Just as a completely positive trace preserving (CPTP) map can always be considered as the reduction of unitary evolution on a system with an initially unentangled auxiliary, quantum instruments are the reductions of projective measurement with a conditional unitary, and also reduce to CPTP maps and POVMs when ignore measurement outcomes and state evolution, respectively. In John Smolin 's terminology, this is an example of "going to the Church of the Larger Hilbert space".