  The data processing inequality is the most basic requirement for any
meaningful measure of information. It essentially states that
distinguishability measures between states decrease if we apply a quantum
channel and is the centerpiece of many results in information theory. Moreover,
it justifies the operational interpretation of most entropic quantities. In
this work, we revisit the notion of contraction coefficients of quantum
channels, which provide sharper and specialized versions of the data processing
inequality. A concept closely related to data processing is partial orders on
quantum channels. First, we discuss several quantum extensions of the
well-known less noisy ordering and relate them to contraction coefficients. We
further define approximate versions of the partial orders and show how they can
give strengthened and conceptually simple proofs of several results on
approximating capacities. Moreover, we investigate the relation to other
partial orders in the literature and their properties, particularly with regard
to tensorization. We then examine the relation between contraction coefficients
with other properties of quantum channels such as hypercontractivity. Next, we
extend the framework of contraction coefficients to general f-divergences and
prove several structural results. Finally, we consider two important classes of
quantum channels, namely Weyl-covariant and bosonic Gaussian channels. For
those, we determine new contraction coefficients and relations for various
partial orders.
