Theorem: Complete complement at the fixed interface

A complete complementary channel is

Tracing its environmental factor gives the retained-source channel of Theorem thm:source.

The finite participating calculation of Theorem thm:normal gives the same accessible channel. Its canonical finite-background purification is embedded isometrically into this full environmental interface.

Proof. The isometry follows from unitarity of $U_G$ and normalization of $\Omega$. Taking the two respective partial traces gives the accessible channel and (eq:complement). A further trace gives ${\mathcal{B}}_G$.

For the last assertion, factor the finite participating CCR space from the full regular representation. Its vector restriction is a purification of ${\rho }_{\mathfrak{k}}$. A Schmidt decomposition embeds the canonical purification space into the remaining representation space. Since $U_G$ acts only on the participating physical modes and the source, this embedding intertwines the dilations. No source-dependent information is removed by that replacement of purification coordinates.