Mixed-State Quantum Anomaly and Multipartite Entanglement

Abstract

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 Hd+2[G,U(1)]. We show that any mixed state ρ that is strongly symmetric under G, in the sense that Gρ∝ρ 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≤1 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 find 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 a 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.