Quasi-local form for $α$--$z$ Rényi QNEC from fixed-ray escorts

Abstract

The fixed-ray escort integral representation expresses $α$--$z$ Rényi divergence as an average over ordinary relative entropy of a family of escort states. Working in the standard UV-regulated density-matrix description of QFT subregions, we use this representation to derive an escort-averaged entanglement first law, an escort-averaged representation of the $α$--$z$ information kernel, and an escort-averaged Bekenstein-type bound for ball-shaped regions in conformal field theories. For the conjectural $α$--$z$ quantum null energy condition (QNEC), we obtain a quasi-local form in which the null energy is evaluated in an escort-averaged state and is corrected by an escort-transport term encoding the failure of escort formation to commute with restriction to a null-deformed region. The $z=α$ specialization gives a similar quasi-local form of the Rényi QNEC for sandwiched Rényi divergence. We explicitly compute the Rényi QNEC, including the explicit escort transport term, for coherent-state excitations in a free scalar field theory. For the same coherent family, we obtain a positive $α$--$z$ null Hessian, verifying the conjectured diagonal $α$--$z$ QNEC for this family.