Recovering the QNEC from the ANEC

Understand

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.