Complete channels and intervention-based influence

For finite input factors indexed by $X$, write

Classical registers have a distinguished basis and diagonal observable algebra. Their canonical matrix extension first pinches that basis. Every controlled branch map is fixed independently of the unknown input density operator.

A channel is completely positive and trace preserving (CPTP). We use

Standard finite-channel conventions may be found in Watrous. Instruments retain their complete records DaviesLewis.