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 Rényi 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.

Figures (6)

• Figure 1: Behavior of $\mathcal{M}_2$ for the SSH/dimerized XX model in Eq. \ref{['eqn:Hamiltonian_SSH']} as a function of the dimerization parameter $\delta$ for different system sizes under OBCs. The maximum lies in the SPTP and shifts toward the critical point $\delta = 0$ as the chain length increases.
• Figure 2: Behavior of $\Delta \mathcal{M}_2(\delta) = \mathcal{M}_2(\delta) - \mathcal{M}_2(-\delta)$, as function of $\delta>0$ for the model in Eq. \ref{['eqn:Hamiltonian_SSH']}. For small $L$, $\Delta \mathcal{M}_2$ shows a size dependence, but as $L$ increases, the curves start to converge to a non-constant limiting function of $\delta$. This finite, parameter-dependent asymmetry is inconsistent with the behavior of a genuine topological invariant. The self-dual point $\delta=0$ is excluded, since $\Delta \mathcal{M}_2=0$ by definition.
• Figure 3: Behavior of the magic $\mathcal{M}_2$ for the Cluster Ising model in Eq. \ref{['eqn:Hamiltonian_CI']} as a function of the parameter $\lambda$.
• Figure 4: Behavior of the magic difference between dual points $\lambda \leftrightarrow 1/\lambda$ for the cluster-Ising model in Eq. \ref{['eqn:Hamiltonian_CI']} as a function of $\lambda$, with OBCs. For large system sizes $L$, the curve converges to a finite, $\lambda$-dependent value, indicating a non-topological origin.
• Figure 5: Behavior of $\mathcal{M}_2$ for the Ising model in eq. \ref{['eq:Hamiltoniam_Ising']}, as a function of the transverse magnetic field $h$, with OBCs. Although this model does not support topological order, quantum magic behaves surprisingly similarly to the topological model considered before.