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

Entropy distribution of localised states