Non-locality is a curious feature, yet essentially a quantum attribute that is linked to the violation of Bell inequality of any form. It arises from the impossibility of simultaneous joint measurements of observables. The Clauser-Horne-Shimony-Holt (CHSH) inequality is the only extremal Bell inequality with two settings and two outcomes per site. This inequality provides a basis to compare predictions of quantum theories with those linked local realism.
During non-Markovian dynamics of open quantum systems, there is break down of the well known Markovian model. This may occur due to strong system-environment coupling or when un-factorized initial conditions exist between the system and environment. Notably, a statistical interpretation of the density matrix is not defined for non-Markovian evolution.
My question is: Is there increased non-locality when a system undergoes non-Markovian dynamics and if so, how can this be quantified. I used the word "increased" because non-locality may be present in the case of Markovian dynamics, and the query focusses on whether certain aspects of non-Markovian dynamics accentuates non-locality.
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.
@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.
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