We construct a finite rational two-player nonlocal game with a binary winning predicate that has a perfect strategy on ℓ²⊗ℓ² and no perfect finite-dimensional strategy. The game has ten questions for Alice, eleven for Bob, at most six answers per question, and a uniform distribution on fifty-two question pairs. We glue two copies of Mančinska's Flagged Magic-Square game along complementary labels, adding a boundary answer to the second copy. An inverse-free sector argument forces every perfect strategy to contain a Schmidt-coefficient chain closed under halving, which rules out finite rank. An explicit normalized spatial strategy has twenty coefficients at each geometric level. Its compression to 20K dimensions has loss exactly 1/[624(4^K−1)], giving D(ε) ≤ 10 log2(1/ε) + 20 for general local POVMs and 0 < ε < 1. We prove a weaker algebraic lower bound and state a matching logarithmic lower bound as an open problem. Thus a binary-scoring game distinguishes spatial from finite-dimensional perfect attainment even though both values are one.
We construct a finite rational two-player nonlocal game with a binary winning predicate that has a perfect strategy on ℓ²⊗ℓ² and no perfect finite-dimensional strategy. The game has ten questions for Alice, eleven for Bob, at most six answers per question, and a uniform distribution on fifty-two question pairs. We glue two copies of Mančinska's Flagged Magic-Square game along complementary labels, adding a boundary answer to the second copy. An inverse-free sector argument forces every perfect strategy to contain a Schmidt-coefficient chain closed under halving, which rules out finite rank. An explicit normalized spatial strategy has twenty coefficients at each geometric level. Its compression to 20K dimensions has loss exactly 1/[624(4^K−1)], giving D(ε) ≤ 10 log2(1/ε) + 20 for general local POVMs and 0 < ε < 1. We prove a weaker algebraic lower bound and state a matching logarithmic lower bound as an open problem. Thus a binary-scoring game distinguishes spatial from finite-dimensional perfect attainment even though both values are one.
We construct a finite rational two-player nonlocal game with ten Alice questions, eleven Bob questions, at most six answers per question and 52 equally likely supported pairs. It has a perfect spatial strategy and no perfect finite-dimensional strategy. For arbitrary finite-dimensional local POVMs and mixed states, its dimension profile is D_Gamma(epsilon) = 10 log2(1/epsilon) + O(log log(1/epsilon)). The lower bound is 10 log2(1/epsilon) - 10 log2 log2(1/epsilon) - O(1). A direct square-root magic-square extraction, robust flagged sector law and approximate gluing force twenty Schmidt coefficients per appreciable halving level. The explicit strategy compressed to 20K dimensions has loss exactly 1/[624(4^K-1)], yielding the matching leading upper coefficient 10.
A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile DΓ(ε)=10log2(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.362740278498934 ebits, and least mean occupation, 275/18 per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II1 factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit Δ∗=ln(7/6)−ln(2)/7. Type-II∞ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.
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.
A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile DΓ(ε)=10log2(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.362740278498934 ebits, and least mean occupation, 275/18 per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II1 factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit Δ∗=ln(7/6)−ln(2)/7. Type-II∞ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.
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.
Can uncomputability occur in the resource law of one fixed finite nonlocal game, rather than in a problem whose input is a game? We construct a finite two-player Boolean constraint-system game with rational question probabilities, quantum value one, and no finite-dimensional perfect strategy. If D(ε) is the least local dimension needed to win with probability at least 1−ε, then limn→∞log2D(2−n)/n=(3+θK)/4, where θK=∑e∈K4−(e+1) and K is the halting set. This exponent has the Turing degree of K. The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.