Gaussian optimizers and the additivity problem in quantum information theory

This paper surveys two remarkable analytical problems of quantum information theory. The main part is a detailed report on the recent (partial) solution of the quantum Gaussian optimizer problem which establishes an optimal property of Glauber's coherent states-a particular case of pure quantum Gaussian states. The notion of a quantum Gaussian channel is developed as a non-commutative generalization of an integral operator with Gaussian kernel, and it is shown that the coherent states, and under certain conditions only they, minimize a broad class of concave functionals of the output of a Gaussian channel. Thus, the output states corresponding to a Gaussian input are the 'least chaotic', majorizing all the other outputs. The solution, however, is essentially restricted to the gauge-invariant case where a distinguished complex structure plays a special role. Also discussed is the related well-known additivity conjecture, which was solved in principle in the negative some five years ago. This refers to the additivity or multiplicativity (with respect to tensor products of channels) of information quantities related to the classical capacity of a quantum channel, such as the (1 -> p)-norms or the minimal von Neumann or Renyi output entropies. A remarkable corollary of the present solution of the quantum Gaussian optimizer problem is that these additivity properties, while not valid in general, do hold in the important and interesting class of gauge-covariant Gaussian channels.