Beyond islands: A free probabilistic approach

TL;DR

This paper introduces a free probabilistic approach to compute radiation entropy in quantum gravity models, providing a new analytical framework that extends beyond traditional island formula limitations.

Contribution

It develops a free convolution method for radiation entropy calculation, connecting free probability, random matrix theory, and quantum extremal surfaces in a novel way.

Findings

The free convolution formula accurately computes radiation entropy where island formula fails.

The approach models quantum information as free random variables in a von Neumann algebra.

The PSSY model is shown to be a generalized Page's model via random matrix theory.

Abstract

We give a free probabilistic proposal to compute the fine-grained radiation entropy for an arbitrary bulk radiation state, in the context of the Penington-Shenker-Stanford-Yang (PSSY) model where the gravitational path integral can be implemented with full control. We observe that the replica trick gravitational path integral is combinatorially matching the free multiplicative convolution between the spectra of the gravitational sector and the matter sector respectively. The convolution formula computes the radiation entropy accurately even in cases when the island formula fails to apply. It also helps to justify this gravitational replica trick as a soluble Hausdorff moment problem. We then work out how the free convolution formula can be evaluated using free harmonic analysis, which also gives a new free probabilistic treatment of resolving the separable sample covariance matrix spectrum. The free convolution formula suggests that the quantum information encoded in competing quantum extremal surfaces can be modelled as free random variables in a finite von Neumann algebra. Using the close tie between free probability and random matrix theory, we show that the PSSY model can be described as a random matrix model that is essentially a generalization of Page's model. It is then manifest that the island formula is only applicable when the convolution factorizes in regimes characterized by the one-shot entropies. We further show that the convolution formula can be reorganized to a generalized entropy formula in terms of the relative entropy.