Linear system games are a generalization of Mermins 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 Bobs measurement operators act on a joint Hilbert space, and Alices operators must commute with Bobs 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.
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