Distilling topological entropy from a single measurement of entanglement on projected systems

Abstract

Entanglement measures find frequent application in the study of topologically ordered systems, where the presence of topological order is reflected in an additional contribution to the entanglement of the system. Obtaining this topological entropy from analytical calculations or numerical simulations is generally difficult due to the fact that it is an order-one correction to leading terms that scale with the size of the system. In order to distill the topological entropy, one resorts to extrapolation as a function of system size or to clever subtraction schemes that allow one to cancel out the leading terms. Both approaches have the disadvantage of requiring multiple (accurate) calculations of the entanglement of the system. Here we propose a modification of conventional entanglement calculations that allows one to obtain the topological entropy of a system from a single measurement of entanglement. In our approach, we replace the conventional trace over the degrees of freedom of a partition of the system with a projection onto a given state (which need not be known). We show that a proper choice of partition and projective measurement allows one to rid the entanglement measures of the typical boundary terms, thus exposing the topological contribution alone. We consider specifically the measures known as von Neumann entropy and entanglement negativity and we discuss their application to both models that exhibit quantum as well as classical topological order.