Manuscript Hawk02-E@1 · Hawk02-E.tex · lemma · 4841592b

Lemma: Normalized Choi-to-diamond bound

Complete-residual certificates and decoder certificates › The dimension factor · lines 1163–1193 · proof included or linked

Hawk02-E@1#lem:choi

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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} .

The output may be infinite dimensional provided the channel outputs are normal states.

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

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

Its channel-output difference is

(L \otimes I) [J_{n} (Φ) - J_{n} (Ψ)] (L^{†} \otimes I) .

Its trace norm is at most $d$ times the Choi trace norm because $∥ L ∥^{2} \leq d$ . Convexity and the finite-reference characterization of channel diamond distance complete the proof.

Displayed formulas in this passage

unlabeled · line 1166
$d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$
unlabeled · line 1178
$∣ ψ ⟩ = (L \otimes I) ∣ Φ_{d} ⟩, Tr (L^{†} L) = d .$
unlabeled · line 1184
$(L \otimes I) [J_{n} (Φ) - J_{n} (Ψ)] (L^{†} \otimes I) .$

Lemma: Normalized Choi-to-diamond bound

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