Highly Entangled Stationary States from Strong Symmetries

Abstract

We find that the presence of strong non-Abelian symmetries can lead to highly entangled stationary states even for unital quantum channels. We derive exact expressions for the bipartite logarithmic negativity, Rényi 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 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>SU</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>N</a:mi> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:math> symmetries, both logarithmic and Rényi negativities scale logarithmically with system size. Finally, we prove that, while Rényi negativities with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>n</e:mi> <e:mo>&gt;</e:mo> <e:mn>2</e:mn> </e:math> scale logarithmically with system size, the logarithmic negativity (as well as generalized Rényi negativities with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>n</g:mi> <g:mo>&lt;</g:mo> <g:mn>2</g:mn> </g:math> ) 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.