  We illustrate the mathematical theory of entropy production in repeated
quantum measurement processes developed in a previous work by studying examples
of quantum instruments displaying various interesting phenomena and
singularities. We emphasize the role of the thermodynamic formalism, and give
many examples of quantum instruments whose resulting probability measures on
the space of infinite sequences of outcomes (shift space) do not have the
(weak) Gibbs property. We also discuss physically relevant examples where the
entropy production rate satisfies a large deviation principle but fails to obey
the central limit theorem and the fluctuation-dissipation theorem. Throughout
the analysis, we explore the connections with other, a priori unrelated topics
like functions of Markov chains, hidden Markov models, matrix products and
number theory.
