A broader proof links quantum information to horizon geometry

What changes when a boundary moves?

A quantum state can look different depending on which region is accessible. Relative entropy measures how distinguishable that state is from a reference state using observations within the region. Rényi relative entropies provide a family of related measures, controlled by an adjustable order.

In this paper, Thomas Faulkner considers regions described by a mathematical structure called a half-sided modular inclusion. One physical setting is a quantum field outside a black hole horizon: moving a cut along the horizon changes which observables belong to the exterior region.

The question is not merely whether distinguishability decreases as the accessible region shrinks. An existing information principle already guarantees that. The stronger question concerns the shape of the decrease: does the relative entropy form a convex curve, whose downward slope becomes less steep as the cut advances?

A continuous range of orders

Faulkner's November 9, 2026 preprint presents a proof that it does, subject to the paper's mathematical assumptions. The result is the diagonal Rényi quantum null energy condition, or Rényi QNEC.

The paper labels its family with a parameter p. Ordinary relative entropy appears in the limit p = 2; the more usual Rényi order is α = p/2. Earlier results discussed in the paper covered free quantum fields and, for general theories within this algebraic setting, integer values α = 2, 3, and so on.

The new proof covers every finite p greater than 2, including noninteger orders. For example, p = 3 corresponds to α = 1.5. It also connects continuously to the ordinary p = 2 result. At the infinite-order endpoint, the relative entropy is constant along the family of cuts for the states admitted by that endpoint's assumptions.

The theorem requires a normalized state with finite relevant Rényi divergence at an initial cut. Beyond that cut, the divergence is continuous, non-increasing and convex, with a continuous derivative in the interior.

How the proof works

The work is mathematical, not experimental: it reports no measured energy densities, observations of horizons or numerical estimates of an effect.

Its main construction is a weighted conditional norm—a way of assigning a size to a quantum-state vector while accounting for nested sets of observables and a reference state. The associated conditional entropy supplies bounds on how much Rényi relative entropy can change between two cuts.

Faulkner uses Hölder's inequality to establish a convexity property of this conditional norm. He then proves that a generalized “quantum expansion” equals the negative derivative of the relative entropy, divided by 2π. Establishing sufficient differentiability and regularity turns the bounds into the desired convexity theorem. These steps are developed in Sections 3–5.

What the result does not establish

“Diagonal” is an essential restriction. The theorem concerns variation associated with the same location on a horizon cut, not mixed variations at distinct locations. The paper notes existing counterexamples to the corresponding off-diagonal Rényi statement, as well as counterexamples below p = 2. It therefore does not establish an unrestricted Rényi version of quantum focusing.

The black hole interpretation also remains incomplete. At p = 2, the conditional quantity connects to a previously defined dynamical black hole entropy in a semiclassical approximation. Extending that physical interpretation to other orders is an open problem, partly because gravitational backreaction can matter.

The contribution is thus a general proof within a specified mathematical setting, presented as a preprint—not a measurement or a demonstrated new law of black hole thermodynamics.