On the stabilizer complexity of Hawking radiation

TL;DR

The paper investigates the stabilizer complexity of Hawking radiation by computing Wigner negativity in holographic toy models of black hole evaporation. Using the PSSY model and a dynamical BR evaporation setup, it shows that negativity remains $O(1)$ before the Page time but becomes exponentially large after, with a universal information-theoretic formula governing the late-time behavior. A gravity-based, basis-independent interpolation formula and a Haar-integrated, integral-representation approach reveal that the negativity is controlled by second Rényi entropy data and bulk wormhole saddles, connecting quantum randomness, chaos, and computability. The authors further propose a geometric formula for negativity in general holographic states and argue that a python's lunch in the entanglement wedge implies exponentially large stabilizer complexity, linking boundary complexity to bulk geometric data through fixed-area/RNT constructs. These results illuminate how gravitational nonlocality and entanglement structure translate into computational complexity of simulating Hawking radiation.

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.

Figures (10)

• Figure 1: (a): Boundary condition to evaluate the overlap $\langle \psi_{\ell}|\psi_k\rangle$:The solid line represents the asymptotic boundary of Euclidean $\text{AdS}_2$ with renormalized length $\beta$. The arrow indicates the direction of time from ket to bra. Indices attached to the dashed lines represent the states of the EOW branes. The boundary condition suggests that EOW branes of type $\ell$ or $k$ should intersect the asymptotic $\text{AdS}_2$ boundary at the appropriate endpoints. (b): The bulk gravity path integral corresponding to the overlap, subject to the boundary conditions illustrated on the left.
• Figure 2: (a) The boundary conditions for computing the Wigner function $W$. (b) The gravitational path integral with the given boundary conditions: the same EOW brane gives $\delta_{kl}$, and so the diagram evaluates to $e^{S_0} Z_1\; \text{Tr}(A)$
• Figure 3: Schematic diagram representing the boundary conditions used to evaluate $W^{2n}$, shown here for $n=3$. Six copies of figure \ref{['fig:W']} are arranged in a circular layout, with each $A$ denoting the insertion of a phase point operator. A gravitational path integral with these boundary conditions generally admits multiple saddle point solutions, corresponding to different ways of filling in the bulk region.
• Figure 4: The replica trick for $n=3$. (a) In the regime $D\ll e^{S_0}$, the fully disconnected geometry dominates the gravitational path integral. Each disconnected piece contributes $e^{S_0} Z_1 \text{Tr}(A)$, leading to a total contribution of $(e^{S_0}Z_1)^6$. (b) In contrast, for $D\gg e^{S_0}$, the dominant saddle is a pairwise connected geometry. Each connected component connects two asymptotic boundaries. Each pair gives a factor $\text{Tr}(A^2)=D$ from the sum over EOW brane indices, and the full diagram evaluates to $(D e^{S_0}Z_2)^3$.
• Figure 5: All possible geometries (at disk level) contributing to the gravitational path integral for $\overline{W^3}$.