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.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.

math.OACross-listedarXiv:2608.30507Sep 1, 20260 citations

Memory Functions, Projection Operators, and the Defect Technique

V. M. Kenkre

Lecture notes in physicsJan 1, 202123 citations

On Time-Dependent Projectors and a Generalization of the Thermodynamical Approach in the Theory of Open Quantum Systems

Khazret Shumafovich Meretukov, Alexander Evgen’evich Teretenkov

Abstract We develop a consistent perturbative technique for obtaining a time-local linear master equation based on projection methods in the case where the projection operator depends on time. Then we introduce a generalization of the Kawasaki–Gunton projection operator, which allows us to use this technique to derive, generally speaking, nonlinear master equations in the case of arbitrary ansatzes consistent with some set of observables. Most of the results obtained are of a very general nature, but when discussing them, we put emphasis on the application of these results to the theory of open quantum systems.Abstract We develop a consistent perturbative technique for obtaining a time-local linear master equation based on projection methods in the case where the projection operator depends on time. Then we introduce a generalization of the Kawasaki–Gunton projection operator, which allows us to use this technique to derive, generally speaking, nonlinear master equations in the case of arbitrary ansatzes consistent with some set of observables. Most of the results obtained are of a very general nature, but when discussing them, we put emphasis on the application of these results to the theory of open quantum systems.

arXiv:2307.00607Proceedings of the Steklov Institute of MathematicsMar 1, 20244 citations

\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.