Hamza Fawzi

Abstract In the analysis of quantum information processing tasks, the choice of distance measure between states or channels often plays a crucial role. This submission introduces new quantum Rnyi divergences for states and channels that are based on a convex optimization program involving the matrix geometric mean. These divergences have mathematical and computational properties that make them applicable to a wide variety of problems. We use these Rnyi divergences to obtain semidefinite programming lower bounds on the key rates for device-independent cryptography, and in particular we find a new bound on the minimal detection efficiency required to perform device-independent quantum key distribution without additional noisy preprocessing. Furthermore, we give several applications to quantum Shannon theory, in particular proving that adaptive strategies do not help in the strong converse regime for quantum channel discrimination and obtaining improved bounds for quantum capacities.

This tutorial will cover some applications of convex optimization, and more particularly semidefinite programming, to quantum information theory. In particular, I will explain how semidefinite programming can be used to formulate certain optimization problems involving quantum entropies. I will also discuss semidefinite relaxations to polynomial optimization problems, with a focus on the separability problem.