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

Equality conditions for internal entropies of certain classical and quantum models