Conditional Mutual Information and Information-Theoretic Phases of Decohered Gibbs States

TL;DR

This paper investigates how local dissipation affects the emergence of long-range conditional mutual information in Gibbs states, revealing conditions for phase transitions and long-range correlations in quantum and classical systems.

Contribution

It provides the first rigorous analysis of conditions under which dissipation induces long-range CMI in Gibbs states, highlighting phase transitions at finite temperature.

Findings

Exponential decay of CMI in high-temperature Gibbs states with local dissipation.

Low-temperature hidden Markov networks can sustain long-range CMI.

Existence of finite-temperature information-theoretic phase transitions without thermodynamic phase transitions.

Abstract

Classical and quantum Markov networks -- including Gibbs states of commuting local Hamiltonians -- are characterized by the vanishing of conditional mutual information (CMI) between spatially separated subsystems. Adding local dissipation to a Markov network turns it into a \emph{hidden Markov network}, in which CMI is not guaranteed to vanish even at long distances. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for teleportation, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this work we provide the first rigorous results in this direction. We establish that CMI in high-temperature Gibbs states subject to local dissipation decays exponentially, (i) for classical Hamiltonians subject to arbitrary local transition matrices, and (ii) for commuting local Hamiltonians subject to unital channels that obey certain mild restrictions. Conversely, we show that low-temperature hidden Markov networks can sustain long-range CMI. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions. We also show several applications in quantum information and many-body physics.