Mixed-state quantum anomaly and multipartite entanglement

Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here we explore a surprising connection between mixed state entanglement and 't Hooft anomaly. More specifically, we consider lattice systems in $d$ space dimensions with anomalous symmetry $G$ where the anomaly is characterized by an invariant in the group cohomology $H^{d+2}(G,U(1))$. We show that any mixed state $\rho$ that is strongly symmetric under $G$, in the sense that $G\rho\propto\rho$, is necessarily $(d+2)$-nonseparable, i.e. is not the mixture of tensor products of $d+2$ states in the Hilbert space. Furthermore, such states cannot be prepared from any $(d+2)$-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in $d\leq1$, and plausibility arguments in $d>1$. The anomaly-nonseparability connection thus allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in $d=1$ we found an example of intrinsically mixed quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly in general does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states.