Proposition: Finite channel realization

Formula (eq:block-channel) is CPTP and preserves every reference. Conversely, every CPTP map between these finite direct sums has such a representation.

A unitary spectral realization of Theorem thm:category transports these channels bijectively by transporting their Kraus maps.

Proof. After any reference extension, each Kraus expression is positive. The displayed completeness identity gives trace preservation separately on each input block.

Conversely, composing a channel with the injection of input block $k$ and projection onto output block $l$ gives a CP map. Its positive Choi matrix decomposes into rank-one positive summands, giving the Kraus maps. Trace preservation gives the completeness identity. This is finite-dimensional Choi theory Choi,Watrous.

Unitary changes of input and output realizations replace $K$ by $U_{\mathrm{out}}K{U}_{\mathrm{in}}^{†}$. They preserve the completeness identity and have inverse transport. Composition is preserved.