  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.
