Abstract
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.
Funder
H2020 European Research Council
Natural Sciences and Engineering Research Council of Canada
Deutsche Forschungsgemeinschaft
Severo Ochoa project
Subject
General Physics and Astronomy,Mathematical Physics,Modelling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
5 articles.
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