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arXiv:2504.02235

Clustering of Conditional Mutual Information via Quantum Belief-Propagation Channels

Kohtaro Kato, Tomotaka Kuwahara

Published Apr 3, 2025• 0 citations
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Abstract

Conditional mutual information (CMI) has recently attracted significant attention as a key quantity for characterizing quantum correlations in many-body systems. While it is conjectured that CMI decays rapidly in finite-temperature Gibbs states, a complete and general proof remains elusive. In this work, we introduce a new formulation of the problem based on the \emph{belief propagation (BP) channel}, namely a completely positive trace-preserving (CPTP) map that realizes local perturbations of the Hamiltonian. Within this framework, we prove that establishing the quasi-locality of BP channels implies the decay of CMI, thereby reducing the original conjecture to a more tractable problem. We show that such quasi-local BP channels can be constructed under natural physical conditions, such as uniform rapid mixing or uniform clustering. Under these assumptions, we obtain conditional proofs of CMI decay valid at all temperatures. Moreover, because these assumptions are automatically satisfied at high temperatures, our results in that regime yield unconditional proofs of CMI decay. At the same time, in order to better understand the high-temperature behavior of Gibbs states, we revisit the cluster expansion method. Contrary to common intuition, we demonstrate that when multipartite correlations such as CMI are considered, the cluster expansion suffers from intrinsic divergence problems rooted in the Baker--Campbell--Hausdorff formula, revealing fundamental limitations of this traditional approach.

DOI
10.48550/arxiv.2504.02235

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Conditional mutual information (CMI) is a widely studied multipartite correlation measure with a connection to the Markov property. Intriguingly, CMI can be generated by local dissipation, a phenomenon closely related to teleportation. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for phase transitions, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this Letter, we provide the first rigorous results in this direction. We establish that CMI in high-temperature commuting Gibbs states subject to local dissipation decays exponentially under certain mild restrictions. Conversely, we show that the same setup can sustain long-range CMI at low temperatures. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions.

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Conditional mutual information (CMI) has recently attracted significant attention as a key quantity for characterizing quantum correlations in many-body systems. While it is conjectured that CMI decays rapidly in finite-temperature Gibbs states, a complete and general proof remains elusive. In this work, we introduce a new formulation of the problem based on the \emph{belief propagation (BP) channel}, namely a completely positive trace-preserving (CPTP) map that realizes local perturbations of the Hamiltonian. Within this framework, we prove that establishing the quasi-locality of BP channels implies the decay of CMI, thereby reducing the original conjecture to a more tractable problem. We show that such quasi-local BP channels can be constructed under natural physical conditions, such as uniform rapid mixing or uniform clustering. Under these assumptions, we obtain conditional proofs of CMI decay valid at all temperatures. Moreover, because these assumptions are automatically satisfied at high temperatures, our results in that regime yield unconditional proofs of CMI decay. At the same time, in order to better understand the high-temperature behavior of Gibbs states, we revisit the cluster expansion method. Contrary to common intuition, we demonstrate that when multipartite correlations such as CMI are considered, the cluster expansion suffers from intrinsic divergence problems rooted in the Baker--Campbell--Hausdorff formula, revealing fundamental limitations of this traditional approach.
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 DDD-dimensional lattices can be prepared by a quantum circuit of depth eO(log⁡D(n/ε))e^{O(\log^D(n/ε))}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.
Belief propagation (BP) provides a scalable heuristic for contracting tensor networks on loopy graphs, but its success in quantum many-body settings has largely rested on empirical evidence. Developing upon a recently introduced cluster-expansion framework for tensor networks, we rigorously study the applicability of BP to many-body quantum systems. For a state represented as a PEPS satisfying a ``loop-decay" condition, we prove that BP supplemented by cluster corrections approximates local observables with exponentially small relative error, and we give explicit formulas expressing local expectation values as BP predictions dressed by connected clusters intersecting the observable region. This representation establishes a direct link between cluster corrections and physical correlation functions. As a result, we show that ``loop-decay" \emph{necessarily implies} exponential decay of connected correlations, yielding sharp, rigorous criteria for when BP can and cannot succeed, and ruling out its validity at critical points. Numerical simulations of the two- and three-dimensional transverse field Ising model at zero and finite temperature confirm our analytical predictions, demonstrating quantitative accuracy deep in gapped phases and systematic failure near criticality.
Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.
We establish improved locally Markovian properties of quantum Gibbs states at every fixed positive temperature. For finite-range interactions of bounded interaction degree, the conditional mutual information (CMI) decays exponentially with separation, with a prefactor exponential in the boundary size of only one of the two regions and independent of the size of the other region and the total system. This improves the exponential volume dependence of the local Markov bounds established by Chen and Rouzé \cite{Chen2025}. For exponentially and stretched-exponentially decaying interactions, we obtain stretched-exponential CMI decay with a boundary-dependent prefactor. For power-law interactions, we prove inverse-logarithmic or algebraic decay with polynomial region-size dependence. Our proof is static, in contrast to the dynamical approach of Chen and Rouzé. We compare the Gibbs state with a product state obtained by removing boundary interactions, reducing the CMI bound to a relative-entropy loss that we control using locality estimates.
Abstract Quantum systems in thermal equilibrium are described using Gibbs states. The correlations in such states determine how difficult it is to describe or simulate them. In this article, we show that if the Gibbs state of a quantum system satisfies that each of its marginals admits a local effective Hamiltonian with short-range interactions, then it satisfies a mixing condition, that is, for any regions A , C the distance of the reduced state ρAC\rho _{AC}ρAC​ ρ AC on these regions to the product of its marginals, $$ \left\| \rho _{AC} \rho _A^{-1} \otimes \rho _C^{-1} - \mathbbm {1}_{AC} \right\| \, , $$ ρ AC ρ A - 1 ⊗ ρ C - 1 - 1 AC , decays exponentially with the distance between regions A and C . This mixing condition is stronger than other commonly studied measures of correlation. In particular, it implies the exponential decay of the mutual information between distant regions. The mixing condition has been used, for example, to prove positive log-Sobolev constants. On the way, we prove that the the condition regarding local effective Hamiltonian is satisfied if the Hamiltonian only has commuting interactions which also commute with every marginal of their products. The proof of these results employs a variety of tools such as Araki’s expansionals, quantum belief propagation and cluster expansions.
Conditional mutual information (CMI) is a widely studied multipartite correlation measure with a connection to the Markov property. Intriguingly, CMI can be generated by local dissipation, a phenomenon closely related to teleportation. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for phase transitions, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this Letter, we provide the first rigorous results in this direction. We establish that CMI in high-temperature commuting Gibbs states subject to local dissipation decays exponentially under certain mild restrictions. Conversely, we show that the same setup can sustain long-range CMI at low temperatures. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions.