Towards a classification of mixed-state topological orders in two dimensions

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 quasi-local 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 quasi-local 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.