Manuscript Oai283-A@1 · Oai283-A.tex · lemma · 6fb75895

Lemma: Normalized Choi bounds

Operational certification of the logical channel › A constructive, dimension-explicit conversion · lines 805–832 · proof included or linked

Oai283-A@1#lem:choi

For channels with input dimension $d$ ,

d_{tr} (J_{n} (Φ), J_{n} (Ψ)) \leq d_{⋄} (Φ, Ψ) \leq d d_{tr} (J_{n} (Φ), J_{n} (Ψ)) .

Proof. The first inequality uses the maximally entangled input.

For the second, any pure input with a reference can be written

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

Here $B$ may map into a larger reference, and $∥ B ∥_{\infty} \leq d$ . Consequently the output difference has trace norm at most

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

Use convexity and the channel diamond-distance characterization.

Displayed formulas in this passage

unlabeled · line 808
$d_{tr} (J_{n} (Φ), J_{n} (Ψ)) \leq d_{⋄} (Φ, Ψ) \leq d d_{tr} (J_{n} (Φ), J_{n} (Ψ)) .$
unlabeled · line 819
$∣ ψ ⟩ = (B \otimes I) ∣ Ω_{d} ⟩, Tr (B^{†} B) = d .$
unlabeled · line 826
$∥ B ∥_{\infty}^{2} ∥ J_{n} (Φ - Ψ) ∥_{1} \leq d ∥ J_{n} (Φ - Ψ) ∥_{1} .$