A No-Cloning Trade-off Between Black Hole No-Hair and Horizon Smoothness

TL;DR

This paper establishes a quantitative trade-off between black hole exterior distinguishability (quantum hair) and horizon smoothness, showing that observable exterior quantum hair implies a violation of horizon smoothness under unitarity.

Contribution

It derives a formal inequality linking horizon smoothness violation to the presence of quantum hair, revealing their incompatibility in semiclassical black hole models.

Findings

Exterior distinguishability is bounded by horizon smoothness violation.

Observable quantum hair requires horizon smoothness to be broken.

Pre-existing entanglement enables quantum hair without violating unitarity.

Abstract

The black hole no-hair theorem is traditionally derived from the uniqueness theorems of general relativity. We show that a quantitative form follows from unitarity together with the standard semiclassical assumptions of horizon causality and interior accessibility. For a semiclassical black hole, we prove that the trace distance between exterior states corresponding to two same-charge infalling states is bounded by $2\sqrt{2\varepsilon}$, where $\varepsilon$ quantifies the diamond norm departure of the interior channel from a perfect isometry which is a quantitative measure of horizon-smoothness violation that upper-bounds $1 - F_I$, where $F_I$ is the interior fidelity capturing how faithfully the infalling state is retained. Inverting this relation yields a trade-off inequality, $\varepsilon \geq D_{\max}^2/8$, between the maximum exterior distinguishability $D_{\max}$ and the degree of horizon smoothness. This establishes that observable exterior quantum hair is quantitatively incompatible with exact horizon smoothness under unitary evolution: any model predicting nonzero exterior hair must violate the equivalence principle at the horizon by a quantifiable amount. Pre-existing entanglement with the infalling system is the only channel for quantum hair compatible with both unitarity and horizon smoothness.