ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 201–213.
Pleiades Publishing, Inc., 2016.
Original Russian Text
F. Buscemi, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 3–16.
Degradable Channels, Less Noisy Channels,
and Quantum Statistical Morphisms:
An Equivalence Relation
Department of Computer Science and Mathematical Informatics, Nagoya University, Nagoya, Japan
Received February 9, 2016
Abstract—Two partial orderings among communication channels, namely “being degradable
into” and “being less noisy than,” are reconsidered in the light of recent results about statistical
comparisons of quantum channels. Though our analysis covers at once both classical and
quantum channels, we also provide a separate treatment of classical noisy channels and show
how in this case an alternative self-contained proof can be constructed, with its own particular
merits with respect to the general result.
Given two channels, it is natural to ask which one is “better.” As one soon realizes, ordering
channels according to their capacity would be too limited in scope, since there may be other
measures of “goodness” that are more relevant than capacity for the task at hand. This is even
more so for quantum channels, for which many inequivalent capacities exist . Indeed, it does not
require too much imagination to come up with uncountably many such “comparisons,” and each
comparison will only induce a partial, rather than a total, ordering between channels. This fact
should not come as a worrying surprise: channels are highly dimensional objects, and any total
ordering can only be extremely coarse—often too coarse to be of any use in practice. Nonetheless,
it is true that some comparisons are more natural, more compelling, or just mathematically simpler
than others, so that they received more attention in the literature. This paper actually deals with
two much studied comparisons, namely the partial orderings “being degradable into” and “being
less noisy than,” introduced in Deﬁnitions 1 and 2 below (for a compendium of many comparisons
among discrete noisy channels, see ).
The goal of this work is to exhibit a connection between degradable channels and less noisy
channels, beyond the obvious one “degradable implies less noisy.” More explicitly, we show how
a formal (but not substantial) modiﬁcation in the deﬁnition of less noisy channels is suﬃcient to
make the two orderings equivalent. Our result is proved in a general scenario, where channels
are modeled as CPTP maps between operator algebras, thus covering quantum channels, classical
channels (when input and output are commutative algebras), and also hybrid classical-to-quantum
and quantum-to-classical channels.
Central to our approach is the notion of quantum statistical morphisms, i.e., linear maps between
operator algebras that generalize in a statistical sense the idea of “post-processing” or “coarse-
graining” (see Deﬁnition 5). The use of statistical morphisms allows us to prove our results under
very mild assumptions, so that the quantum and classical cases are recovered as special cases of
Supported in part by the JSPS KAKENHI, grant no. 26247016.