Directed-completeness of quantum statistical experiments in the randomization order

TL;DR

This paper establishes that the set of equivalence classes of quantum statistical experiments and channels is mathematically complete in the randomization order, allowing for the existence of least upper and greatest lower bounds in this framework.

Contribution

It introduces the concept of directed-completeness for the set of statistical experiments and channels under the randomization order, extending classical and finite-dimensional quantum results.

Findings

Set of equivalence classes is upper and lower directed-complete.

Supremum and infimum exist for nets of experiments and channels.

Explicit derivation of infima for specific quantum channels.

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.