Topological Mixed States: Phases of Matter from Axiomatic Approaches

TL;DR

This paper proposes an axiomatic framework for classifying topological phases in open quantum systems, extending the concept of topological order to mixed states and demonstrating robustness through numerical simulations.

Contribution

It introduces a set of axioms defining fixed points and phases for mixed states, establishing a foundation for axiomatic classification of topological matter in open systems.

Findings

States satisfying axioms are fixed points with rich topological data.

Topological data are robust under relaxation of axioms.

Numerical simulations confirm stability of weakly decohered fixed points.

Abstract

For closed quantum systems, topological orders are understood through the equivalence classes of ground states of gapped local Hamiltonians. The generalization of this conceptual paradigm to open quantum systems, however, remains elusive, often relying on operational definitions without fundamental principles. Here, we fill this gap by proposing an approach based on three axioms: ($i$) local recoverability, ($ii$) absence of long-range correlations, and ($iii$) spatial uniformity. States that satisfy these axioms are fixed points; requiring the axioms only after coarse-graining promotes each fixed point to an equivalence class, i.e., a phase, presenting the first step towards the axiomatic classification of mixed-state phases of matter: mixed-state bootstrap program. From these axioms, a rich set of topological data naturally emerges; importantly, these data are robust under relaxation of axioms. For example, each topological mixed state supports locally indistinguishable classical and/or quantum logical memories with distinct responses to topological operations. These data label distinct mixed-state phases, allowing one to distinguish them. We further uncover a hierarchy of secret-sharing constraints: in non-Abelian phases, reliable recovery-even of information that looks purely classical-demands a specific coordination among spatial subregions, a requirement different across non-Abelian classes. This originates from non-Abelian fusion rules that can stay robust under decoherence. Finally, we performed large-scale numerical simulations to corroborate stability: weakly decohered fixed points respect the axioms once coarse-grained. These results lay the foundation for a systematic classification of topological states in open quantum systems.