Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature

TL;DR

This paper proves that quantum Gibbs states at sufficiently high temperatures exhibit rapid decay of conditional mutual information, enabling efficient algorithms for their simulation and analysis.

Contribution

It establishes the decay properties of quantum Gibbs states above a threshold temperature and links these to efficient sampling algorithms and area laws.

Findings

Exponential decay of mutual information in short-ranged systems

Power-law decay in long-ranged systems

Polynomial-time classical simulation algorithms

Abstract

We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasi-locality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for classical Gibbs state simulations.