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

Abstract

We compute Araki's relative entropy associated with 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 nonpure 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 (-∞,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,R) symmetries and obtain the modular group, the 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′′≥0 (derivative with respect to the length of the interval, with its center fixed). Finally, we extend the results to the free massless boson in 1+1 dimensions with analogous bounds.