Mathematical methods of quantum key distribution

Quantum key distribution and, more generally, quantum cryptography is a modern branch of science where methods of secure communication based on principles of quantum mechanics are studied. The rigorous proof of the security of quantum key distribution gave rise to a complex and beautiful mathematical theory, which is based on methods of quantum information theory, namely, quantum entropic measures and entropic uncertainty relations. In particular, to estimate secret key rate, one needs to minimize the quantum relative entropy (a convex function) subject to linear constraints. The problem is, in general, infinite-dimensional, but symmetry properties of the problem reduces the dimensionality and allows one to solve this problem analytically. However, currently, an important task is to prove the security of quantum key distribution with imperfect (i.e., practical) devices. Imperfections introduce asymmetries and thus make the problem more complicated. In the talk, estimations for the secret key rate in the case of detection-efficiency mismatch will be presented. Using entropic uncertainty relations, an infinite-dimensional problem is reduced to a one-dimensional one.