Discrete Max-Focusing

TL;DR

This paper introduces a discrete nonexpansion concept in quantum focusing, simplifying the axiomatic structure of semiclassical gravity and deriving key bounds and conditions without relying on continuous values or von Neumann entropy.

Contribution

It formulates a discrete quantum nonexpansion applicable at non-smooth points, unifies several quantum bounds under a single conjecture, and establishes new entropy inequalities.

Findings

Discrete max-QFC suffices for QNEC, Bousso bound, and entanglement wedge properties.

New discrete nonexpansion is evaluable at non-smooth points.

Proves strong subadditivity of generalized smooth conditional max and min entropies.

Abstract

The Quantum Focusing Conjecture (QFC) lies at the foundation of holography and semiclassical gravity. The QFC implies the Bousso bound and the Quantum Null Energy Condition (QNEC). The QFC also ensures the consistency of the quantum extremal surface prescription and bulk reconstruction in AdS/CFT. However, the central object in the QFC -- the expansion of lightrays -- is not defined at points where geodesics enter or leave a null congruence. Moreover, the expansion admits three inequivalent quantum extensions in terms of the conditional max, min, and von Neumann entropies. Here we formulate a discrete notion of nonexpansion that can be evaluated even at non-smooth points. Moreover, we show that a single conjecture, the discrete max-QFC, suffices for deriving the QNEC, the Bousso bound, and key properties of both max and min entanglement wedges. Continuous numerical values need not be assigned, nor are the von Neumann or min-versions of the quantum expansion needed. Both our new notion of nonexpansion, and also the properties of conditional max entropies, are inherently asymmetric and outward directed from the input wedge. Thus the framework we develop here reduces and clarifies the axiomatic structure of semiclassical gravity, eliminating redundancies and fixing ambiguities. We also derive a new result: the strong subadditivity of the generalized smooth conditional max and min entropies of entanglement wedges.