Abstract

I’ll present a new scheme to prove separation between a point and an arbitrary convex set S via calls to an oracle able to perform linear optimizations over S. Compared to other methods, it has almost negligible memory requirements and the number of calls to the optimization oracle does not depend on the dimensionality of the underlying space. We study the speed of convergence of the scheme under different promises on the shape of the set S and/or the location of the point, validating the accuracy of the theoretical bounds with numerical examples.

I will then present some applications of the scheme in quantum information theory. The algorithm out-performs existing linear programming methods for certain large scale problems, allowing us to certify nonlocality in bipartite scenarios with upto 42 measurement settings. I’ll show how to use the algorithm to upper bound the visibility of two-qubit Werner states, hence improving known lower bounds on Grothendieck’s constant KG(3). Similarly, we compute new upper bounds on the visibility of GHZ states and on the steerability limit of Werner states for a fixed number of measurement settings.