Solution of the propeller conjecture in R3
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Quantum strategies are better than classical in almost any XOR game
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Multipartite entanglement in XOR games
Quantum Information & Computation
Efficient rounding for the noncommutative grothendieck inequality
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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The classical Grothendieck constant, denoted $K_G$, is equal to the integrality gap of the natural semi definite relaxation of the problem of computing$$\max \{\sum_{i=1}^m\sum_{j=1}^n a_{ij} \epsilon_i\delta_j: \{\epsilon_i\}_{i=1}^m,\{\delta_j\}_{j=1}^n\subseteq \{-1,1\}\},$$a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that $K_G\leq \pi / (2\log(1+\sqrt{2}))$ and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that $K_G 0$. Our main contribution is conceptual: despite dealing with a binary rounding problem, random 2-dimensional projections combined with a careful partition of $R^2$ in order to round the projected vectors, beat the random hyper plane technique, contrary to Krivine's long-standing conjecture.