  Quantum statistical models (i.e., families of normalized density matrices)
and quantum measurements (i.e., positive operator-valued measures) can be
regarded as linear maps: the former, mapping the space of effects to the space
of probability distributions; the latter, mapping the space of states to the
space of probability distributions. The images of such linear maps are called
the testing regions of the corresponding model or measurement. Testing regions
are notoriously impractical to treat analytically in the quantum case. Our
first result is to provide an implicit outer approximation of the testing
region of any given quantum statistical model or measurement in any finite
dimension: namely, a region in probability space that contains the desired
image, but is defined implicitly, using a formula that depends only on the
given model or measurement. The outer approximation that we construct is
minimal among all such outer approximations, and close, in the sense that it
becomes the maximal inner approximation up to a constant scaling factor.
Finally, we apply our approximation formulas to characterize, in a semi-device
independent way, the ability to transform one quantum statistical model or
measurement into another.
