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

Original 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.