Entanglement entropy in gauge theories and the holographic principle for electric strings

Abstract

We consider quantum entanglement between gauge fields in some region of space A and its complement B. It is argued that the Hilbert space of physical states of gauge theories cannot be decomposed into a direct product H A ⊗ H B of Hilbert spaces of states localized in A and B. The reason is that elementary excitations in gauge theories—electric strings—are associated with closed loops rather than points in space, and there are closed loops which belong both to A and B. Direct product structure and hence the reduction procedure with respect to the fields in B can only be defined if the Hilbert space of physical states is extended by including the states of electric strings which can open on the boundary of A. The positions of string endpoints on this boundary are the additional degrees of freedom which also contribute to the entanglement entropy. We explicitly demonstrate this for the confining and deconfining phases of the three-dimensional Z 2 lattice gauge theory both numerically and using a simple trial ground state wave function. The entanglement entropy appears to be saturated almost completely by the entropy of string endpoints, thus reminding of a “holographic principle” in quantum gravity and AdS/CFT correspondence.