  There exists, in general, a convex set of quantum state estimators that
maximize the likelihood for informationally incomplete data. We propose an
estimation scheme, catered to measurement data of this kind, to search for the
exact maximum-likelihood-maximum-entropy estimator using semidefinite
programming and a standard multi-dimensional function optimization routine.
This scheme can be used to infer the expectation values of a set of
entanglement witnesses that can be used to verify the entanglement of the
unknown quantum state for composite systems. Next, we establish an alternative
numerical scheme that is more computationally robust for the sole purpose of
maximizing the likelihood and entropy.
