No off-diagonal quantum focusing for Rényi divergences

Abstract

The quantum focusing conjecture (QFC) is a mathematical expression of the idea that semiclassical gravity remains universally attractive. In its non-gravitational limit, the diagonal inequality is the quantum null energy condition (QNEC), which requires convexity of relative entropy with respect to the vacuum under null deformations. Rényi divergences generalize relative entropy, and positive results for the diagonal Rényi QNEC raise the question of whether they also satisfy the off-diagonal inequalities required by a full focusing principle. We show that classical correlations between distinct null generators obstruct this extension for any non-affine Rényi-type divergence satisfying data processing, tensor additivity, and matched classical-quantum conditioning.

arXiv comment8 pages, 1 figure; v2: expanded introduction and discussion, and coherent-state SRD counterexample added