Quantum channel

In plain terms

Here is the problem. Unitary evolution is what happens to a perfectly isolated system, and nothing is perfectly isolated. Send a photon down a real fibre and it attenuates, dephases, mixes with noise. The general description of any such process -- any physically realisable transformation of quantum states -- is called a channel.

The structure theorem for channels says every one of them can be written as a family of simpler operators applied and summed -- the Kraus form -- subject to one completeness condition that guarantees probabilities still add to one afterwards.

The corpus formalises channels in exactly that form and then proves the certificates: trace preserved, positivity preserved, and the Choi-matrix criterion for complete positivity, in both directions.

In CSD

Channels are the vocabulary of the programme's open-system and coarse-graining tiers: the dephasing and damping semigroups are channel families in time; the mode-tracing coarse-graining of the field arena is a channel built from an explicit isometry; and the renormalisation-step error budget is stated as a distance between channel outputs.

The layer is operational-stratum breadth, formalised to the corpus's usual standard: no channel property is used anywhere without a named theorem carrying it.

Mathematically

QuantumChannel: an indexed Kraus family with the completeness law summing adjoint-products to the identity; apply is the Kraus sum. Proved: trace preservation, PSD preservation, unitality criteria, and choiMatrix_posSemidef -- the easy direction of Choi -- with the converse in the companion module closing the equivalence. Composition and the dilation-side constructions live where their consumers need them.