Quantum non-local games: Quantum relations, projection lattices and rule operators
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 ℛ-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra 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.
Related
- Concept lattices of Q-relations via diagonal and back-diagonal categories
- On the transformation of unbounded convergent sequences by linear operators on Banach lattices
- Applications of the Projection Characterization and Symmetry Bootstrap for Operators that are Nearby Commuting Operators
- DW-L-Weakly Compact Operators Between Banach Lattices
- Norm attainment for multilinear operators and polynomials on Banach Spaces and Banach lattices
BibTeX
@misc{chatzinikolaou2026quantum,
title = {{Quantum non-local games: Quantum relations, projection lattices and rule operators}},
author = {Alexandros Chatzinikolaou},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30507},
eprint = {2608.30507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30507},
}
RIS
TY - UNPB
TI - Quantum non-local games: Quantum relations, projection lattices and rule operators
AU - Alexandros Chatzinikolaou
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30507
UR - https://arxiv.org/abs/2608.30507
AB - 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 ℛ-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra 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.
ER -