  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 $\log(D_\omega^2) \leq
E_{inacc} \leq \log(|G|)$, where $D_\omega$ is the topologically protected edge
mode degeneracy of the SPT phase $\omega$ with symmetry $G$. For particular
phases such as the Haldane phase, $D_\omega = \sqrt{|G|}$ 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.
