We provide a complete structural characterization of perfect quantum strategies for arbitrary quantum magic rectangle games. We derive necessary and sufficient conditions that jointly constrain the shared state and measurement operators, establishing a unified analytical framework for perfect nonlocal strategies in this setting. Our results show that all perfect quantum solution states (PQSS) must exhibit a specific algebraic--combinatorial structure, ruling out a priori assumptions about particular entangled resources and clarifying the full class of states compatible with perfect correlations. We further show that perfect quantum strategies do not exist for $2 \times n$ quantum magic rectangle games with odd $n$, and introduce a corresponding quantum magic rectangle inequality to characterize optimal non-perfect strategies. While our results are structural, they may provide a foundation for future developments in quantum information and quantum cryptography based on perfect nonlocal correlations.
We provide a complete structural characterization of perfect quantum strategies for arbitrary quantum magic rectangle games. We derive necessary and sufficient conditions that jointly constrain the shared state and measurement operators, establishing a unified analytical framework for perfect nonlocal strategies in this setting. Our results show that all perfect quantum solution states (PQSS) must exhibit a specific algebraic--combinatorial structure, ruling out a priori assumptions about particular entangled resources and clarifying the full class of states compatible with perfect correlations. We further show that perfect quantum strategies do not exist for 2×n quantum magic rectangle games with odd n, and introduce a corresponding quantum magic rectangle inequality to characterize optimal non-perfect strategies. While our results are structural, they may provide a foundation for future developments in quantum information and quantum cryptography based on perfect nonlocal correlations.
Quantum entanglement, and the resulting peculiar non-classical correlations are one of the most counter-intuitive aspects of quantum mechanics.
The formalism of interactive games from computational complexity theory provides a useful framework in which to understand the power of entanglement. In an interactive game, a verifier interacts with a number of infinitely powerful provers who are allowed to share quantum entanglement but otherwise can't communicate. The ability of the verifier to extract useful information from the provers, whom he does not trust, provides an interesting measure of the ability of the provers to coordinate using their shared entanglement. Two particular interactive games we'll look at are Magic Square and 3SAT.
The Magic Square game is the iconic example of a game where two classical provers cannot perfectly coordinate their strategies using shared randomness, but quantum provers with shared entanglement can win with probability 1. We show that by adding an extra prover, we disallow perfect cheating. For 3SAT with three provers we also show that perfect cheating is not possible.
We then generalize the results for Magic Square and 3SAT by looking at non-commuting provers, a superset of entangled provers (communication is allowed, but operators applied by different provers must commute). Using this method, we obtain a generalized Tsirelson inequality that we apply to the Magic Square. Hence, we are able to give provably optimal strategies for the general Magic Square with n players. We also recover a similar result for 3SAT as with entangled provers, and we improve on it by showing that the gap is inverse exponential in the input size.
The no-signaling principle, which forbids faster than light communication, is a fundamental constraint on the non-local correlations resulting from quantum entanglement. However, no-signaling allows distributions that are more general than those arising from quantum mechanics. In particular, using a specific unit of general non-local correlation (the Popescu-Rohlich box), we show that there are classes that are equal to NEXP classically, but collapse to AM once such correlations are allowed. We also show that MIP where the verifier only looks at the XOR of the answers collapses to PSPACE. For the second approach, we show that by writing general non-local correlations as linear constraints, MIP is included in EXP under such correlations (vs NEXP classically).
We can extend these results to entangled quantum provers, by formulating an artificial MIP-like class built on a promise problem, that classically is equal to NEXP, but that also collapses to AM when quantum correlations are present.
We develop a method for the transfer of perfect strategies between various classes of two-player, one round cooperative non-local games with quantum inputs and outputs using the simulation paradigm in quantum information theory. We show that such a transfer is possible when canonically associated operator spaces for each game are quantum homomorphic or isomorphic, as defined in the joint work of H. and Todorov (2024). We examine a new class of QNS correlations, needed for the transfer of strategies between games, and characterize them in terms of states on tensor products of canonical operator systems. We define jointly tracial correlations and show they correspond to traces acting on tensor products of canonical C∗-algebras associated with individual game parties. We then make an inquiry into the initial application of such results to the study of concurrent quantum games.
Exact value and rigidity of the 2×3 magic rectangle
The 2×3 magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. Its quantum value was suggested by numerical bounds and known quantum random access codes. We prove that this value is (1+2/3)/2 and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements: up to local isometries and ancillary systems, each contains a maximally entangled pair of four-dimensional systems with fixed measurements on the local state supports. The same value holds for commuting-operator strategies and for the almost-quantum relaxation, confirming the Adamson--Wallden conjecture for this rectangle. For finite-dimensional magic-square strategies, the largest score compatible with perfect prediction of Alice's complete first row by an adversary with quantum side information is one third plus two thirds of the rectangle value. We classify the strategies attaining this threshold with perfect prediction and give linear-order upper and lower bounds on one minus the best compatible guessing probability above it. The rectangle value is also the optimal average success probability of finite-dimensional three-bit random access codes that hide the full input parity, and we classify all optimal codes.
Nonlocal games provide a unified framework for studying the distinction between classical, quantum, and more general no-signaling correlations. In this work, we develop this perspective by connecting the Bell-locality framework to several complementary mathematical representations of nonlocal games and quantum strategies. We begin with local hidden-variable models, the CHSH inequality, and the role of Bell nonlocality as a device-independent witness of entanglement, and then introduce nonlocal games through the standard predicate/verifier formalism.
We next examine a set of representative examples, including XOR games, the GHZ game, graph-based coloring games, the Mermin-Peres magic square game, and Hardy's paradox as a related logical manifestation of nonlocality. Building on this foundation, we compare four closely related representation frameworks: conditional-probability and correlation descriptions, Bell-functional formulations, entangled-value optimization, and the quantum-operator approach together with the Navascues-Pironio-Acin (NPA) hierarchy. These viewpoints are then instantiated for the CHSH, magic square, and GHZ games, showing how each representation emphasizes a different aspect of the same underlying task.
Taken together, these examples show that nonlocal games can be studied simultaneously as geometric objects in correlation space, optimization problems over entangled resources, and operator-theoretic constructions. This multi-representation viewpoint clarifies the relation between Bell inequality violations, perfect quantum strategies, pseudo-telepathy, and semidefinite relaxations of quantum correlations.
First, we consider the problem of deciding whether a nonlocal game admits a perfect entangled strategy that uses projective measurements on a maximally entangled shared state. Via a polynomial-time Karp reduction, we show that independent set games are the hardest instances of this problem. Secondly, we show that if every independent set game whose entangled value is equal to one admits a perfect entangled strategy, then the same holds for all symmetric synchronous games. Finally, we identify combinatorial lower bounds on the classical and entangled values of synchronous games in terms of variants of the independence number of appropriate graphs. Our results suggest that independent set games might be representative of all nonlocal games when dealing with questions concerning perfect entangled strategies.
We identify a structural mechanism by which perfect play in a nonlocal game requires and self-tests a specific non-maximally entangled state. To this end, we construct a Flagged Magic-Square game by extending the standard Magic-Square game with an additional ``flag'' outcome and two additional questions for each player. The game admits a perfect strategy using a state with Schmidt coefficients proportional to (1,2,2,2,2). We give a direct algebraic proof that maximally entangled strategies have winning probability bounded away from one, independently of dimension. Moreover, the game self-tests this state, providing an example of a two-party pseudo-telepathy game that certifies a specific non-maximally entangled state. Our construction illustrates how adding suitable constraints to a rigid measurement geometry can enforce specific Schmidt ratios and provides a modular route to further self-tests based on other rigid pseudo-telepathy games.
We establish a general game-to-channel reduction that transfers separations in the game model to one-shot zero-error communication over classical channels. Applied to the Flagged Magic-Square game, it yields a classical channel for which entanglement permits the zero-error transmission of ten messages in a single use, whereas any finite-dimensional maximally entangled resource permits at most nine. Thus, maximally entangled states are not a universally optimal entanglement resource for zero-error communication over classical noisy channels.