joint work with David de Laat, Monique Laurent
Graph parameters such as the chromatic number can be formulated in several ways. E.g., as polynomial optimization problems or using nonlocal games (in which two separated parties must convince a referee that they have a valid coloring of the graph of a certain size). We show how these formulations lead to quantum analogues of graph parameters, which can be used in quantum information theory to study the power of entanglement. We apply techniques from the theory of noncommutative polynomial optimization to obtain hierarchies of semidefinite programming bounds, unifying existing bounds.
joint work with Ion Nechita
One of the defining properties of quantum mechanics is the existence of incompatible observables such as position and momentum. We will connect the problem of determining whether a given set of measurements is compatible to the inclusion of free spectrahedra, which arise in convex optimization. We show how results from algebraic convexity can be used to quantify the degree of incompatibility of binary quantum measurements. In particular, this new connection allows us to completely characterize the case in which the dimension of the quantum system is exponential in the number of measurements.
joint work with Hamza Fawzi
We consider the problem of maximizing a homogeneous polynomial over the unit sphere and its Sum-of-Square (SOS) hierarchy relaxations. Exploiting the polynomial kernel technique, we show that the convergence rate of the hierarchy is no worse than O(d^2/l^2), where d is the dimension of the sphere and l is the level of the hierarchy. Moreover, we explore the duality relation between the SOS polynomials and the quantum extendable states in details, and further explain the connection of our result with the one by Navascues, Owari & Plenio from quantum information perspective.