A Unified Framework for Locally Stable Phases

TL;DR

This paper introduces a unifying framework for characterizing phases of matter based on local stability, linking correlation decay, reversibility, and phase invariance across different regimes.

Contribution

It defines locally stable states and phases, proving their equivalence to short-range correlations and invariance under local channels, bridging pure and mixed state phases.

Findings

Locally stable states are equivalent to short-range correlations.

Local stability implies decay of nonlinear correlators.

Post-selection on stable states can be implemented via local channels.

Abstract

We propose a unifying framework for characterizing pure and mixed state phases of matter across equilibrium, non equilibrium, and metastable regimes. We introduce the concept of locally stable states, defined by the operational property that any local operation (including post selection) can be reversed by a local channel. We prove that local stability is equivalent to a state being short range correlated, defined by the decay of both correlations and conditional mutual information. We demonstrate that these properties are invariant under locally reversible channels, thus defining locally stable phases. Furthermore, we prove that local stability implies both the decay of a family of nonlinear correlators, including the fidelity correlator, and the decay of correlations in the canonical purification, thus bridging the gap between mixed and pure states. Along the way, we establish two results which may be of independent interest: we show that post-selection on locally stable (short range correlated) states can be implemented via local channels and that quantum Markov chains can be characterized by the local computability of nonlinear observables.