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 Einacc as the entanglement that cannot be used for distillation under G-LOCC. We derive a tight bound on Einacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely , where Dω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G. For particular phases such as the Haldane phase, 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 Einacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.
Highlights
Entanglement is the essential resource which allows for tasks beyond the restrictions of local operations and classical communication (LOCC), that are impossible to implement classically, such as quantum teleportation, dense coding and secure cryptography [1,2,3]
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-symmetric LOCC (G-LOCC), and we define the inaccessible entanglement Einacc as the entanglement that cannot be used for distillation under G-LOCC
We study the effect of reduced G-LOCC, so that the local operations commute with some H ⊂ G where G is still the symmetry protecting the symmetry-protected topological (SPT) phase, as opposed to the effective reduced symmetry scheme
Summary
Entanglement is the essential resource which allows for tasks beyond the restrictions of local operations and classical communication (LOCC), that are impossible to implement classically, such as quantum teleportation, dense coding and secure cryptography [1,2,3]. The motivation of this work is to apply insights about entanglement under SSR from the quantum information perspective to SPT phases This as yet unexplored connection is natural to make, as topological phases describe order beyond the Landau description of symmetry breaking and local order parameters, and are instead determined by global patterns of entanglement [32,33,34]. We study the entanglement of SPT ordered systems under local operations commuting with a global onsite symmetry G, which we call G-LOCC. This setting is naturally suggestive of SPT phases which are only defined with respect to the symmetry group G protecting the order.
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