Anomaly in open quantum systems and its implications on mixed-state quantum phases

In this paper, we develop a systematic approach to characterize the 't Hooft anomaly in open quantum systems. Owing to nontrivial couplings to the environment, symmetries in such systems manifest as either strong or weak type. By representing their symmetry transformation through superoperators, we incorporate them in a unified framework that enables a direct calculation of their anomalies. In the case where the full symmetry group is $K\times G$, with $K$ the strong symmetry and $G$ the weak symmetry, we find that anomalies of bosonic systems are classified by $H^{d+2}(K\times G,U(1))/H^{d+2}(G,U(1))$ in $d$ spatial dimensions. To illustrate the power of anomalies in open quantum systems, we generally prove that anomaly must lead to nontrivial mixed-state quantum phases as long as the weak symmetry is imposed. Analogous to the ``anomaly matching" condition ensuring nontrivial low-energy physics in closed systems, anomaly also guarantees nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. Notably, we identify a novel $(1+1)$-D mixed-state quantum phase that has no counterpart in closed systems, where the steady state shows no nontrivial correlation function in the bulk, but displays spontaneous symmetry breaking order on the boundary, which is enforced by anomalies. We further establish the general relations between mixed-state anomalies and such unconventional boundary correlation. Finally, we explore the generalization of the ``anomaly inflow" mechanism in open quantum systems. We construct $(1+1)$-D and $(2+1)$-D Lindbladians whose steady states have mixed-state symmetry-protected-topological order in the bulk, with corresponding edge theories characterized by nontrivial anomalies.