# Bisynchronous Games and Factorizable Maps

## TL;DR

This paper introduces bisynchronous non-local games, explores their connection to quantum groups, and shows how they relate to factorizable maps in quantum information theory.

## Contribution

It defines bisynchronous games, links them to quantum groups, and demonstrates that associated maps are factorizable, advancing understanding of non-local game symmetries.

## Key findings

- Bisynchronous densities correspond to traces on quantum permutation groups.
- Each bisynchronous density yields a factorizable completely positive map.
- The study reveals symmetries and algebraic structures in bisynchronous games.

## Abstract

We introduce a new class of non-local games, and corresponding densities, which we call bisynchronous. Bisynchronous games are a subclass of synchronous games and exhibit many interesting symmetries when the algebra of the game is considered. We develop a close connection between these non-local games and the theory of quantum groups which recently surfaced in studies of graph isomorphism games. When the number of inputs is equal to the number of outputs, we prove that a bisynchronous density arises from a trace on the quantum permutation group. Each bisynchronous density gives rise to a completely positive map and we prove that these maps are factorizable maps.