Quantum-correlation breaking channels, broadcasting scenarios, and finite Markov chains

One of the classical results concerning quantum channels is the characterization of entanglement-breaking channels [M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math. Phys. 15, 629 (2003)]. We address the question whether there exists a similar characterization on the level of quantum correlations which may go beyond entanglement. The answer is fully affirmative in the case of breaking quantum correlations down to the, so-called, QC (quantum-classical) type, while it is no longer true in the CC (classical-classical) case. The corresponding channels turn out to be measurement maps. Our study also reveals an unexpected link between quantum state and local correlation broadcasting and finite Markov chains. We present a possibility of broadcasting via non von Neumann measurements, which relies on the Perron-Frobenius theorem. Surprisingly, this is not the typical generalized controlled-not (c-not) gate scenario appearing naturally in this context.