Manuscript Hawk02-D@1 · Hawk02-D.tex · lemma · e6e4b72f

Lemma: Normalized Choi-to-diamond estimate

Channels, complete residuals, and reference-stable estimates · lines 367–403 · proof included or linked

Hawk02-D@1#lem:choi

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

J_{n} (Φ - Ψ) = (id_{C} \otimes (Φ - Ψ)) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) .

Then

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

In particular, a complementary Choi-state bound $e_{J}$ yields $δ_{full} \leq d e_{J}$ , when the replacement state in that bound is the state used in the channel comparison.

Proof. A pure input with a $d$ -dimensional reference has the form

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

Hence $∥ A ∥^{2} \leq d$ . Its output difference is

(A \otimes I) J_{n} (Φ - Ψ) (A^{†} \otimes I),

whose trace norm is at most $d ∥ J_{n} (Φ - Ψ) ∥_{1}$ . Convexity and the finite-reference characterization of channel diamond distance complete the proof.

Displayed formulas in this passage

unlabeled · line 370
$J_{n} (Φ - Ψ) = (id_{C} \otimes (Φ - Ψ)) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) .$
unlabeled · line 376
$d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ - Ψ) ∥_{1} .$
unlabeled · line 388
$∣ ψ ⟩ = (A \otimes I) ∣ Φ_{d} ⟩, Tr (A^{†} A) = d .$
unlabeled · line 394
$(A \otimes I) J_{n} (Φ - Ψ) (A^{†} \otimes I),$

Lemma: Normalized Choi-to-diamond estimate

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