On the stabilizer complexity of Hawking radiation

TL;DR

This paper investigates the stabilizer complexity of Hawking radiation in black hole evaporation models by calculating Wigner negativity, revealing a transition from low to exponentially large negativity at the Page time, with implications for holography.

Contribution

It introduces a universal formula for Wigner negativity in black hole models and connects stabilizer complexity to geometric features of holographic states.

Findings

Negativity is O(1) before the Page transition.

Negativity becomes exponentially large after the Page transition.

A geometric formula relates negativity to entanglement wedge features.

Abstract

We study the complexity of Hawking radiation for an evaporating black hole from the perspective of the stabilizer theory of quantum computation. Specifically, we calculate Wigner negativity -- a magic monotone which can be interpreted as a measure of the stabilizer complexity, or equivalently, the complexity of classical simulation -- in various toy models for evaporating black holes. We first calculate the Wigner negativity of Hawking radiation in the PSSY model directly using the gravitational path integral, and show that the negativity is $O(1)$ before the Page transition, but becomes exponentially large past the Page transition. We also derive a universal, information theoretic formula for the negativity which interpolates between the two extremes. We then study the Wigner negativity of radiation in a dynamical model of black hole evaporation. In this case, the negativity shows a sharp spike at early times resulting from the coupling between the black hole and the radiation system, but at late times when the system settles down, we find that the negativity satisfies the same universal formula as in the PSSY model. Finally, we also propose a geometric formula for Wigner negativity in general holographic states using intuition from fixed area states and random tensor networks, and argue that a python's lunch in the entanglement wedge implies a stabilizer complexity which is exponentially large in $\frac{1}{8G_N}$ times the difference between the areas corresponding to the outermost and minimal extremal surfaces.