Quantum Gibbs states are locally Markovian

2025-04-03

AI Summary

This work addresses the local Markov property of quantum Gibbs states at arbitrary temperatures for finite-interaction-range Hamiltonians. Methodologically, it constructs a quasilocal recovery map via detailed-balance Lindblad dynamics for any local region, establishing a universal local Markov structure. The key innovation lies in rigorously connecting Dirichlet-form-based dissipative dynamics with the commutator-based static description under the KMS inner product, combined with imaginary-time evolution regularization, conditional mutual information (CMI) analysis, and explicit quantum circuit construction. The results show that CMI decays exponentially below the screening distance; for D-dimensional lattices, an ε-approximate Gibbs state admits a quantum circuit of depth bounded by $e^{O(log^D(n/varepsilon))}$, improvable under a local spectral gap assumption. These findings provide novel theoretical tools for quantum thermodynamics and mixing-time analysis of quantum many-body systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree, the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on $D$-dimensional lattices can be prepared by a quantum circuit of depth $e^{O(log^D(n/epsilon))}$, which can be further reduced assuming certain local gap condition. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes.

Problem

• Understanding local Markovian property in quantum Gibbs states
• Exponential decay of conditional mutual information in shielded regions
• Efficient quantum circuit preparation for non-commuting Hamiltonians

Innovation

• Quantum Gibbs states exhibit local Markovian properties
• Quasi-local recovery maps from detailed-balanced Lindbladians
• Exponential decay of conditional mutual information