Multiplicativity of $p$-norms of completely positive maps and the additivity problem in quantum information theory

The additivity problem is one of the most profound mathematical problems of quantum information theory. From an analytical point of view it is closely related to the multiplicative property, with respect to tensor products, of norms of maps on operator spaces equipped with the Schatten norms (non-commutative analogue of $l_p$-norms). In this paper we survey the current state of the problem.