Relative entropy of an interval for a massless boson at finite temperature

TL;DR

This paper calculates the relative entropy for a bounded interval in a massless boson at finite temperature, revealing bounds and symmetries, and extends the results to 1+1 dimensions.

Contribution

It introduces a method using PSL(2,R) symmetries to compute the relative entropy for bounded intervals at finite temperature in a massless boson model.

Findings

Derived the modular group and Hamiltonian for bounded intervals at finite temperature.

Established Bekenstein-like and QNEC-like bounds for the relative entropy.

Extended results to the free massless boson in 1+1 dimensions.

Abstract

We compute Araki's relative entropy associated to a bounded interval $I=(a,b)$ between a thermal state and a coherent excitation of itself in the bosonic U(1)-current model, namely the (derivative of the) chiral boson. For this purpose we briefly review some recent results on the entropy of standard subspaces and on the relative entropy of non-pure states such as thermal states. In particular, recently Bostelmann, Cadamuro and Del Vecchio have obtained the relative entropy at finite temperature for the unbounded interval $(-\infty,t)$, using previous results of Borchers and Yngvason, mainly a unitary dilation that provides the modular evolution in the negative half-line. Here we find a unitary rotation in order to make use of the full PSL$(2,\mathbb{R})$ symmetries and obtain the modular group, modular Hamiltonian and the relative entropy $S$ of a bounded interval at finite temperature. Such relative entropy entails both a Bekenstein-like bound and a QNEC-like bound, but violates $S''\geq 0$. Finally, we extend the results to the free massless boson in $1+1$ dimensions with analogous bounds.