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

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 $S^{\prime \prime }\ge 0$. Finally, we extend the results to the free massless boson in $1+1$ dimensions with analogous bounds.

22 pages, 2 figures

References in corpus (3)

• Relative entropy and curved spacetimes
• Entropy of Coherent Excitations
• Modular operator for null plane algebras in free fields

Cited by in corpus (5)

• Modular Hamiltonian for de Sitter diamonds
• Modular Hamiltonian for fermions of small mass
• A numerical analysis of Araki-Uhlmann relative entropy in Quantum Field Theory
• Relative entropy of single-mode squeezed states in Quantum Field Theory
• Investigating the Araki-Uhlmann relative entropy between two coherent states in relativistic Quantum Field Theory