Exponential conditional mutual information decay for Gibbs states on lattices

From papers

Let $\rho$ be a Gibbs state of a short-ranged Hamiltonian defined on a $D$ -dimensional spin lattice. Let $A$ , $B$ , and $C$ be three regions, with $B$ shielding $A$ from $C$ , and let $d(A,C)$ denote the distance between $A$ and $C$ . Exponential decay conjecture. There exist constants $C,c>0$ such that

I(A:C|B)_\rho\leq C e^{-c d(A,C)}.

This conjecture extends the fast conditional-mutual-information decay established for one-dimensional systems and suggests that Gibbs states of short-ranged Hamiltonians form approximate quantum Markov networks on regular $D$ -dimensional lattices. The source gives no resolution, so the conjecture remains open.

Progress summary

Claimed solved

A 2025 paper claims to prove the conjecture at every temperature, but its stated estimate does not clearly establish the exact uniform bound in the problem.

The conjecture says that separating two regions by a buffer makes their remaining dependence decrease exponentially with separation, in any lattice dimension and at any temperature. The problem is attributed to [Kuw24]; a 2016 source recorded it as open.

Known results

High temperature: exponential conditional-mutual-information decay for short-range interactions (Kato, Kuwahara, and Brandão, 2020).
One dimension: related exponential approximate-Markov results (Kato, Kuwahara, and Brandão, 2016).
Commuting Hamiltonians: exponential decay of modified conditional mutual information (2024/2025).

April 2025 claimed resolution

“Quantum Gibbs states are locally Markovian” claims an arbitrary-temperature proof for bounded-degree interaction graphs, including shielded regions. Its displayed estimate contains region-size factors and a positive size-dependent exponential term, so it is not visibly the conjecture’s size-independent bound. A separate 2025 paper proves decay only when $A\cup B\cup C=\Lambda$ and explicitly leaves the subsystem case open.

Current status (as of August 2026): An unverified 2025 preprint claims the general result, while the exact size-independent conjecture remains unsettled by the retrieved evidence.

Sources

Sources & referencesView supporting material

Primary source

Kohtaro Kato and Fernando G. S. L. Brandao, “Quantum Approximate Markov Chains are Thermal”, arXiv:1609.06636 (2019).

1 week ago