Unbounded violation of tripartite Bell inequalities

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck's constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized GHZ states is always bounded so that, in contrast to many other contexts, GHZ states do in this case not lead to extremal quantum correlations. The results are based on tools from the theories of operator spaces and tensor norms which we exploit to prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras.Comment: Substantial changes in the presentation to make the paper more accessible for a non-specialized reade

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

International audienceThis chapter proposes a framework for dealing with two problems related to the analysis of shapes: the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [8], we consider the characteristic functions of the subsets of ℝ2 and their distance functions. The L 2 norm of the difference of characteristic functions and the L∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular that induced by the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of ℝ2 of positive reach in the sense of Federer [12], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem. We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational de.nitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples

Weak-polynomial convergence on spaces lp and Lp

This paper is concerned with the study of the set P-1 (0), when P varies over all orthogonally additive polynomials on l(p) and L (p) spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials

Invariance principle for the random conductance model

We study a continuous time random walk X in an environment of i.i.d. random conductances μe∈[0,∞) in Zd . We assume that P(μe>0)>pc , so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μ e . We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality