# De Finetti theorems and entanglement in large-N theories and gravity

## TL;DR

This paper explores the application of de Finetti theorems to large-N quantum gravity theories, providing insights into entanglement, decoherence, and the structure of symmetric states in models like SYK and gauge theories.

## Contribution

It extends de Finetti theorems to large-N theories and applies them to analyze entanglement and decoherence in quantum gravity models, unifying these concepts.

## Key findings

- Identified a gauge invariant operator for entanglement entropy in large-N theories.
- Applied de Finetti theorems to SYK and gauge theories for a rigorous understanding of entanglement.
- Unified the understanding of Schmidt decompositions and decoherence in large-N contexts.

## Abstract

The de Finetti theorem and its extensions concern the structure of multipartite probability distributions with certain symmetry properties, the paradigmatic original example being permutation symmetry. These theorems assert that such symmetric distributions are well approximated by convex combinations of uncorrelated ones. In this article, we apply de Finetti theorems to quantum gravity theories, such as the Sachdev-Ye-Kitaev (SYK) model or large-N vector and gauge theories. For SYK we put recent studies of information/entanglement dynamics in a general and rigorous basis. For vector and gauge theories, we find a gauge invariant operator whose expectation value provides the leading term in the entanglement entropy in all states close enough to a given classical state. These results can be unified through a generic statement about the nature of Schmidt decompositions and decoherence in large-N theories. In the reverse direction, we extend de Finetti theorems in various ways and provide an independent approach to the theorems only based on the large-N properties of the gauge invariant coherence group.