  It is generally assumed that every process in quantum physics can be
described mathematically by a completely positive map. However, experimentally
reconstructed processes are not necessarily completely positive due to
statistical or systematic errors. In this paper, we introduce a test for
discriminating statistical from systematic errors which is necessary to
interpret experimentally reconstructed, non-completely positive
maps.Wedemonstrate the significance of the test using several examples given by
experiments and simulations. In particular, we demonstrate experimentally how
an initial correlation between the system to be measured and its environment
leads to an experimentally reconstructed map with negative eigenvalues. These
experiments are carried out using atomic 171Yb+ ions confined in a linear Paul
trap, addressed and coherently manipulated by radio frequency radiation.
