  Most existing results in the analysis of quantum reservoir computing (QRC)
systems with classical inputs have been obtained using the density matrix
formalism. This paper shows that alternative representations can provide better
insights when dealing with design and assessment questions. More explicitly,
system isomorphisms are established that unify the density matrix approach to
QRC with the representation in the space of observables using Bloch vectors
associated with Gell-Mann bases. It is shown that these vector representations
yield state-affine systems (SAS) previously introduced in the classical
reservoir computing literature and for which numerous theoretical results have
been established. This connection is used to show that various statements in
relation to the fading memory (FMP) and the echo state (ESP) properties are
independent of the representation, and also to shed some light on fundamental
questions in QRC theory in finite dimensions. In particular, a necessary and
sufficient condition for the ESP and FMP to hold is formulated using standard
hypotheses, and contractive quantum channels that have exclusively trivial
semi-infinite solutions are characterized in terms of the existence of
input-independent fixed points.
