Theorem: Snapshot–channel–adjoint equivalence

The following are equivalent:

1. there exists a UCP map $K:A_{\mathrm{out}}\to A_{\mathrm{in}}$ such that

   $$T{\iota }_J={\iota }_{I}K;$$

2. for all input states,

   $$\omega {\iota }_I=\eta {\iota }_I⟹\omega T{\iota }_J=\eta T{\iota }_J;$$
3. $$T({\iota }_J(A_{\mathrm{out}}))\subseteq {\iota }_I(A_{\mathrm{in}}).$$

The map $K$ is unique. No spatial tensor factor, canonical state extension, or complementary-state channel is used.

If the algebras are von Neumann algebras, the inclusions are normal with von Neumann ranges, and $T$ is normal, then the same equivalence holds with all normal states and a normal $K$.

Proof. Factorization implies state comparison and containment. State comparison applied to every self-adjoint output element and Lemma lem:snapshot gives containment.

Under containment define

It is unital. At every matrix level, ${\iota }_I$ preserves and reflects positivity on its range, so complete positivity of $T$ proves complete positivity of $K$. Injectivity proves uniqueness.

For the normal version, a self-adjoint element outside the weak-$\ast$ closed range of ${\iota }_I$ can be separated from it by a normal Hermitian functional. Its normal Jordan decomposition gives two normal equal-snapshot states as above. The same argument proves containment, and the inverse of the normal von Neumann algebra isomorphism onto the range is normal.