  Comparison of statistical models (experiments) is an important branch of
mathematical statistics, which gives deep insights in many aspects of
foundation of statistics. So far, there are two quantum versions of the
concept: Comparison with respect to classical tasks and full quantum tasks. In
the latter version, a quantum statistical model is more informative than
another if and only if a trace preserving positive map sends the former to the
latter. On the other hand, in the former version, the existence of a trace
preserving positive map sending the former to the latter is a useful sufficient
condition. A natural question is whether this is necessary or not. We resolve
this question negatively by giving a counter example.
