Complementarity and the unitarity of the black hole $S$-matrix

TL;DR

This paper investigates the unitarity of the black hole S-matrix within a holographic model involving postselection, showing it remains approximately unitary under certain conditions and has polynomial complexity.

Contribution

It demonstrates that the black hole S-matrix is nearly unitary and computationally manageable under assumptions about the internal dynamics and the infaller's operations.

Findings

S-matrix unitarity is maintained to high precision with suitable conditions.

Black hole S-matrix has polynomial computational complexity.

Unitarity is preserved despite postselection effects.

Abstract

Recently, Akers et al. proposed a non-isometric holographic map from the interior of a black hole to its exterior. Within this model, we study properties of the black hole $S$-matrix, which are in principle accessible to observers who stay outside the black hole. Specifically, we investigate a scenario in which an infalling agent interacts with radiation both outside and inside the black hole. Because the holographic map involves postselection, the unitarity of the $S$-matrix is not guaranteed in this scenario, but we find that unitarity is satisfied to very high precision if suitable conditions are met. If the internal black hole dynamics is described by a pseudorandom unitary transformation, and if the operations performed by the infaller have computational complexity scaling polynomially with the black hole entropy, then the $S$-matrix is unitary up to corrections that are superpolynomially small in the black hole entropy. Furthermore, while in principle quantum computation assisted by postselection can be very powerful, we find under similar assumptions that the $S$-matrix of an evaporating black hole has polynomial computational complexity.