Theorem: Tensor-free snapshot comparison

The following are equivalent:

1. $T{|}_{A_{\mathrm{out}}}$ factors through a UCP map $K:A_{\mathrm{out}}\to A_{\mathrm{in}}$;

2. equal input restrictions to $A_{\mathrm{in}}$ imply equal output restrictions to $A_{\mathrm{out}}$;

3. $T(A_{\mathrm{out}})\subseteq A_{\mathrm{in}}$.

The factor $K$ is unique. No spatial complementary factor or state-extension channel is required.

Proof. Factorization implies the other two assertions. State comparison applied to every self-adjoint output element, followed by Lemma lem:snapshot, gives containment. Under containment, restriction defines $K$. Positivity at every matrix level is reflected by the subalgebra inclusion, so $K$ is completely positive; it is unital and unique.