  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 briefly discuss mixed anomaly
involving both strong and weak symmetries, including systems constrained by the
Lieb-Schultz-Mattis type of anomaly.
