A general proof of integer R\'enyi QNEC

TL;DR

This paper proves the integer Re9nyi quantum null energy condition (QNEC) for certain von Neumann algebras, establishing log-convexity of the Kosaki $L^n$ norm of positive functionals under null translation.

Contribution

It provides a general proof of Re9nyi QNEC for all integer parameters $n \u2265 2$ in the context of von Neumann algebras with half-sided modular inclusion.

Findings

Proves Re9nyi QNEC for all integer $n \u2265 2$.

Establishes log-convexity of the Kosaki $L^n$ norm under null translation.

Requires only finiteness of the SRD relative to the vacuum.

Abstract

The R\'enyi quantum null energy condition conjectures that the second null shape variation of the sandwiched R\'enyi divergence (SRD) of an excited state relative to the vacuum is non-negative in local Poincar\'e-invariant quantum field theory, giving a one-parameter generalization of the quantum null energy condition (QNEC). We prove R\'enyi QNEC for all integer R\'enyi parameters $n\geq 2$ for von Neumann algebras carrying a half-sided modular inclusion structure. The only assumption on the excited state is finiteness of its SRD relative to the vacuum. Concretely, for any $\sigma$-finite von Neumann algebra with such an inclusion, we prove log-convexity, under the associated null-translation semigroup, of the Kosaki $L^n$ norm of any normal positive functional with finite $L^n$ norm.