  Mathematical models use information from past observations to generate
predictions about the future. If two models make identical predictions the one
that needs less information from the past to do this is preferred. It is
already known that certain classical models (certain Hidden Markov Models
called \epsilon-machines which are often optimal classical models) are not in
general the preferred ones. We extend this result and show that even optimal
classical models (models with minimal internal entropy) in general are not the
best possible models (called ideal models). Instead of optimal classical models
we can construct quantum models which are significantly better but not yet the
best possible ones (i.e. they have a strictly smaller internal entropy). In
this paper we show conditions when the internal entropies between classical
models and specific quantum models coincide. Furthermore it turns out that this
situation appears very rarely. An example shows that our results hold only for
the specific quantum model construction and in general not for alternative
constructions. Furthermore another example shows that classical models with
minimal internal entropy need not to be related to quantum models with minimal
internal entropy.
