  The minimum Kullback entropy principle (mKE) is a useful tool to estimate
quantum states and operations from incomplete data and prior information. In
general, the solution of a mKE problem is analytically challenging and an
approximate solution has been proposed and employed in different context.
Recently, the form and a way to compute the exact solution for finite
dimensional systems has been found, and a question naturally arises on whether
the approximate solution could be an effective substitute for the exact
solution, and in which regimes this substitution can be performed. Here, we
provide a systematic comparison between the exact and the approximate mKE
solutions for a qubit system when average data from a single observable are
available. We address both mKE estimation of states and weak Hamiltonians, and
compare the two solutions in terms of state fidelity and operator distance. We
find that the approximate solution is generally close to the exact one unless
the initial state is near an eigenstate of the measured observable. Our results
provide a rigorous justification for the use of the approximate solution
whenever the above condition does not occur, and extend its range of
application beyond those situations satisfying the assumptions used for its
derivation.
