The outer spectral radius and dynamics of completely positive maps

We examine a special case of an approximation the joint spectral radius given by Blondel and Nesterov, which we call outer radius. The is square root ordinary n2 matrix $$\sum \overline {{X_i}} \otimes {X_i}$$ ∑ X i ¯ ⊗ . give analogue formula for can be used to quickly obtain error bounds in methods based on work Nesterov. analyze iterates completely positive map, including quantum channels. average map approach where Kraus operators span ideal algebra generated original map. also elementary treatment Popescu’s theorems similarity row contractions case, describe connections Parrilo-Jadbabaie relaxation, detailed analysis maximal spectrum