Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · lemma · 94d46352

Lemma: Normalized Choi conversion

Quantitative certificates and the additional endpoint obligations › Complement proximity is an upper certificate · lines 1634–1663 · proof included or linked

Hawk04-C@1#lem:choi, Hawk04-C@1#eq:choi-bound

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For channels $Φ, Ψ$ with input dimension $d$ ,

d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .

Proof. Every pure input with a $d$ -dimensional reference can be written

∣ ψ ⟩ = (L \otimes I_{C}) ∣ Φ_{d} ⟩, Tr (L^{†} L) = d .

Its output difference is obtained from the normalized Choi difference by multiplying on the left by $L \otimes I$ and on the right by $L^{†} \otimes I$ . Its trace norm is therefore at most

∥ L ∥_{\infty}^{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} \leq d ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .

Convexity handles mixed inputs. Taking the referenced supremum and dividing by two proves the claim.

Displayed formulas in this passage

Hawk04-C@1#eq:choi-bound · line 1637
$d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$
unlabeled · line 1647
$∣ ψ ⟩ = (L \otimes I_{C}) ∣ Φ_{d} ⟩, Tr (L^{†} L) = d .$
unlabeled · line 1655
$∥ L ∥_{\infty}^{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} \leq d ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$

Lemma: Normalized Choi conversion

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