Inaccessible entanglement in symmetry protected topological phases

Abstract

We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group G, which we call G-LOCC, and we define the inaccessible entanglement E inacc as the entanglement that cannot be used for distillation under G-LOCC. We derive a tight bound on E inacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely log ( D ω 2 ) ⩽ E i n a c c ⩽ log ( | G | ) , where D ω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G. For particular phases such as the Haldane phase, D ω = | G | so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of E inacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.

Rapid thermalization of spin chain commuting Hamiltonians

“…Our result does not apply however in the presence of a strong symmetry, a key condition identified in [35,37,39,41,44,45] to preserve SPT in open quantum systems, which emphasizes even more the totally different behavior between weak and strong symmetries in the context of SPT phases in non-zero temperature regimes. The proof of our main result has two main steps.…”

“…The result also has implications in the context of Symmetry Protected Topological (SPT) phases [31][32][33]. There has been a quite intensive study of SPT in open quantum systems [34][35][36][37][38][39][40][41][42][43][44][45][46] and there is yet no consensus on what is the fate of SPT in the presence of temperature (see e.g. [36,41] for negative results and [40,47] for positive ones).…”

Symmetry-resolved entanglement in symmetry-protected topological phases

“…In 1D, similar results can be obtained somewhat more easily from matrix product state (MPS) considerations 4,14 . Our approach, however, offers some hope of generalization to higher dimensions.…”

“…However the degeneracy always persists in the topological sector. This implies a minimal value of the entanglement entropy: Since there are at least |G| eigenvalues ≤ 1/|G|, the entropy of nontrivial SPTs has a lower bound at 14 S non trivial SPT ≥ log |G|, (equipartition) (15) as illustrated by the dashed lines in the histogram plots in Fig. 3(b).…”

“…The method allows to study nonuniversal features that vary withing topological phases, by studying co-boundary transformations. While the entanglement entropy itself in nonuniversal, it has minimal value which is a property of each SPT 14 , and is intimately connected with SRE equipartition.…”