Relative entropy of coherent states on general CCR algebras

TL;DR

This paper derives a unified formula for the relative entropy between quasifree states and their coherent excitations on CCR algebras, analyzing its behavior along symplectic subspaces with applications to conformal field theory.

Contribution

It provides a general formula for relative entropy in CCR algebras using single-particle modular data and studies its properties under subspace variations.

Findings

Unified entropy formula in terms of modular data

Convexity replaced with lower bounds for second derivatives

Applications to thermal states in conformal field theory

Abstract

For a subalgebra of a generic CCR algebra, we consider the relative entropy between a general (not necessarily pure) quasifree state and a coherent excitation thereof. We give a unified formula for this entropy in terms of single-particle modular data. Further, we investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces; here convexity of the entropy (as usually considered for the Quantum Null Energy Condition) is replaced with lower estimates for the second derivative, composed of "bulk terms" and "boundary terms". Our main assumption is that the subspaces are in differential modular position, a regularity condition that generalizes the usual notion of half-sided modular inclusions. We illustrate our results in relevant examples, including thermal states for the conformal $U(1)$-current.