An example of a quantum statistical model which cannot be mapped to a   less informative one by any trace preserving positive map
  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. 
Keiji Matsumoto
quant-ph math.ST stat.TH