Quantum Gibbs states are locally Markovian

TL;DR

This work proves that quantum Gibbs states of Hamiltonians with bounded interaction degree are locally Markov at any temperature by constructing a quasi-local recovery map implemented as a time-averaged detailed-balanced Lindbladian acting on a region. The authors establish exponential decay of the quantum conditional mutual information $I(A:C|B)$ with shielding distance, and show that Gibbs states can be efficiently prepared via quasi-local patches under uniform clustering, with potential improvements under a local gap condition. Central to the approach are a regularization scheme for imaginary-time evolution using operator-Fourier decompositions, and a Dirichlet-form–commutator correspondence that ties dissipative dynamics to static static properties. The results advance the quantum Gibbs sampling and quantum thermodynamics toolkit, providing foundations for low-temperature analyses and practical Gibbs-state preparation with locality-controlled dynamics.

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 $e^{O(\log^D(n/ε))}$, 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.

Figures (5)

• Figure 1: Classical Gibbs distributions of local Hamiltonians are always Markovian at any non-zero temperatures. For the 2D Ising model with nearest neighbour interactions, the regions $A$ and $C$ are independent conditioned on the separating set $B$. Precise notions of pairwise Markov property (when $\left\vert {A} \right\vert,\left\vert {C} \right\vert= 1$), local Markov property (when $\left\vert {A} \right\vert=1$), and global Markov property (when $A$ is any subset) can be defined, and, are equivalent for positive distributions. In the quantum case, these three are not known to be equal, even allowing for approximations.
• Figure 2: Our main result says that local disturbance to the Gibbs state can be recovered locally. Suppose we trace out a region $A$ and replace it with the maximally mixed state $\bm{\tau}_A$, the recovery map $\mathcal{R}$ is quasi-local with radius growing with the region size $\left\vert {A} \right\vert$ and the inverse temperature $\beta.$ In fact, the recovery map is a time-averaged detailed-balanced Lindbladian based on single-Pauli jumps on $A$.
• Figure 3: Any operator can be decomposed by the Bohr frequencies $\bm{A} = \sum_{\nu} \bm{A}_{\nu}$. When the operator $\bm{A}$ acts on a single site, the amplitudes concentrate around $\nu = \mathcal{O}(1) \sim \Vert {[\bm{H},\bm{A}]} \Vert.$ However, when the Hamiltonian is non-commuting and beyond one dimensions, there could be an exponentially small amplitude for large energy changes $\nu$, which causes divergence for the imaginary time conjugation $e^{\beta \bm{H}}\bm{A} e^{-\beta \bm{H}}$ at a large constant $\beta$. The operator Fourier transform $\hat{\bm{A}}(\omega)$ with Gaussian weights selects the amplitudes near $\omega\pm \mathcal{O}(\sigma)$. The Gaussian tail is particularly effective for mitigating the exponential divergence due to imaginary time conjugation.
• Figure 4: Combining local indistinguishability (\ref{['thm:localindistinguishability']}) and local Markov property (\ref{['thm:main']}) to recover the Gibbs state from a restricted Hamiltonian. There are two length scales: $\ell'$ as the correlation length for local indistinguishability and $\ell$ as the quasi-locality of the recovery map.
• Figure 5: Building the global Gibbs states from quasi-local patches (see \ref{['fig:zoomin']} for the specification of each patch) in parallel. The patching argument proceeds by punching point-like holes (white squares) until the whole lattice is covered. Note that we assumed local indistinguishability holds for all intermediate Hamiltonians (supported on colored regions), which may have lots of punched holes and a nontrivial topology.

Theorems & Definitions (71)

• Remark 2.0.1
• Lemma 2.1: Operator norm controls weighted norms and inner-product
• Theorem 2.1: chen2023efficient
• Theorem 2.2: chen2023efficientchen2023quantum
• Remark 2.2.1
• Remark 2.2.2
• Theorem 3.1: Quasi-local recovery maps via time-averaged Gibbs sampling
• Remark 3.1.1
• Remark 3.1.2
• Corollary 3.1: Quasi-locality estimates