Relative Entropy of Coherent States on General CCR Algebras

For a subalgebra of a generic CCR algebra, we consider the relative entropy between a general (not necessarily pure) quasifree state and a coherent excitationthereof. 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.

Automated summary

The study examines the relative entropy between a general quasifree state and a coherent excitation within a subalgebra of a generic CCR algebra, presenting a unified formula based on single-particle modular data. Changes in relative entropy along subalgebras arising from an increasing family of symplectic subspaces are investigated, with convexity of entropy replaced by lower estimates for the second derivative. The assumption of subspaces in differential modular position is crucial, and the results are illustrated in examples such as thermal states for the conformal U(1)-current.