  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.
