Highly entangled stationary states from strong symmetries

TL;DR

This paper demonstrates that strong non-Abelian symmetries in quantum channels lead to highly entangled stationary states, with explicit formulas and bounds for entanglement measures, contrasting with Abelian cases that produce separable states.

Contribution

It provides exact expressions and bounds for entanglement in stationary states under strong symmetries, highlighting differences between Abelian and non-Abelian symmetry effects.

Findings

Non-Abelian symmetries induce logarithmic scaling of entanglement measures.

Abelian symmetries lead to separable stationary states.

Volume law scaling occurs for certain non-Abelian symmetry cases.

Abstract

We find that the presence of strong non-Abelian conserved quantities can lead to highly entangled stationary states even for unital quantum channels. We derive exact expressions for the bipartite logarithmic negativity, R\'enyi negativities, and operator space entanglement for stationary states restricted to one symmetric subspace, with focus on the trivial subspace. We prove that these apply to open quantum evolutions whose commutants, characterizing all strongly conserved quantities, correspond to either the universal enveloping algebra of a Lie algebra or to the Read-Saleur commutants. The latter provides an example of quantum fragmentation, whose dimension is exponentially large in system size. We find a general upper bound for all these quantities given by the logarithm of the dimension of the commutant on the smaller bipartition of the chain. As Abelian examples, we show that strong U($1$) symmetries and classical fragmentation lead to separable stationary states in any symmetric subspace. In contrast, for non-Abelian SU$(N)$ symmetries, both logarithmic and R\'enyi negativities scale logarithmically with system size. Finally, we prove that while R\'enyi negativities with $n>2$ scale logarithmically with system size, the logarithmic negativity (as well as generalized R\'enyi negativities with $n<2$) exhibits a volume law scaling for the Read-Saleur commutants. Our derivations rely on the commutant possessing a Hopf algebra structure in the limit of infinitely large systems, and hence also apply to finite groups and quantum groups.