Directed-completeness of quantum statistical experiments in the randomization order

Abstract

A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.