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

arXiv:2610.06512

The right entanglement for perfect nonlocality and zero-error communication

Laura Mančinska

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Related papers

Maximally entangled states in pseudo-telepathy games

Laura Mančinska

A pseudo-telepathy game is a nonlocal game which can be won with probability one using some finite-dimensional quantum strategy but not using a classical one. Our central question is whether there exist two-party pseudo-telepathy games which cannot be won with probability one using a maximally entangled state. Towards answering this question, we develop conditions under which maximally entangled states suffice. In particular, we show that maximally entangled states suffice for weak projection games which we introduce as a relaxation of projection games. Our results also imply that any pseudo-telepathy weak projection game yields a device-independent certification of a maximally entangled state. In particular, by establishing connections to the setting of communication complexity, we exhibit a class of games $G_n$ for testing maximally entangled states of local dimension $Ω(n)$. We leave the robustness of these self-tests as an open question.

arXiv:1506.07080•Jun 23, 2015•0 citations

Robust self-testing of nonmaximal entanglement from a reduced inner product game

Ranyiliu Chen

We reduce Lalonde's pseudo-telepathy inner product game from six to five dimensions, keeping its four Alice questions and three Bob questions and reducing each answer alphabet to five. We prove that the reduced game self-tests its nonmaximally entangled state and all 35 measurement effects. The self-test is also robust: for a strategy winning with probability at least $1-\varepsilon$ we obtain $d_{\mathrm{ext}}\le\min\{1,2^{21}\sqrt{\varepsilon}\}$, where $d_{\mathrm{ext}}=\sqrt{1-Ξ}$ and $Ξ$ is the optimal squared fidelity with the reference state under local extraction channels. A compactness argument then gives simultaneous qualitative robustness for all measurement actions on the state under common local isometries. As a consequence of the self-test, no maximally entangled state of any finite dimension can win the game perfectly.

quant-pharXiv:2609.24349•Sep 22, 2026•0 citations

Self-testing high dimensional states using the generalized magic square game

Matthew McKague

To date all self-tests for high dimensional systems are confined to many-qubit states. This is due to two restrictions in the literature: the standard techniques for proving self-testing results apply only to qubits, and there are few suitable non-local games that are known to require high dimensional non-qubit states to achieve their optimal quantum value. In this paper we address these two problems. Specifically, we generalize the usual self-testing framework to q$d$its and then apply it to a generalization of the magic square game, giving a self-test for two maximally entangled pairs of q$d$its. This self-test has two parties, perfect completeness for honest quantum players, and requires only a constant number of measurement settings, regardless of the dimension $d$.

arXiv:1605.09435•May 30, 2016•4 citations

Maximally entangled states are not complete for pseudo-telepathy

Olivier Lalonde

One of the longstanding open problems in quantum nonlocality is to determine if maximally entangled states are complete for bipartite pseudo-telepathic games: namely, if every nonlocal game which admits a perfect entangled strategy admits such a strategy which uses a maximally entangled state. We exhibit a counterexample to this in the form of a bipartite nonlocal game with input sets of size 4 and 3 and output sets both of size 6. This game is part of a new class of nonlocal games, which we call inner product games, which could be of independent interest.

quant-pharXiv:2608.05378•Aug 7, 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

Glued magic games self-test maximally entangled states

Laura Mančinska, Thor Gabelgaard Nielsen, Jitendra Prakash

Self-testing results allow us to infer the underlying quantum mechanical description of states and measurements from classical outputs produced by non-communicating parties. The standard definition of self-testing does not apply in situations when there are two or more inequivalent optimal strategies. To address this, we introduce the notion of self-testing convex combinations of reference strategies, which is a generalisation of self-testing to multiple strategies. We show that the Glued Magic Square game [Quantum 4 (2020), p. 346] self-tests a convex combination of two inequivalent strategies. As a corollary, we obtain that the Glued Magic square game self-tests two EPR pairs thus answering an open question from [Quantum 4 (2020), p. 346]. Our self-test is robust and extends to natural generalisations of the Glued Magic Square game.

arXiv:2105.10658•May 22, 2021•3 citations
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.
A pseudo-telepathy game is a nonlocal game which can be won with probability one using some finite-dimensional quantum strategy but not using a classical one. Our central question is whether there exist two-party pseudo-telepathy games which cannot be won with probability one using a maximally entangled state. Towards answering this question, we develop conditions under which maximally entangled states suffice. In particular, we show that maximally entangled states suffice for weak projection games which we introduce as a relaxation of projection games. Our results also imply that any pseudo-telepathy weak projection game yields a device-independent certification of a maximally entangled state. In particular, by establishing connections to the setting of communication complexity, we exhibit a class of games GnG_nGn​ for testing maximally entangled states of local dimension Ω(n)Ω(n)Ω(n). We leave the robustness of these self-tests as an open question.
We reduce Lalonde's pseudo-telepathy inner product game from six to five dimensions, keeping its four Alice questions and three Bob questions and reducing each answer alphabet to five. We prove that the reduced game self-tests its nonmaximally entangled state and all 35 measurement effects. The self-test is also robust: for a strategy winning with probability at least 1−ε1-\varepsilon1−ε we obtain dext≤min⁡{1,221ε}d_{\mathrm{ext}}\le\min\{1,2^{21}\sqrt{\varepsilon}\}dext​≤min{1,221ε​}, where dext=1−Ξd_{\mathrm{ext}}=\sqrt{1-Ξ}dext​=1−Ξ​ and ΞΞΞ is the optimal squared fidelity with the reference state under local extraction channels. A compactness argument then gives simultaneous qualitative robustness for all measurement actions on the state under common local isometries. As a consequence of the self-test, no maximally entangled state of any finite dimension can win the game perfectly.
To date all self-tests for high dimensional systems are confined to many-qubit states. This is due to two restrictions in the literature: the standard techniques for proving self-testing results apply only to qubits, and there are few suitable non-local games that are known to require high dimensional non-qubit states to achieve their optimal quantum value. In this paper we address these two problems. Specifically, we generalize the usual self-testing framework to qdddits and then apply it to a generalization of the magic square game, giving a self-test for two maximally entangled pairs of qdddits. This self-test has two parties, perfect completeness for honest quantum players, and requires only a constant number of measurement settings, regardless of the dimension ddd.
One of the longstanding open problems in quantum nonlocality is to determine if maximally entangled states are complete for bipartite pseudo-telepathic games: namely, if every nonlocal game which admits a perfect entangled strategy admits such a strategy which uses a maximally entangled state. We exhibit a counterexample to this in the form of a bipartite nonlocal game with input sets of size 4 and 3 and output sets both of size 6. This game is part of a new class of nonlocal games, which we call inner product games, which could be of independent interest.
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.
Self-testing results allow us to infer the underlying quantum mechanical description of states and measurements from classical outputs produced by non-communicating parties. The standard definition of self-testing does not apply in situations when there are two or more inequivalent optimal strategies. To address this, we introduce the notion of self-testing convex combinations of reference strategies, which is a generalisation of self-testing to multiple strategies. We show that the Glued Magic Square game [Quantum 4 (2020), p. 346] self-tests a convex combination of two inequivalent strategies. As a corollary, we obtain that the Glued Magic square game self-tests two EPR pairs thus answering an open question from [Quantum 4 (2020), p. 346]. Our self-test is robust and extends to natural generalisations of the Glued Magic Square game.