Monogamy of nonlocal quantum correlations

We describe a new technique for obtaining Tsirelson bounds, or upper bounds on the quantum value of a Bell inequality. Since quantum correlations do not allow signaling, we obtain a Tsirelson bound by maximizing over all no-signaling probability distributions. This maximization can be cast as a linear program. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities, relevant for interactive proof systems and cryptography.Comment: This is the submitted version. The refereed version, which contains an additional result about strong parallel repetition and corrects some typos, is available on my personal web site at http://bentoner.com/papers/monogamyrs.pdf [PDF

Simulating Quantum Correlations with Finite Communication

Assume Alice and Bob share some bipartite $d$-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension $d$. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.Comment: 19 pages, 3 figures, preliminary version in IEEE FOCS 2007; to appear in SICOM

A generalized Grothendieck inequality and entanglement in XOR games

Suppose Alice and Bob make local two-outcome measurements on a shared entangled state. For any d, we show that there are correlations that can only be reproduced if the local dimension is at least d. This resolves a conjecture of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing the first lower bounds on a new generalization of Grothendieck's constant.Comment: Version submitted to QIP on 10-20-08. See also arxiv:0812.1572 for related results, obtained independentl

Entangled Games Are Hard to Approximate

We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1-1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games

New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion

In this paper we present our detailed investigations on the nature of the phase transition in the scalar noise model (SNM) of collective motion. Our results confirm the original findings of Vicsek et al. [Phys. Rev. Lett. 75 (1995) 1226] that the disorder-order transition in the SNM is a continuous, second order phase transition for small particle velocities ($v\leq 0.1$). However, for large velocities ($v\geq 0.3$) we find a strong anisotropy in the particle diffusion in contrast with the isotropic diffusion for small velocities. The interplay between the anisotropic diffusion and the periodic boundary conditions leads to an artificial symmetry breaking of the solutions (directionally quantized density waves) and a consequent first order transition like behavior. Thus, it is not possible to draw any conclusion about the physical behavior in the large particle velocity regime of the SNM.Comment: 13 pages, 11 figure

An Historical Analysis of the Legal Status of the North Carolina Cherokees

We investigate the existence of secure bit commitment protocols in the convex framework for probabilistic theories. The framework makes only minimal assumptions, and can be used to formalize quantum theory, classical probability theory, and a host of other possibilities. We prove that in all such theories that are locally non-classical but do not have entanglement, there exists a bit commitment protocol that is exponentially secure in the number of systems used