Theorem: Uniqueness and coherent environment

For a triple $B$, every CPTP map satisfying $D{{\mathcal{N}}_{\mathcal{B}}}^V={\mathrm{id}}_2$ equals $D_B$ on the whole matrix algebra $\mathcal{B}(({\mathbb{C}}^2{)}^{\otimes 3})$. Its Choi rank is four. Every pure-environment Stinespring implementation therefore requires environment dimension at least four, and a coherent output-plus-environment realization needs at least three qubits in total.

Proof. Let $K_a$ be Kraus operators of $D$. Kraus operators of $D{{\mathcal{N}}_{\mathcal{B}}}^V$ are

Because the identity channel has Choi rank one, each is scalar times $I_2$. Thus

for suitable vectors $v_a$. Trace preservation gives ${\sum }_a|v_a⟩⟨v_a|=I_4$, whence $D={\mathrm{Tr}}_{{\mathbb{C}}^4}\circ {\mathrm{Ad}}_{{U_B}^{†}}$.

The four Kraus operators $(I_2\otimes ⟨e|){U_B}^{†}$ are linearly independent, so the Choi rank is four. A Stinespring environment dimension bounds the number of linearly independent Kraus operators. The lower bound is attained by applying ${U_B}^{†}$ and retaining the two-qubit environment.