Theorem: Finite hybrid channel structure

The map (eq:hybrid-map) is completely positive and trace preserving on ${\mathfrak{X}}_{\mathsf{G}}$. It preserves the classical–quantum state space and has adjoint

Conversely, every completely positive trace-preserving map between two finite direct sums has this branch form. If

then

A canonical extension to the full matrix space on ${\mathbb{C}}^{\mathsf{G}}\otimes \mathcal{H}$ has Kraus operators

This extension discards off-diagonal input label blocks and has block-diagonal output.

Proof. After adjoining any matrix ancilla, a positive direct-sum input has positive blocks. Each amplified branch map is positive, and sums of its outputs are positive. Thus $\mathfrak{F}$ is completely positive. Furthermore,

The block formula proves preservation of classicality, and pairing with $B$ gives (eq:hybrid-adjoint).

For the converse, compose the given map with the completely positive injection into input block $g$ and projection onto output block $h$. These compositions define completely positive ${\Psi }_{h|g}$. Linearity gives the branch formula, and trace preservation on each input block gives (eq:instrument-complete).

Finite-dimensional completely positive maps have Kraus representations: the positive Choi matrix can be decomposed into rank-one positive terms and each vector unvectorized into a Kraus operator Choi1975. Completeness gives (eq:kraus-complete). The operators (eq:lift-kraus) have sum of squared moduli equal to the full identity. Their action reads only the diagonal input blocks and writes only diagonal output blocks, proving the final assertion.