Perfect Commuting-Operator Strategies for Linear System Games

TL;DR

This paper explores perfect strategies for linear system games within the commuting-operator model, linking them to potentially infinite-dimensional operator solutions and using group theory to analyze their structure.

Contribution

It extends the understanding of linear system games by characterizing perfect strategies in the commuting-operator model through group-theoretic methods.

Findings

Perfect strategies correspond to possibly-infinite-dimensional operator solutions.

A finitely-presented group is associated with the linear system and the non-commutative equations.

The approach generalizes the tensor-product model to the commuting-operator setting.

Abstract

Linear system games are a generalization of Mermin's magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bob's measurement operators act on a joint Hilbert space, and Alice's operators must commute with Bob's operators. We show that perfect strategies in this model correspond to possibly-infinite-dimensional operator solutions of the non-commutative equations. The proof is based around a finitely-presented group associated to the linear system which arises from the non-commutative equations.