Quantum non-local games: Quantum relations, projection lattices and rule operators

• URL: http://arxiv.org/abs/2608.30507v1
• Date: Mon, 31 Aug 2026 09:41:14 GMT
• Title: Quantum non-local games: Quantum relations, projection lattices and rule operators
• Abstract summary: (mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.<n>(mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.<n>(mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.<n>(mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.
• Score: 51.56484100374058
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum non-local games with quantum inputs or outputs have been formulated in several different languages, including rank-one and quantum XOR games, support maps between projection lattices, probabilistic quantum hypergraphs, and Frobenius-algebraic rule operators on finite quantum sets. We give a unified operator-algebraic comparison of these models by assigning to each rule its winning transformation space: the operator space of transformations accepted with certainty. We introduce \(\mathcal R\)-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra \(\mathcal R\). Their winning transformation spaces are precisely operator spaces with a natural bimodule structure, equivalently rectangular quantum relations. We compare this formalism with projection-lattice games, hypergraph quantum games, and the graphical rule-operator definition. Projection-lattice games capture exactly the reflexive winning bimodules. Hypergraph quantum games admit value-preserving projection-test realisations, and, after passing to perfect transformations, describe the same reflexive part as projection-lattice games. Using Daws' technique, we also translate rule operators to projections in tensor products of finite-dimensional von Neumann algebras. This identification places graphical rules in the same operator-bimodule framework and preserves both values and perfectness. Finally, we compare concurrency with the synchronicity conditions of Goldberg and of Bochniak--Kasprzak--Sołtan. The framework is illustrated by classical, rank-one, quantum XOR, colouring, and quantum graph homomorphism and isomorphism games.

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