Su ciency in quantum statistical inference

This paper attempts to develop a theory of su ciency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Su ciency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper su cient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only in nite Hilbert space dimension, but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity.