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arXiv:2410.09525

Perfect Quantum Approximate Strategies for Imitation Games

Liang Hao, T. Yu, Lihong Zhi

Published Oct 12, 2024• 0 citations
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Abstract

We prove that an imitation game has a perfect quantum approximate strategy if and only if there exists a bi-tracial state on the minimal tensor product of two universal C${^*}$-algebras, which induces the perfect correlation. Moreover, we are trying to relate imitation games to the minimal tensor product of two universal C${^*}$-algebras and demonstrate that an imitation game has a perfect quantum approximate strategy if and only if there exist a von Neumann algebra and an amenable tracial state on it, such that the perfect correlation can be induced by the tracial state. However, We encountered some difficulties regarding continuity in the proof process. In section 2 we get some results for special cases, and in section 3 we list our problems.

DOI
10.48550/arxiv.2410.09525

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We prove that an imitation game has a perfect quantum approximate strategy if and only if there exists a bi-tracial state on the minimal tensor product of two universal C∗{^*}∗-algebras, which induces the perfect correlation. Moreover, we are trying to relate imitation games to the minimal tensor product of two universal C∗{^*}∗-algebras and demonstrate that an imitation game has a perfect quantum approximate strategy if and only if there exist a von Neumann algebra and an amenable tracial state on it, such that the perfect correlation can be induced by the tracial state. However, We encountered some difficulties regarding continuity in the proof process. In section 2 we get some results for special cases, and in section 3 we list our problems.
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∗{\rm C}^{*}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.
We show that the ∗^*∗-algebra of the product of two synchronous games is the tensor product of the corresponding ∗^*∗-algebras. We prove that the product game has a perfect C∗^*∗-strategy if and only if each of the individual games does, and that in this
We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.
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 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×n2 \times n2×n quantum magic rectangle games with odd nnn, 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 give an operator-algebraic formulation of robust self-testing in terms of states on C*-algebras. We show that a quantum correlation p is a robust self-test only if among all (abstract) states, there is a unique one achieving p. We show that the "if" direction of this statement also holds, provided that p is optimal/perfect for a nonlocal game that has a robust game algebra. This last condition applies to many nonlocal games of interest, including all XOR games, synchronous games, and boolean constrained system (BCS) games. For those nonlocal games with robust game algebras, we prove that self-testing is equivalent to the uniqueness of finite-dimensional tracial states on the associated game algebra, and robust self-testing is equivalent to the uniqueness of amenable tracial states. Applying this tracial-state characterization of self-testing to parallel repetition, we show that a synchronous game is a self-test for perfect quantum strategies if and only if its parallel repeated version is a self-test for perfect quantum strategies. As a proof approach, we give the first quantitative Gower-Hatami theorem that is applicable to C*-algebras. Here "quantitative" means there is a constructive bound on the distance between the approximate representations and exact representations. We also demonstrate how this quantitative Gowers-Hatami theorem can be used to calculate the explicit robustness function of a self-test.