DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a lattice L⊆ Rⁿ satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebras A of operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to Autbr(DHR(A)) , containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry D , we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center Z(D) . We use this to show that, for the double spin-flip action Z/2Z× Z/2Z C²⊗ C² , the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy of S₃ ; hence, it is non-abelian, in contrast to the case with no symmetry.