Prof. Dr. André Nies (University of Auckland): Two non-classical versions of the Shannon-McMillan-Breiman theorem
Thursday, 25.01.2018 16:30 im Raum M5
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).