In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value $ω^*$ . We show that the value $ω^*$ can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of $ω^*$ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication $ω_{qow}$ . In particular, we show that $ω_{qow}$ can be computed efficiently and it satisfies a perfect parallel repetition theorem.In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value $ω^*$. We show that the value $ω^*$ can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of $ω^*$ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication $ω_{qow}$. In particular, we show that $ω_{qow}$ can be computed efficiently and it satisfies a perfect parallel repetition theorem.

arXiv:1112.3563Dec 15, 20114 citations

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

Alexandros Chatzinikolaou

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

Quantum games

J.M.A Figueiredo

Physica A Statistical Mechanics and its ApplicationsJan 27, 20041 citation

A Separation between Full-Rank PVM and Assumption-free Self-Testing

Ranyiliu Chen

We construct a nonlocal game that self-tests a maximally entangled qubit strategy among pure full-Schmidt-rank projective strategies, but admits an inequivalent optimum using one nonprojective measurement. This resolves a conjecture of Baptista et al. on imposing full rank and projectivity simultaneously. The construction combines CHSH with an auxiliary game $G$ that forces deterministic answers under these assumptions and admits a trine POVM optimum without them. We classify the optimal correlations of $G$ as a line segment parametrized by the tracial states of $\mathbb{C}\oplus M_2(\mathbb{C})$ . Projectivity on the state support removes the matrix summand, whereas projective dilations with nonzero nonabort probability have a nontracial local state whose zero left ideal is not two-sided. Finally, a family in local dimension six shows that the full-rank PVM self-test is not robust.We construct a nonlocal game that self-tests a maximally entangled qubit strategy among pure full-Schmidt-rank projective strategies, but admits an inequivalent optimum using one nonprojective measurement. This resolves a conjecture of Baptista et al. on imposing full rank and projectivity simultaneously. The construction combines CHSH with an auxiliary game $G$ that forces deterministic answers under these assumptions and admits a trine POVM optimum without them. We classify the optimal correlations of $G$ as a line segment parametrized by the tracial states of $\mathbb{C}\oplus M_2(\mathbb{C})$. Projectivity on the state support removes the matrix summand, whereas projective dilations with nonzero nonabort probability have a nontracial local state whose zero left ideal is not two-sided. Finally, a family in local dimension six shows that the full-rank PVM self-test is not robust.

quant-pharXiv:2609.10013Sep 10, 20260 citations

Quantum games

Παναγιώτης Αποστολίδης, Εύαγγελος Κ. Κοντονικολάου

Jan 1, 20250 citations

Do classical (or quantum) transitive preferences always result in indifferent divisions

Marcin Makowski, Edward W. Piotrowski, Jan Sładkowski

The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural sciences. The paper discusses a simple model of sequential game in which two players in each iteration of the game choose one of the two elements. They make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive! (the so-called \textsl{relevant intransitive strategies}.) On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in importance of intransitive strategies is observed -- there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences.The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural sciences. The paper discusses a simple model of sequential game in which two players in each iteration of the game choose one of the two elements. They make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive! (the so-called \textsl{relevant intransitive strategies}.) On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in importance of intransitive strategies is observed -- there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences.

arXiv:1501.04063Dec 30, 20140 citations

In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value

ω^*

. We show that the value

ω^*

can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of

ω^*

under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication

ω_{qow}

. In particular, we show that

ω_{qow}

can be computed efficiently and it satisfies a perfect parallel repetition theorem.

\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.

G

that forces deterministic answers under these assumptions and admits a trine POVM optimum without them. We classify the optimal correlations of

G

as a line segment parametrized by the tracial states of

\mathbb{C}\oplus M_2(\mathbb{C})

. Projectivity on the state support removes the matrix summand, whereas projective dilations with nonzero nonabort probability have a nontracial local state whose zero left ideal is not two-sided. Finally, a family in local dimension six shows that the full-rank PVM self-test is not robust.