  We present a framework that formulates the quest for the most efficient
quantum state tomography scheme as an optimization problem which can be solved
numerically. This approach can be applied to a broad spectrum of relevant
setups including measurements restricted to a subsystem. To illustrate the
power of this method we present results for the six-dimensional Hilbert space
constituted by a qubit-qutrit system, which could be realized e.g. by the N-14
nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in
diamond. Measurements of the qubit subsystem are expressed by projectors of
rank three, i.e., projectors on half-dimensional subspaces. For systems
consisting only of qubits, it was shown analytically that a set of projectors
on half-dimensional subspaces can be arranged in an informationally optimal
fashion for quantum state tomography, thus forming so-called mutually unbiased
subspaces. Our method goes beyond qubits-only systems and we find that in
dimension six such a set of mutually-unbiased subspaces can be approximated
with a deviation irrelevant for practical applications.
