a fixed constant, and any initial state $\rho$, the Lindbladian evolution runs for $\bigO{\log(n/\varepsilon)}$ time and outputs a stat

Theorems & Definitions (111)

  • Theorem 1.1: Rapid mixing of Lindbladian evolution (informal)
  • Theorem 1.2: Exponential decay of CMI (informal)
  • Remark 2.1: Prior work on Wasserstein norm
  • Remark 2.2: Prior work on mixing conditions
  • Remark 2.3: Prior work with related techniques
  • Definition 2.1: Hamiltonian
  • Definition 2.4: Graphs and distances induced by a Hamiltonian
  • Definition 2.5: Locality $\mathfrak{K}$, degree ${\mathfrak{d}}$, and growth parameters $\mathfrak{b}$, $\mathfrak{a}$ of a Hamiltonian
  • Theorem 2.1: Rapid mixing for high temperature Gibbs states
  • Theorem 2.2: Rapid mixing with a discrete channel
  • ...and 101 more