Bounding quantumclassical separations for classes of nonlocal games
Abstract
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 multiparty 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 socalled Schmidt state. We generalize this class to mod$m$ games and show that their classical value is always at least $\frac{1}{m} + \frac{m1}{m} t^{1t}$. Secondly, for free XOR games, in which the input distribution is of product form, we show $\beta(G) \geq \beta^*(G)^{2^t}$ where $\beta(G)$ and $\beta^*(G)$ are the classical and entangled biases of the game respectively. We also introduce socalled 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 twoplayer unique games and show that if the entangled value is $1\epsilon$ then the classical value is at least $1\mathcal{O}(\sqrt{\epsilon \log k})$ where $k$ is the number of outputs in the game. Our proofs use semidefiniteprogramming techniques, the Gowers inverse theorem and hypergraph norms.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.11068
 Bibcode:
 2018arXiv181111068B
 Keywords:

 Quantum Physics
 EPrint:
 35 pages, 1 figure