Inaccessible entanglement in symmetry protected topological phases
Abstract
We study the entanglement structure of symmetryprotected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that nontrivial SPT phases in onedimension 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 GLOCC, and we define the inaccessible entanglement E_{inacc} as the entanglement that cannot be used for distillation under GLOCC. We derive a tight bound on E_{inacc} which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely $\mathrm{log}\left({D}_{\omega }^{2}\right){\leqslant}{E}_{\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{c}\mathrm{c}}{\leqslant}\mathrm{log}\left(\vert G\vert \right)$?> , 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, ${D}_{\omega }=\sqrt{\vert G\vert }$?> 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 August 2020
 DOI:
 10.1088/17518121/ab98c7
 arXiv:
 arXiv:2003.06830
 Bibcode:
 2020JPhA...53G5302D
 Keywords:

 quantum information;
 symmetry protected topological phases;
 tensor networks;
 quantum entanglement;
 Quantum Physics;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 16 pages, 7 figures