# Quantum Gibbs states are locally Markovian

arXiv: https://arxiv.org/abs/2504.02208 · HTML: https://mlsanity.com/paper/2504.02208/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2025.

## Built on by (most cited first, 2 of 2)

- [Learning and Generating Mixed States Prepared by Shallow Channel Circuits](https://mlsanity.com/paper/2604.01197/) (2026)
- [Stability of mixed-state phases under weak decoherence](https://mlsanity.com/paper/2511.01976/) (2025)

## Abstract (arXiv)

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 eO(logD(n/ε)), 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.