In this talk I will sketch a project which aims at the
design of systematic and efficient procedures to infer quantum models from
measured data. Progress in experimental control have enabled an increasingly
fine tuned probing of the quantum nature of matter, e.g., in superconducting
qubits. Such experiments have shown that we not always have a good
understanding of how to model the experimentally performed measurements via
POVMs. It turns out that the ad hoc postulation of POVMs can lead to
inconsistencies. For example, when doing asymptotic state tomography via linear
inversion, one sometimes recovers density operators which are significantly not
positive semidefinite. Assuming the asymptotic regime, we suggest an
alternative procedure where we do not make a priori assumptions on the quantum
model, i.e., on the Hilbert space dimension, the prepared states or the
measured POVMs. In other words, we simultaneously estimate the dimension of the
underlying Hilbert space, the quantum states and the POVMs. We are guided by
Occam's razor, i.e., we search for the minimal quantum model consistent with
the data.