A finite entanglement entropy and the c-theorem

TL;DR

This paper introduces a finite, well-defined measure of entanglement entropy in quantum field theory, constrained by symmetries, and demonstrates its use in establishing an entropic version of the c-theorem in 1+1 dimensions.

Contribution

It defines a finite entanglement measure F(A,B) constrained by symmetries and proves an entropic c-theorem in 1+1 dimensional QFT.

Findings

F(A,B) is uniquely determined in 2D conformal field theories.

F(A,B) provides a finite, meaningful entanglement measure.

An entropic c-theorem is established for 1+1 dimensional QFT.

Abstract

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with non zero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincare symmetry and the mathematical properties of the entropy. In particular, for one component sets in two dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for 1+1 dimensional QFT. We propose this well defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.