  We study quantum many-body mixed states with a symmetry from the perspective
of separability, i.e., whether a mixed state can be expressed as an ensemble of
short-range entangled (SRE) symmetric pure states. We provide evidence for
'symmetry-enforced separability transitions' in a variety of states, where in
one regime the mixed state is expressible as a convex sum of symmetric SRE pure
states, while in the other regime, such a representation is not feasible. We
first discuss Gibbs state of Hamiltonians that exhibit spontaneous breaking of
a discrete symmetry, and argue that the associated thermal phase transition can
be thought of as a symmetry-enforced separability transition. Next, we study
cluster states in various dimensions subjected to local decoherence, and
identify several distinct mixed-state phases and associated separability phase
transitions, which also provides an alternate perspective on recently discussed
'average SPT order'. We also study decohered p+ip superconductors, and find
that if the decoherence breaks the fermion parity explicitly, then the
resulting mixed state can be expressed as a convex sum of non-chiral states,
while a fermion-parity preserving decoherence results in a phase transition at
a non-zero threshold that corresponds to spontaneous breaking of fermion
parity. Finally, we briefly discuss systems that satisfy NLTS (no low-energy
trivial state) property, such as the recently discovered good LDPC codes, and
argue that the Gibbs state of such systems exhibits a temperature-tuned
separability transition.
