  We study asymmetric binary channel discrimination, for qantum channels acting
on separable Hilbert spaces. We establish quantum Stein's lemma for channels
for both adaptive and parallel strategies, and show that under finiteness of
the geometric R\'enyi divergence between the two channels for some $\alpha >
1$, adaptive strategies offer no asymptotic advantage over parallel ones. One
major step in our argument is to demonstrate that the geometric R\'enyi
divergence satisfies a chain rule and is additive for channels also in infinite
dimensions. These results may be of independent interest. Furthermore, we not
only show asymptotic equivalence of parallel and adaptive strategies, but
explicitly construct a parallel strategy which approximates a given adaptive
$n$-shot strategy, and give an explicit bound on the difference between the
discrimination errors for these two strategies. This extends the finite
dimensional result from [B. Bergh et al., arxiv:2206.08350]. Finally, this also
allows us to conclude, that the chain rule for the Umegaki relative entropy in
infinite dimensions, recently shown in [O. Fawzi, L. Gao, and M. Rahaman,
arxiv:2212.14700v2] given finiteness of the max divergence between the two
channels, also holds under the weaker condition of finiteness of the geometric
R\'enyi divergence. We give explicit examples of channels which show that these
two finiteness conditions are not equivalent.
