  We address the problem of compressing density operators defined on a finite
dimensional Hilbert space which assumes a tensor product decomposition. In
particular, we look for an efficient procedure for learning the most likely
density operator, according to Jaynes' principle, given a chosen set of partial
information obtained from the unknown quantum system we wish to describe. For
complexity reasons, we restrict our analysis to tree-structured sets of
bipartite marginals. We focus on the tripartite scenario, where we solve the
problem for the couples of measured marginals which are compatible with a
quantum Markov chain, providing then an algebraic necessary and sufficient
condition for the compatibility to be verified. We introduce the generalization
of the procedure to the n-partite scenario, giving some preliminary results. In
particular, we prove that if the pairwise Markov condition holds between the
subparts then the choice of the best set of tree-structured bipartite marginals
can be performed efficiently. Moreover, we provide a new characterisation of
quantum Markov chains in terms of quantum Bayesian updating processes.
