Symmetry-Protected Topological Entanglement

TL;DR

This paper introduces a new entanglement-based order parameter for identifying Symmetry-Protected Topological phases, demonstrating its invariance within phases and connecting it to traditional string order parameters, with implications for quantum computation.

Contribution

It defines the SPT-entanglement as a novel order parameter for SPT phases protected by Abelian symmetries, proving its invariance and non-zero nature in non-trivial phases, and linking it to string order parameters.

Findings

SPT-entanglement remains constant within a given SPT phase.

It is zero in trivial phases and non-zero in non-trivial phases.

The order parameter is invariant under symmetry-respecting low-depth quantum circuits.

Abstract

We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the "SPT-entanglement", is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A, B and the boundaries of the system. In the case of 1-dimensional systems we prove that in the limit where A and B are large and far from each other compared to the correlation length, the SPT-entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases. Moreover, we show that the SPT-entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT-entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to a new algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.