  We study the geometric distribution of the relative entropy of a charged
localised state in Quantum Field Theory. With respect to translations, the
second derivative of the vacuum relative entropy is zero out of the charge
localisation support and positive in mean over the support of any single
charge. For a spatial strip, the asymptotic mean entropy density is $\pi E$,
with $E$ the corresponding vacuum charge energy. In a conformal QFT, for a
charge in a ball of radius $r$, the relative entropy is non linear, the
asymptotic mean radial entropy density is $\pi E$ and Bekenstein's bound is
satisfied. We also study the null deformation case. We construct, operator
algebraically, a positive selfadjoint operator that may be interpreted as the
deformation generator, we thus get a rigorous form of the Averaged Null Energy
Condition that holds in full generality. In the one dimensional conformal
$U(1)$-current model, we give a complete and explicit description of the
entropy distribution of a localised charged state in all points of the real
line; in particular, the second derivative of the relative entropy is strictly
positive in all points where the charge density is non zero, thus the Quantum
Null Energy Condition holds here for these states and is not saturated in these
points.
