  We introduce a graphical language for coherent control of general quantum
channels inspired by practical quantum optical setups involving polarising beam
splitters (PBS). As standard completely positive trace preserving maps are
known not to be appropriate to represent coherently controlled quantum
channels, we propose to instead use purified channels, an extension of
Stinespring's dilation. We characterise the observational equivalence of
purified channels in various coherent-control contexts, paving the way towards
a faithful representation of quantum channels under coherent control.
