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Quantum statistical models (i.e., families of normalized density matrices)and quantum measurements (i.e., positive operator-valued measures) can beregarded as linear maps: the former, mapping the space of effects to the spaceof probability distributions; the latter, mapping the space of states to thespace of probability distributions. The images of such linear maps are calledthe testing regions of the corresponding model or measurement. Testing regionsare notoriously impractical to treat analytically in the quantum case. Ourfirst result is to provide an implicit outer approximation of the testingregion of any given quantum statistical model or measurement in any finitedimension: namely, a region in probability space that contains the desiredimage, but is defined implicitly, using a formula that depends only on thegiven model or measurement. The outer approximation that we construct isminimal among all such outer approximations, and close, in the sense that itbecomes the maximal inner approximation up to a constant scaling factor.Finally, we apply our approximation formulas to characterize, in a semi-deviceindependent way, the ability to transform one quantum statistical model ormeasurement into another.

Tight conic approximation of testing regions for quantum statistical models and measurements