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

2025-10-09

AI Summary

A unified theoretical framework linking structural properties and dynamical behavior—particularly convergence rates—in quantum many-body systems, both at and out of equilibrium, has long been lacking. Method: We develop a structural theory for quantum Markovian dynamics at high temperature: (i) we introduce the quantum Dobrushin condition, extending classical path-coupling methods to quantum channels; (ii) we recast quantum dynamical coupling via optimal transport, yielding linear differential equations whose rapid mixing is rigorously established; (iii) we derive exponential decay of conditional mutual information, independent of subsystem size. Contribution: This work establishes the first quantitative equivalence between the convergence rate of quantum Markovian dynamics and strong locality in Gibbs states—including area-law entanglement scaling and exponential correlation decay—thereby bridging a fundamental theoretical gap between dynamical relaxation and emergent structure in open quantum systems. It provides a foundational framework for quantum thermalization, noise-resilient quantum computation, and analog quantum simulation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Developing a quantum Dobrushin condition for rapid mixing of Markov chains
• Establishing exponential decay of conditional mutual information in quantum systems
• Bridging dynamic and structural properties of quantum Gibbs states at high temperature

Innovation

• Quantum Dobrushin condition enables rapid mixing
• Path-coupling proves exponential decay of CMI
• Optimal transport couples dynamics to linear equations