Linearity of Holographic Entanglement Entropy

TL;DR

This paper examines whether holographic entanglement entropy is truly linear in the expectation value of an operator, revealing nonlinearity in certain superpositions and the role of the homology constraint in the bulk.

Contribution

It demonstrates that the homology constraint introduces nonlinearity in the entanglement entropy calculation for superpositions of geometries.

Findings

Linear expectation value approximation holds for simple superpositions

Nonlinearity emerges when superpositions grow exponentially with central charge

The homology constraint enforces nonlinearity in the bulk geometry

Abstract

We consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.