Inaccessible entanglement in symmetry protected topological phases

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 groupG, which we callG-LOCC, and we define the inaccessible entanglement E(inacc)as the entanglement that cannot be used for distillation underG-LOCC. We derive a tight bound on E(inacc)which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely log(D-omega(2)) <= E-inacc <= log(vertical bar G vertical bar), whereD(omega)is the topologically protected edge mode degeneracy of the SPT phase omega with symmetryG. For particular phases such as the Haldane phase,D omega = root|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.