Corollary: Recovery of the supplied hybrid comparison

For finite tensor-product hybrid interfaces, use the local inclusions by identity extension and let $T={\Phi }^{\ast }$. Then (eq:dual-factor) is precisely

Thus Theorem thm:comparison recovers lem:comparison in En05-A, En05-B, and En06-A without their complementary-state proof.

Proof. Under finite-dimensional trace duality, the dual of identity extension is partial trace together with classical marginalization. The dual of a UCP map is a complete trace-preserving channel. Apply Theorem thm:comparison.