Recasting your question in mathematical terms,  it would be a matter of looking at the relevant wave functions satisfying the Schrödinger equation pertaining to the system . In Markovian dynamics, you'll just have a Ψ of the variables at hand. In non Markovian dynamics you'll have a hierarchy of wave functions, which will involve, alongside the usual variables, other historically present wave functions, yielding something reducible to a Ψ(Ψ1, Ψ2, Ψ3, x, y, z ...) form.

Now the challenge would be to establish a metric for non locality and compare the Ψ's in both cases - challenging, to put it mildly. Since a purely mathematical resolution would be fiendishly complicated, you are right to seek other avenues, via density matrices etc. I fear a resolution might not be easily attainable. Instinctively, the answer would not be obvious at all either.

However, essentially, the difference between M and non M has to do with the time variable. Different ways to model time could perhaps be the way to go. If you were to take the view that time is an illusory non observable that emerges from other sub-components, then you could say that time per se has no influence and that non locality is unchanged as a function of time. (Interestingly, there is something akin to that in a totally different area: free will. The Free Will theorem takes a Markovian view of free will, but irrespective of whether free will is Markovian or not, the issue of free will remains essentially unsolved.) Good luck with it, let us know if you are making progress....

I agree that time makes the real difference. But interestingly system dynamics should be temporal with good cybernetics.

Alagu,

Non-locality is an effect of entanglement between two or more quantum systems. We can notice the non-locality if we are able to do measurements on all the involved systems. It is not so clear from your question what are the measurable systems. I may be misunderstanding you, but your question makes the impression that you have ONE quantum system which can be observed, and it is entangled with an environment. How can we observe the environment, do measurements on it?

@Sophia  If you look up on the basics of Markovian models (PRA, google), you will note that at least two systems are involved. So there is nothing confusing about my question, it is aimed at gathering feedback on a challenging subject from experts in this field.

@ Ans  Are you sure this forum is the right venue for your views?

@Bernd Schmeikal    Interesting point as it implies that  Markovian &  non-Markovian elements are cluttering up the road to clear  understanding of non-locality. If it is the case, then we don't need these heavy vehicles that cause the traffic jams, maybe just zero in on Minkowski algebra. Have I interpreted you correctly?

Certainly, since non-Markovian dynamics, by construction, introduces non-local effects. Now these effects can have physical significance, that's why they're introduced in the first place. 

I believe the question is of considerable interest. It might therefore be of interest to rephrase it more clearly. As it stands, it assumes that there is a quantitative measure associated with Non-Markovianity and another associated to non-locality, the question being whether the one grows when the other does? Now, before addressing that question, one should ask:   which quantities do you have in mind? I guess, for entanglement the purity of a subsystem might be a measure: if the whole system is in a pure state, the less pure the subsystem is, the more entangled the entire state. But what do you have in mind for Non-Markovianity? In particular, one of the criteria for Non-Markovianity is, that the subsystem's purity can grow with time, which sounds rather different from entanglement being large. Similarly, when we consider Markovianity or lack thereof, we usually make a distinction between a central system and an external bath. This distinction is usually not made when looking at entanglement, though we certainly may ask how entangled a central system and the bath become. I just am not sure that there are any straightforward connections, nor how to phrase them if there are.

@Leyvraz  Non-Markovianity has been linked quantitatively to different distance measures (e.g., trace distance, Bures distance, Hilbert-Schmidt distance). However these measures vary  due to inherently differet characteristics,  and as such there is no  unique definition of non-Markovianity in quantum systems. In a sense, if there is a measure that responds to deviations from the continuous, memoryless, completely positive semi-group feature of Markovian evolution, then it is may qualify as a tool to quantify non-Markovianity. The concepts of divisibility and distinguishability  have been very useful  in identifying  measures which violate the complete positivity during the evolution of a  system. So to answer your question: there is no single quantity that I have in mind.....as long as "it" is linked to the breakdown of the complete positivity during  evolution, it will do ok. Even the increase of trace distance during time intervals (for instance) may be  taken as a sufficient but not necessary signature of  non-Markovianity. This shows the complexities of non-Markovian dynamics.

It is challenging as well to quantify Non-locality, and there are attempts to involve  the  Bell inequality and variations thereof  to quantify the boundaries of non-local events. So we have hurdles in defining the problem even before resorting to solving it!

Perhaps it is a good start to seek a link (implicit or otherwise) between non-Markovianity and non-locality. To this end, it would be interesting to seek  the role of mathematical mappings known as the assignment or extension maps within the context of this problem. Assignment maps provide description as to how a subsystem is embedded within a larger system. These maps can take on negative values in some instances, and the  quantum system proceeds to evolve with signatures of non-Markovian dynamics for a given period of time. Will further examination of the Assignment maps help establish a link  between increased non-Markovianity and non-locality? What is the role of the Minkowski space & geometric algebra in this regard? Are they better platforms with which to examine this problem, which appears ill defined at the moment.