Generalizations of logarithmic Sobolev inequality arising from quantum information theory

We suggest a method of computation of the classical capacity of quantum measurement channel based on principles of convex programming. It is targeted to solve the problem of Gaussian Maximizers for models out of the scope of the "threshold condition" which ensures that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable on Gaussian encodings. The method is illustrated by the cases of noisy Gaussian homodyning and heterodyning in quantum optics. Rather remarkably, for both models the method reduces the solution of the optimization problem to new generalizations of the celebrated log-Sobolev inequality.