Computing mixed Schatten norm of completely positive maps

Computing $p \rightarrow q$ norm for matrices is a classical problem in computational mathematics and power iteration is a well-known method for computing $p \rightarrow q$ norm for a matrix with nonnegative entries. Here we define an equivalent iteration method for computing $S_p \rightarrow S_q$ norm for completely positive maps where $S_p$ is the Schatten $p$ norm. We generalize almost all of the definitions, properties, lemmas, etc. in the matrix setting to completely positive maps and prove an important theorem in this setting.