A unified approach to classical and quantum Hidden Markov processes

Hidden Markov processes (HMP) (also called Hidden Markov models (HMM)) are very popular in contemporary research in applied mathematics including a large set of disciplines, such as machine learning, speech or handwriting recognition, image reconstruction, mathematical finance, ion channels, $\dots$.
Several different quantum extensions of HMP have been proposed in the literature, especially in quantum information, but they are not very satisfactory for several reasons that will be discussed during the talk. In my talk I will briefly recall the notion of quantum Markov chain (QMC) and show that, restricting to different diagonal algebras the simplest class of quantum Markov chains one obtains all the classical HMP.
This shows that, for model building in any discipline, quantum Markov chains are a much more flexible and powerful tool than classical HMP because they allow, with the same (finite) number of parameters, to deal simultaneously with infinitely many classical HMP.
Then I will give a new definition of quantum HMP and show that:
1) in the classical case, this definition covers all HMP;
2) Restricting quantum HMP to to different diagonal algebras one obtains new classes of classical processes with a nice a rich structure which opens new possibilities for model building.
This is joint work with Soueidy El Gheteb, Yun Gang Lu and Abdessatar Souissi.