Symmetry-Protected Topological Entanglement

Understand

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.