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arXiv:2506.04221 · Physical Review X

Topological Mixed States: Phases of Matter from Axiomatic Approaches

Tai-Hsuan Yang, Bowen Shi, Jong Yeon Lee

Published Jul 20, 2026• 0 citations
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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 toward the axiomatic classification of mixed-state phases of matter: the . From these axioms, a rich set of topological data naturally emerges; importantly, these data are 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. Surprisingly, the axiom violations at different length scales exhibit an exact crossing at the critical point, enabling its identification without any rescaling or data-collapse ambiguities. These results lay the foundation for a systematic classification of topological states in open quantum systems.

DOI
10.1103/3bpt-f9pd

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The classification and characterization of topological phases of matter is well understood for ground states of gapped Hamiltonians that are well isolated from the environment. However, decoherence due to interactions with the environment is inevitable—thus motivating the investigation of topological orders in the context of mixed states. Here, we take a step toward classifying mixed-state topological orders in two spatial dimensions by considering their (emergent) generalized symmetries. We argue that their 1-form symmetries and the associated anyon theories lead to a partial classification under two-way connectivity by quasilocal quantum channels. This allows us to establish mixed-state topological orders that are intrinsically mixed, i.e., that have no ground-state counterpart. We provide a wide range of examples based on topological subsystem codes, decohering G -graded string-net models, and “classically gauging” symmetry-enriched topological orders. One of our main examples is an Ising string-net model under the influence of dephasing noise. We study the resulting space of locally indistinguishable states and compute the modular transformations within a particular coherent space. Based on our examples, we identify two possible effects of quasilocal quantum channels on anyon theories: (1) anyons can be incoherently proliferated—thus reducing to a commutant of the proliferated anyons, or (2) the system can be “classically gauged,” resulting in the symmetrization of anyons and an extension by transparent bosons. Given these two mechanisms, we conjecture that mixed-state topological orders are classified by premodular anyon theories, i.e., those for which the braiding relations may be degenerate.
This paper aims to provide a unified physical origin explanation for topological order that transcends the Landau paradigm, starting from principled axioms.Based on two fundamental axioms—Information Conservation (A1) and Finite-StepComputability (A2)—we argue that topological order is not an accidental productof specific models, but rather a macroscopic quantum structure that inevitablyemerges from quantum many-body systems under the joint action of A1 (globalinformation constraints) and A2 (local interaction rules) in order to achieve robustnon-local information storage and processing. This paper systematically demonstrates that core features such as topological degeneracy, anyonic fractional statistics, and topological entanglement entropy can be logically derived from these twoaxioms, and integrates topological order with many-body dynamics from a unifiedspatiotemporal perspective, further proposing numerical and experimental testingschemes. This framework provides a new principled foundation for understandingthe nature, classification, and role of topological order in quantum computing andquantum gravity
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