Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Finite games that read the type of a von Neumann algebra

Seth Douglas, Nidhal Mghirbi

Published Oct 30, 2026• 1 citation
Publisher

Abstract

We construct finite two-player games with binary scoring whose normal strategies distinguish operator-algebraic type. A 243-pair game has perfect play in type III and none when either ambient algebra is semifinite. For factors with separable predual, hosting it is equivalent to an invariant normal state on the flow of weights, or to an expected hyperfinite type III1 subfactor. Prepared-state hosting is therefore insensitive to the ratio invariant throughout the injective flow-finite class. A different quantifier recovers that invariant: every faithful normal state has value one for the flagged magic-square game exactly when 1/2 belongs to the Connes invariant; a nine-question conference-basis game gives the corresponding criterion for 1/3. Together these two response properties characterize type III1. One prepared state already wins both games in dimension 25. We prove that every positive rational is forced as a modular ratio by some finite game, and identify the exact rational orthogonal-pair single-bridge parameters. For the type-III-only game, a robust sorted spectral law holds for every normal semifinite strategy. It yields matching entropy, deficit, dimension and variance exponents, including an exponential inverse-square-root dimension law. No validated numerical constant is given; the sharp constant is conjectural. We give explicit finite templates, retain their boundary penalties, and distinguish sorted trace position from log density. The specified normal semifinite perturbative gravitational models cannot attain perfect play, whereas suitable type III quantum-field-theory hosts can. This conditional algebraic consequence gives no preparation or energy bound.

DOI
10.5281/zenodo.23787123

Related papers

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

A finite game that normal hyperfinite strategies approach but never win

Seth Douglas, Nidhal Mghirbi

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.

•Oct 30, 2026•0 citations

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•2 citations

The right entanglement for perfect nonlocality and zero-error communication

Laura Mančinska

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 $(1,\sqrt{2},\sqrt{2},\sqrt{2},\sqrt{2})$. 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.

quant-pharXiv:2610.06512•Oct 6, 2026•0 citations

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

Seth Douglas, Nidhal Mghirbi

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.

•Oct 29, 2026•0 citations

Exact value and rigidity of the $2\times3$ magic rectangle

Isaac Barouch Essayag, Aryeh Lev Zabokritskiy

The $2\times3$ 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 $(1+\sqrt{2/3})/2$ 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.

quant-pharXiv:2610.12001•Oct 9, 2026•0 citations
We construct finite two-player games with binary scoring whose normal strategies distinguish operator-algebraic type. A 243-pair game has perfect play in type III and none when either ambient algebra is semifinite. For factors with separable predual, hosting it is equivalent to an invariant normal state on the flow of weights, or to an expected hyperfinite type III1 subfactor. Prepared-state hosting is therefore insensitive to the ratio invariant throughout the injective flow-finite class. A different quantifier recovers that invariant: every faithful normal state has value one for the flagged magic-square game exactly when 1/2 belongs to the Connes invariant; a nine-question conference-basis game gives the corresponding criterion for 1/3. Together these two response properties characterize type III1. One prepared state already wins both games in dimension 25. We prove that every positive rational is forced as a modular ratio by some finite game, and identify the exact rational orthogonal-pair single-bridge parameters. For the type-III-only game, a robust sorted spectral law holds for every normal semifinite strategy. It yields matching entropy, deficit, dimension and variance exponents, including an exponential inverse-square-root dimension law. No validated numerical constant is given; the sharp constant is conjectural. We give explicit finite templates, retain their boundary penalties, and distinguish sorted trace position from log density. The specified normal semifinite perturbative gravitational models cannot attain perfect play, whereas suitable type III quantum-field-theory hosts can. This conditional algebraic consequence gives no preparation or energy bound.
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 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.
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 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.
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.
Exact value and rigidity of the 2×32\times32×3 magic rectangle
The 2×32\times32×3 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(1+\sqrt{2/3})/2(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.