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I am currently involved in some understanding on this matter with a colleague of mine. I know all the literature about but I do not know the state of art. Please, could you provide some relevant recent literature about? Also explanations are much appreciated.

My present view is that there exists a time scale that defines the limit of validity of a Chapman-Kolmogorov like equation. It is a situation very similar to quantum chaos but I do not know if this has been currently caught.

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Hmm, this could be an interesting question, but can you provide more background and expand the question a little bit? – Marcin Kotowski Nov 9 '11 at 20:43
Your Question at present brings to mind Edward Nelson's "Quantum Fluctuations", from 1985, Princeton University Press. That's usually taken to establish that QM is non-Markovian. Can you give some examples of the literature you mean? – Peter Morgan Nov 9 '11 at 21:00
There are some papers appeared in the '90s where the question was debated for the two-level model as that generated several comments and replies. But I think that the father of all these works is the one from Hanggi et al. What I need is a more up to date understanding of the situation. Yes, surely Nelson was one of the pioneers on this matter but I would like to know if it has been settled and, if yes, how. – Jon Nov 9 '11 at 21:12

A short well-written starting point is Carlton Caves' on-line notes "Completely positive maps, positive maps, and the Lindblad form", see in particular Caves' discussion following (eq. 21), and the references provided.

It helps also to have a physical appreciation of "unravelling" as applied to quantum operations (the alternate spelling "unraveling" is common too); this term was (AFAICT) first introduced in Howard Carmichael's An Open Systems Approach to Quantum Optics (1993, see Section 7.4, p. 122).

In summary, a clear appreciation of the physical process of unravelling quantum trajectories, and the mathematical description of quantum trajectory unravelling as a Markov processes of Lindblad form, will go far to answer your questions.

No doubt other folks can recommend favorite references; please do so.

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