Manuscript Oai272-A@1 · Oai272-A.tex · corollary · a40828a0

Corollary: Witness for the completed channel

Stage 3: finite witness construction under an exact support promise › Transporting the witness to the channel square · lines 1613–1635 · proof included or linked

Oai272-A@1#cor:channel-witness

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Under Theorem thm:completion, suppose $Z = J (GF) \neq = 0$ has a witness $W = Q - ϵ I$ from Proposition prop:support-witness. For $D = 2 d + 1$ , set

W = (V_{0} \otimes V_{0}) W (V_{0}^{†} \otimes V_{0}^{†}) .

Then $W$ is nonnegative on the full separable cone and

Tr [W j (Θ^{2})] = - \frac{s t ϵ Tr Z}{D} =: - g < 0.

Proof. Compression of a product vector is a product vector, possibly zero. This proves block positivity of $W$ . Equation (eq:corner) gives the expectation and the factor $1/ D$ from normalized Choi convention.

Displayed formulas in this passage

unlabeled · line 1619
$W = (V_{0} \otimes V_{0}) W (V_{0}^{†} \otimes V_{0}^{†}) .$
unlabeled · line 1623
$Tr [W j (Θ^{2})] = - \frac{s t ϵ Tr Z}{D} =: - g < 0.$