  We study relative entropy in QFT, comparing the vacuum state to a special
family of purifications determined by an input state and constructed using
relative modular flow. We use this to prove a conjecture by Wall that relates
the shape derivative of relative entropy to a variational expression over the
averaged null energy of possible purifications. This variational expression can
be used to easily prove the quantum null energy condition. We formulate Wall's
conjecture as a theorem pertaining to operator algebras satisfying the
properties of a half-sided modular inclusion, with the additional assumption
that the input state has finite averaged null energy. We also give a new
derivation of the strong superadditivity property of relative entropy in this
context. We speculate about possible connections to the recent methods used to
strengthen monotonicity of relative entropy with recovery maps.
