  Herein we study the problem of recovering a density operator from a set of
compatible marginals, motivated from limitations of physical observations.
Given that the set of compatible density operators is not singular, we adopt
Jaynes' principle and wish to characterize a compatible density operator with
maximum entropy. We first show that comparing the entropy of compatible density
operators is QSZK-complete, even for the simplest case of 3-chains. Then, we
focus on the particular case of quantum Markov chains and trees and establish
that for these cases, there exists a quantum polynomial circuit that constructs
the maximum entropy compatible density operator. Finally, we extend the
Chow-Liu algorithm to the same subclass of quantum states.
