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Abstract: A family of probability distributions i.e. a statistical model is said tobe sufficient for another, if there exists a transition matrix transforming theprobability distributions in the former to the probability distributions in thelatter. The Blackwell-Sherman-Stein BSS theorem provides necessary andsufficient conditions for one statistical model to be sufficient for another,by comparing their information values in statistical decision problems. In thispaper we extend the BSS theorem to quantum statistical decision theory, wherestatistical models are replaced by families of density matrices defined onfinite-dimensional Hilbert spaces, and transition matrices are replaced bycompletely positive, trace-preserving maps i.e. coarse-grainings. Theframework we propose is suitable for unifying results that previously wereindependent, like the BSS theorem for classical statistical models and itsanalogue for pairs of bipartite quantum states, recently proved by Shmaya. Animportant role in this paper is played by statistical morphisms, namely, affinemaps whose definition generalizes that of coarse-grainings given by Petz andinduces a corresponding criterion for statistical sufficiency that is weaker,and hence easier to be characterized, than Petz-s.



Author: Francesco Buscemi

Source: https://arxiv.org/







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