Hamza Fawzi

The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. Our rate calculations are valid for systems on general separable Hilbert spaces and we also investigate the convergence of our method to the actual rate, proving convergence in certain situations. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a new minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is able to converge rapidly by recovering instances of all known tight analytical bounds. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.

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.