Minimal sufficient Jordan algebras characterize sufficiency for positive trace-preserving maps on quantum states, with Neyman-Pearson tests generating them and equality in data-processing inequalities implying Petz recovery.
Matsumoto
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abstract
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.
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Sufficiency and Petz recovery for positive maps
Minimal sufficient Jordan algebras characterize sufficiency for positive trace-preserving maps on quantum states, with Neyman-Pearson tests generating them and equality in data-processing inequalities implying Petz recovery.