A sharper proof links quantum energy to information at black-hole horizons

An energy bound expressed through information

Quantum physics complicates the classical idea that energy density should always be nonnegative. The quantum null energy condition, or QNEC, instead constrains energy along a lightlike direction using changes in quantum entropy. It connects what a quantum field carries with how its information is distributed across a boundary.

In a preprint submitted to arXiv on November 9, 2026, Thomas Faulkner strengthens a proof of this condition within a particular mathematical framework. The work is theoretical: it presents theorems and proofs, not experimental measurements or observations of black holes.

Its central quantity is relative entropy, which compares a quantum state with a reference state using the observables available in a chosen region. For the stationary-black-hole interpretation, the reference is the Hartle–Hawking state, describing thermal equilibrium. The region is the exterior associated with a chosen cut, or cross-section, of the horizon.

Moving that cut changes which observables belong to the region. The paper proves that relative entropy is a convex function of the cut parameter within the stated framework. This is the algebraic form of the QNEC studied here.

Why convexity is stronger than information loss

The construction uses a nested family of observable algebras: mathematical collections of the measurements available in successive regions. As the cut parameter increases, the accessible algebra becomes smaller.

Relative entropy already obeys a familiar information principle: restricting access to observables cannot increase distinguishability from the reference state. Consequently, relative entropy cannot increase along this family.

Convexity adds another restriction. Between two cuts, the relative-entropy curve lies at or below the straight line joining its endpoint values. Wherever its derivative exists, its slope cannot decrease. Because the curve is non-increasing, this means a negative slope becomes less negative or stays constant.

The relevant structure is called a half-sided translation or half-sided modular inclusion. It combines nested observables, translations generated by a positive operator, and a special evolution associated with the reference state called modular flow. These are substantive mathematical requirements. Removing an assumption about the states does not mean proving the result without any framework assumptions.

Closing the finite-energy gap

Earlier algebraic proofs required the regional state to admit a purification with finite averaged null energy. A purification represents the state as part of a larger pure quantum state while preserving all its regional measurement predictions.

The difficulty was not merely finding many states that satisfy this requirement. Relative entropy can be unbounded, so proving an inequality for a dense collection of states does not automatically establish it for their limits.

Faulkner constructs approximating states with finite relative entropy and finite quantum expansion, while controlling how both quantities approach their original values. Here, quantum expansion is defined by minimizing the expectation of the positive translation generator over compatible purifications.

The resulting theorem establishes convexity without the earlier finite-energy-purification assumption. It also relates changes in relative entropy to the integral of quantum expansion. Where relative entropy is finite on an open interval, it is continuously differentiable. Infinite relative-entropy values are handled explicitly rather than silently excluded.

A useful consequence is that finite relative entropy at one cut guarantees finite quantum expansion at every strictly later cut. A possible exceptional boundary cut therefore does not invalidate the subsequent evolution.

Replacing an equilibrium state with a weight

The paper also generalizes the reference object from a normalized state to a weight. A weight assigns nonnegative values to positive observables but may assign infinity; it need not describe a normalized probability distribution.

This matters because the quantum-field description proposed for an evaporating black hole can live in a different Hilbert-space representation from the equilibrium Hartle–Hawking state. A suitable weight could provide a reference where that equilibrium state is unavailable as a normalizable state.

The mathematical extension is more delicate. Relative entropy against a weight can be negative or infinite in either direction. The general theorem establishes convexity on intervals where relative entropy is finite. Stronger conclusions connecting its derivative to quantum expansion retain an additional finiteness requirement.

What remains unresolved

The evaporating-black-hole application is motivation, not an established consequence. The paper offers non-rigorous physical arguments for a required property called semifiniteness on the relevant subalgebras, but does not prove it there.

That discussion also treats quantum fields on a fixed Schwarzschild background, neglecting their back-reaction on spacetime. It is not a calculation of a black hole shrinking self-consistently through evaporation.

The advance is therefore a strengthened mathematical energy–information constraint and a broader reference framework. Whether the genuinely new weight setting describes the proposed evaporating-black-hole system remains open.