Lemma: Choi and matrix-unit certification

Let $\Phi ,\Psi$ be channels with input dimension $d$, and write

Then

For the qubit-input extractor, a valid certificate is

Proof. The Bell input gives the lower bound.

Every pure referenced input can be written

Thus ${\Vert B\Vert }_{\infty }^2\le d$, and

Convexity covers mixed inputs and gives the Choi upper bound.

Alternatively, write $|\psi ⟩={\sum }_i|x_i{⟩}_Q\otimes |i{⟩}_M$, where ${\sum }_i{\Vert x_i\Vert }^2=1$. Then

Divide by two and take the referenced supremum. The channel-distance cap one and the specialization $d=2$ complete the proof.