Recovering the QNEC from the ANEC

Fikret Ceyhan, Thomas Faulkner

Research output: Contribution to journalArticlepeer-review


We study the 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 (ANE) of possible purifications. This variational expression can be used to easily prove the quantum null energy condition (QNEC). 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.

Original languageEnglish (US)
Pages (from-to)999-1045
Number of pages47
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Jul 1 2020

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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