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Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Authors Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, Troy Lee

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Author Details

Tom Bannink
  • CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Jop Briët
  • CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Harry Buhrman
  • CWI, University of Amsterdam & QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Farrokh Labib
  • CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Troy Lee
  • Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia


We thank Peter Høyer, Serge Massar, and Henry Yuen for useful discussions. We thank Shravas Rao for providing a proof of one of the lemmas.

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Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum communication complexity
  • Nonlocal games
  • communication complexity
  • bounded separations
  • semidefinite programming
  • pseudorandomness
  • Gowers norms


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