Resource complexity of Symmetry Protected Topological phases

Abstract

We pursue the identification of quantum resources carried by topological order, by evaluating quantum magic, quantified through the rank-$2$ Stabilizer R\'enyi entropy $\mathcal{M}_2$, in one-dimensional systems hosting symmetry-protected topological phases (SPTP). Focusing on models with an exact duality between an SPTP and a trivial one, namely the dimerized XX and the Cluster-Ising chains, we show that dual points exhibit identical amounts of magic, even thought they belong to distinct topological sectors. A subextensive asymmetry arises only under open boundary conditions, where edge effects break the duality, but this correction is non-topological and depends on microscopic parameters. These results stand in contrast to the case of topological frustration, where delocalized excitations enhance the magic logarithmically with system size. They also complement recent analyses in the literature, showing that the total magic is largely insensitive to the presence of topological order, hence suggesting that topological order is not necessarily a genuine computational resource.