Multiplayer Xor Games and Quantum Communication Complexity with Clique-wise Entanglement

XOR games are a simple computational model with connections to many areas of complexity theory including interactive proof systems, hardness of approximation, and communication complexity. Perhaps the earliest use of XOR games was in the study of quantum correlations, as an experimentally realizable setup which could demonstrate non-local effects as predicted by quantum mechanics. XOR games also have an interesting connection to Grothendieck’s inequality, a fundamental theorem of analysis—Grothendieck’s inequality shows that two players sharing entanglement can achieve at most a constant factor advantage over players following classical strategies in an XOR game. The case of multiplayer XOR games is much less well understood. Pérez-Garćıa et al. show the existence of entangled states which allow an unbounded advantage for players in a three-party XOR game over their classical counterparts. On the other hand, they show that when the players share GHZ states, a well studied multiparty entangled state, this advantage is bounded by a constant. We use a multilinear generalization of Grothendieck’s inequality due to Blei and Tonge to simplify the proof of the second result and extend it to the case of so-called Schmidt states, answering an open problem of Pérez-Garćıa et al. Via a reduction given in that paper, this answers a 35-year-old problem in operator algebras due to Varopoulos, showing that the space of compact operators on a Hilbert space is a Q-algebra under Schur product. A further generalization of Grothendieck’s inequality due to Carne lets us show that the gap between the entangled and classical value is at most a constant in any multiplayer XOR game in which the players are allowed to share combinations of GHZ states and EPR pairs of any dimension. Based on a result by Bravyi et al. this implies that in a three-party XOR game, players sharing an arbitrary stabilizer state cannot achieve more than a constant factor advantage over unentangled players. Finally, we discuss applications of our results to communication complexity. We show that the discrepancy method in communication complexity remains a lower bound in the multiparty model where the players have quantum communication and any of the kinds of entanglement discussed above. This answers an open question of Lee, Schechtman, and Shraibman who showed that discrepancy was a lower bound on multiparty communication complexity but were unable to handle the case of entanglement. Date: November 20, 2009. CWI and University of Amsterdam. CWI and University of Amsterdam. Rutgers University. Computer Science Division, University of California, Berkeley. 0