Abstract:
The mathematical tradition in the information theory goes back to A. N. Kolmogorov and A. Ya. Khintchine. For a mathematician, the information theory is a source of fresh ideas and hard problems with decent motivation. This applies also to quantum information theory — a new scientific discipline studying the general laws of transmission, storage and transformation of information in the systems obeying quantum mechanics. Its problematic is closely related to the fundamental structures of positivity and tensor product in operator algebras, noncommutative probability theory, and asymptotic methods of random matrices. One of the difficult analytical problems of quantum information theory is the “additivity hypothesis” which was discussed at the International Congress of Mathematicians in 2006 and at the European Congress of Mathematicians in 2008.
The talk will highlight the following issues:

The general concept of the channel as a morphism of the category of $C^*$-algebras;

Non-commutative analogue of the Shannon coding theorem;

The additivity problem for the channel capacity and entanglement of
quantum states;