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Measurements of consecutive sites in a many body qudit system (e.q. a spin chain) can be interpreted as generating a probabilistic sequence of numbers $X_1 X_2 X_3 \ldots$, where $X_i\in \{0,1,\ldots,d-1 \}$.

Are there any studies on that approach, in particular - exploring predictability of such systems or constructing a Markov model of some order simulating it?

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Maybe I'm missing something (I'm about to go to sleep). You take the spin-spin correlation functions and build (say) whatever order transition matrix you like, no? –  S Huntsman Jan 28 '12 at 3:23
    
@SHuntsman In the one ways (state -> sequence) it is straightforward. I am interested what can one deduce about the state (or Hamiltonian, if it an ground/eigenstate) knowing only the sequence. –  Piotr Migdal Jan 28 '12 at 10:12
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I am not aware of anything specific, but if the system is one-dimensional and the state you are investigating can be represented as a matrix product state, then it is essentially a (quantum) Markov model where the state space is that of the "virtual space" of the matrix product state.

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True, Matrix Product States are in some sense a hidden Markov process for quantum states. And here, in fact, I am interested how many sequence properties are preserved (or lost) when converting amplitudes to probabilities. –  Piotr Migdal Apr 10 '13 at 21:07
    
I am not entirely sure about what you mean with "sequence properties". If, for example, you are talking about properties of the Hamiltonian as above, the probability for a given sequence of observations is conditional on the Hamiltonian parameters and as such the probability for such a sequence is the likelihood function for the Hamiltonian parameters (see for example dx.doi.org/10.1103/PhysRevA.87.032115 - this paper is for continous measurements, but applies equally well to your discrete process). –  S. Gammelmark Apr 11 '13 at 7:38
    
Well, "sequence properties" is vague, as it is still an open-ended question (I was thinking about something in line of correlations between particles (esp. adjacent), e.g. in the line of arxiv.org/abs/0812.5079). Thanks for your reference. –  Piotr Migdal Apr 11 '13 at 9:27
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