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Perfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy

Seth Douglas, Nidhal Mghirbi

Published Oct 29, 2026• 2 citations
Publisher

Abstract

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 $D_\Gamma(\varepsilon)=10\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$. 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-II$_1$ factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit $\Delta_* = \ln(7/6)-\ln(2)/7$. Type-II$_\infty$ 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.

DOI
10.5281/zenodo.23772890

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Perfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy

Seth Douglas, Nidhal Mghirbi

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 $D_\Gamma(\varepsilon)=10\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$. 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-II$_1$ factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit $\Delta_* = \ln(7/6)-\ln(2)/7$. Type-II$_\infty$ 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.

•Oct 29, 2026•0 citations

Perfect play in a binary-scoring nonlocal game requires infinite dimension

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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.

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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.

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quant-pharXiv:2610.06512•Oct 6, 2026•0 citations
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)=10log⁡2(1/ε)+O(log⁡log⁡(1/ε))D_\Gamma(\varepsilon)=10\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))DΓ​(ε)=10log2​(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.3627402784989345.3627402784989345.362740278498934 ebits, and least mean occupation, 275/18275/18275/18 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_11​ 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\Delta_* = \ln(7/6)-\ln(2)/7Δ∗​=ln(7/6)−ln(2)/7. Type-II∞_\infty∞​ 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.
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)=10log⁡2(1/ε)+O(log⁡log⁡(1/ε))D_\Gamma(\varepsilon)=10\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))DΓ​(ε)=10log2​(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.3627402784989345.3627402784989345.362740278498934 ebits, and least mean occupation, 275/18275/18275/18 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_11​ 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\Delta_* = \ln(7/6)-\ln(2)/7Δ∗​=ln(7/6)−ln(2)/7. Type-II∞_\infty∞​ 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 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.
In a nonlocal game of quantum value one that no finite-dimensional strategy wins, near-perfect play needs growing dimension, which separable states can have. We ask how much usable entanglement it needs: the demand at loss ε\varepsilonε is the least one-way distillable entanglement of the original state of a strategy of loss at most ε\varepsilonε. For synchronous games with tested diagonals and for constraint-variable consistency games, demand diverges exactly when no finite-dimensional strategy is perfect; then, for any number of Bell pairs and any fidelity below one, every sufficiently good mixed strategy yields that many at that fidelity by local isometries. For rational synchronous or binary constraint games, a loss guaranteeing this is computable under a promise, but boundedness of demand is undecidable among games of value one. Demand is at least logarithmic if near-winners have a sign relation and an even dilation mirrored by both players, as in a game from Abels' group. For agreement games the directions agree up to a factor three in loss, optimal among Bob-fixed deterministic local symmetrizations. The geometry of the group behind a game, not its presentation, sets the rates. Fixed games demand αlog⁡(1/ε)\alpha\log(1/\varepsilon)αlog(1/ε) for every α=12∑ilog⁡miqi\alpha=\frac12\sum_i\log_{m_i}q_iα=21​∑i​logmi​​qi​ with qi≥2q_i\ge2qi​≥2 a power of two dividing mim_imi​, a dense set containing all positive rationals and some transcendentals, such as 12log⁡62\frac12\log_6221​log6​2; a configuration-lamp game demands ε−1+o(1)\varepsilon^{-1+o(1)}ε−1+o(1); Pauli lamps on the points of Houghton's three rays give demand between ε−0.0690\varepsilon^{-0.0690}ε−0.0690 and ε−1/4\varepsilon^{-1/4}ε−1/4. If group relations hold exactly and Alice matches Bob to error eee, demand counts qubits addressable by short words: Θ(1/e)\Theta(1/e)Θ(1/e), Θ(e−k)\Theta(e^{-k})Θ(e−k) for integers k≥1k\ge1k≥1, or 2Θ(1/e)2^{\Theta(1/e)}2Θ(1/e). Conditional on our attainment and tail bounds, the I3322I_{3322}I3322​ optimum needs bounded usable entanglement.
We exhibit non-closedness at perfect play for normal injective hosts: an explicit finite two-player game with 0/1 scoring has a perfect commuting-operator correlation approached by finite-dimensional quantum correlations, while no perfect strategy has a normal marginal on an injective von Neumann algebra of either player. The game has 561 Alice questions, 157 Bob questions, 3,696 sampled pairs and at most 20 answers per question. Perfect play is realized on a matrix amplification of the Bellaterra critical-KMS factor of type III with parameter 1/2. After restricting to the support of the chosen normal marginal, every perfect strategy yields a modular Cuntz pair and six self-similar relations that force the canonical trace on the free product of three groups of order two. The resulting noninjective group von Neumann algebra is an expected subalgebra of that support corner, with a faithful normal state-preserving expectation. A two-sided POVM lift through a matrix ultraproduct proves approximation by finite systems. Consequently correlations admitting normal injective hosts are not closed. Under the explicit hypotheses of Halvorson's Proposition 2.31, locally normal quantum-field-theory strategies cannot win exactly. We also formulate a KMS-face criterion for complete modular presentations. Exact arithmetic certificates check every supported losing event; the forcing proof applies to arbitrary normal 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)(1,\sqrt{2},\sqrt{2},\sqrt{2},\sqrt{2})(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.