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arXiv:2408.10114

Topics in Non-local Games: Synchronous Algebras, Algebraic Graph Identities, and Quantum NP-hardness Reductions

Entong He

Published Aug 19, 2024• 0 citations
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Abstract

We review the correspondence between synchronous games and their associated $*$-algebra. Building upon the work of (Helton et al., New York J. Math. 2017), we propose results on algebraic and locally commuting graph identities. Based on the noncommutative Nullstellensätze (Watts, Helton and Klep, Annales Henri Poincaré 2023), we build computational tools that check the non-existence of perfect $C^*$ and algebraic strategies of synchronous games using Gröbner basis methods and semidefinite programming. We prove the equivalence between the hereditary and $C^*$ models questioned in (Helton et al., New York J. Math. 2017). We also extend the quantum-version NP-hardness reduction $\texttt{3-SAT}^* \leq_p \texttt{3-Coloring}^*$ due to (Ji, arXiv 2013) by exhibiting another instance of such reduction $\texttt{3-SAT}^* \leq_p \texttt{Clique}^*$.

DOI
10.48550/arxiv.2408.10114

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We review the correspondence between synchronous games and their associated ∗*∗-algebra. Building upon the work of (Helton et al., New York J. Math. 2017), we propose results on algebraic and locally commuting graph identities. Based on the noncommutative Nullstellensätze (Watts, Helton and Klep, Annales Henri Poincaré 2023), we build computational tools that check the non-existence of perfect C∗C^*C∗ and algebraic strategies of synchronous games using Gröbner basis methods and semidefinite programming. We prove the equivalence between the hereditary and C∗C^*C∗ models questioned in (Helton et al., New York J. Math. 2017). We also extend the quantum-version NP-hardness reduction 3-SAT∗≤p3-Coloring∗\texttt{3-SAT}^* \leq_p \texttt{3-Coloring}^*3-SAT∗≤p​3-Coloring∗ due to (Ji, arXiv 2013) by exhibiting another instance of such reduction 3-SAT∗≤pClique∗\texttt{3-SAT}^* \leq_p \texttt{Clique}^*3-SAT∗≤p​Clique∗.
We establish several strong equivalences of synchronous non-local games, in the sense that the corresponding game algebras are ∗*∗-isomorphic. We first show that the game algebra of any synchronous game on nnn inputs and kkk outputs is ∗*∗-isomorphic to the game algebra of an associated bisynchronous game on nknknk inputs and nknknk outputs. As a result, we show that there are bisynchronous games with equal question and answer sets, whose optimal strategies only exist in the quantum commuting model, and not in the quantum approximate model. Moreover, we show that there are bisynchronous games with equal question and answer sets that have non-zero game algebras, but no winning quantum commuting strategies, resolving a problem of V.I. Paulsen and M. Rahaman. We also exhibit a ∗*∗-isomorphism between any synchronous game algebra with nnn questions and k>3k>3k>3 answers and a synchronous game algebra with n(k−2)n(k-2)n(k−2) questions and 333 answers.
We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other perfect strategies. when applied to the graph coloring game, this leads to characterizations in terms of properties of an algebra of various quantum chromatic numbers that have been studied in the literature. This allows us to develop a correspondence between various chromatic numbers of a graph and ideals in this algebra which can then be approached via various Grobner basis methods.
In the past few decades, the connections between non-local games arising in quantum information theory and the theory of operator algebras have undergone a phase of significant development. Operator algebras provide a particularly fruitful framework for approaching questions of non-locality in quantum systems, as the input-output behavior of measurements on bipartite quantum systems can be encoded through noncommutative operator algebras and their state spaces. This thesis is the compilation of a series of papers, written by the author in collaboration with Ivan G. Todorov, utilizing this framework by using the theory of operator algebras and completely positive maps to study questions involving non-local games. ☐ Using the simulation paradigm from information theory, we define a quantized version of homomorphisms and isomorphisms between classical hypergraphs, generalizing quantum homomorphisms and isomorphisms of graphs from the literature. We show that these quantum homomorphisms and isomorphisms constitute pre-orders and equivalence relations, respectively. We use quantum homomorphisms of hypergraphs to provide multiple examples of strict separation between correlation classes of varying types. Specializing to the case when our underlying hypergraphs arise from classical non-local games, we define quantum non-local game homomorphisms and isomorphisms. We also show how the existence of a homomorphism or isomorphism of some fixed type between games is reflected in a comparison of optimal and asymptotic game values with respect to this type. These non-local game homomorphisms are defined via a new class of non-signalling correlations which we introduce here. We develop the multivariate tensor product theory in the category of operator systems, which we then use to provide a characterization of game homomorphisms in terms of state spaces for various tensor products of canonically associated operator systems. We then apply these results to the study of synchronous non-local games, defining jointly synchronous correlations and showing how they correspond to tracial states of tensor products of C*-algebras canonically associated to each game party. ☐ We then move to a second level of quantization by proposing a "quantized" notion of hypergraphs (where "quantum" here is in the sense of a non-commutative analogue for a discrete combinatorial object), and introduce quantum homomorphisms between these quantum objects. We provide an initial foray into the properties quantum morphisms between quantum hypergraphs display, showing they satisfy analogous properties to quantum homomorphisms between classical hypergraphs. We also indicate initial connections these quantum homomorphisms between quantum hypergraphs have to the study of quantum input-output games. We end the thesis showing that homomorphisms of a local type between quantum hypergraphs is closely related, and in some cases identical, to the TRO equivalence of finite-dimensionally acting operator spaces canonically associated with each hypergraph. This suggests a quantum information theoretic approach to Morita equivalence in the category of operator spaces.
We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies. This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton and collaborators. An algorithm based on noncommutative Gröbner basis computation and semidefinite programming is given for certifying that a given mirror game has no perfect commuting operator strategies.
Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum–classical game,” that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analog of the notion of non-commutative graph homomorphisms due to Stahlke [IEEE Trans. Inf. Theory 62(1), 554–577 (2016)]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by Helton et al. [New York J. Math. 25, 328–361 (2019)]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the four coloring game for a quantum graph is always non-trivial, extending a result of Helton et al. [New York J. Math. 25, 328–361 (2019)].
In the work of Paulsen et al. [J. Funct. Anal. (in press); preprint arXiv:1407.6918], the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation matrices arose in their study of various versions of quantum chromatic numbers of graphs and other quantum versions of graph theoretic parameters. In this paper, we develop these ideas further, focusing on the relations between synchronous correlation matrices and microstates. We prove that Connes’ embedding conjecture is equivalent to the equality of two families of synchronous quantum correlation matrices. We prove that if Connes’ embedding conjecture has a positive answer, then the tracial rank and projective rank are equal for every graph. We then apply these results to more general non-local games.