Quantum-correlation breaking channels, quantum conditional probability and Perron-Frobenius theory

Using the quantum analog of conditional probability and classical Bayes theorem we discuss some aspects of quantum-classical and classical-classical channels in connection to the recent paper J. Korbicz, P. Horodecki, and R. Horodecki, "Quantum-correlation breaking channels, broadcasting scenarios, and finite Markov chains". Applying the quantum analog of Perron-Frobenius theorem we generalize the result of Korbicz et. al. on full and spectrum broadcasting from quantum-classical channels to arbitrary channels.