Theorem: Exact named complementary-channel distance

The complete complement obeys

At prefixes zero and one, comparison with the respective complementary output on $I_2/2$ gives distance $3/4$.

For $\eta =1$,

This equality is specific to this model and these metrics.

Proof. The key blocks and fuel sectors in (eq:fullcomplement) are orthogonal, so their trace norms add. On a failed sector $W_{ab}$ is an isometry. Its comparison with the replacement output on $I_2/2$ is therefore the comparison of $\mathrm{id}$ with $\mathcal{T}$, of distance $3/4$. On success use Lemma lem:dephleak. This proves the weighted upper bound.

The same Bell input attains the $3/4$ value in every failure block and the value in Lemma lem:dephleak in every success block. Hence it attains the weighted sum. Independent tensor factors do not change trace norm.

At prefix zero the whole purified evolution is an isometric input channel. At prefix one the complement is an orthogonal mixture of isometric channels with known independent labels. The same calculation gives $3/4$. The final displayed equality follows from Theorem thm:recovery at $\eta =1$.