  A family of probability distributions (i.e. a statistical model) is said to
be sufficient for another, if there exists a transition matrix transforming the
probability distributions in the former to the probability distributions in the
latter. The Blackwell-Sherman-Stein (BSS) theorem provides necessary and
sufficient conditions for one statistical model to be sufficient for another,
by comparing their information values in statistical decision problems. In this
paper we extend the BSS theorem to quantum statistical decision theory, where
statistical models are replaced by families of density matrices defined on
finite-dimensional Hilbert spaces, and transition matrices are replaced by
completely positive, trace-preserving maps (i.e. coarse-grainings). The
framework we propose is suitable for unifying results that previously were
independent, like the BSS theorem for classical statistical models and its
analogue for pairs of bipartite quantum states, recently proved by Shmaya. An
important role in this paper is played by statistical morphisms, namely, affine
maps whose definition generalizes that of coarse-grainings given by Petz and
induces a corresponding criterion for statistical sufficiency that is weaker,
and hence easier to be characterized, than Petz's.
