A well-known result from the theory of discrete dynamical system, commonly referred to as Shannon-McMillan-Breiman theorem, is an improvement due to Breiman (1959) of a theorem of Shannon (1948) and McMillan (1953). It states intuitively that that given an ergodic discrete dynamical system, for almost every trajectory, the Shannon entropy rate of the system asymptotically equals the mean entropy of longer and longer blocks of the trajectory. We first look at computable versions of the theorem. A result implicit in Hochman (2009), and explicitly stated by Hoyrup (2012), shows that when the system is computable, the randomness in the sense of Martin-Loef (1967) of the trajectory is sufficient for the statement. Another nonclassical version emerges when the dynamical systems are taken in the quantum setting and modelled by certain UHF C*-algebras (Bjelakovich et al, Inventiones, 2004). In work in progress with Marco Tomamichel (and others) we combine the two approaches, using the recently defined version of Martin-Loef- randomness in the quantum setting (Nies and Scholz, arxiv.org/abs/1709.08422).