Manuscript Hawk05-A@1 · Hawk05-A.tex · lemma · 449f85bb

Lemma: Normalized Choi comparison

Approximate and same-model endpoint certificates › A finite Choi certificate · lines 1435–1465 · proof included or linked

Hawk05-A@1#lem:choi

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For channels $F, Q$ with input dimension $d$ ,

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

Proof. The first inequality uses the Bell input. For a pure referenced input, write

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

allowing $T$ to be rectangular for a larger reference. Then $∥ T ∥_{\infty}^{2} \leq d$ , and its output difference is

(T \otimes I) [J_{n} (F) - J_{n} (Q)] (T^{†} \otimes I) .

Its trace norm is at most $d$ times the Choi trace norm. Convexity handles mixed referenced states. The state characterization of channel diamond distance gives the result Watrous (Chapter 3).

Displayed formulas in this passage

unlabeled · line 1438
$\frac{1}{2} ∥ J_{n} (F) - J_{n} (Q) ∥_{1} \leq d_{⋄} (F, Q) \leq \frac{d}{2} ∥ J_{n} (F) - J_{n} (Q) ∥_{1} .$
unlabeled · line 1449
$∣ ψ ⟩ = (T \otimes I_{C}) ∣ Φ_{d} ⟩, Tr (T^{†} T) = d,$
unlabeled · line 1456
$(T \otimes I) [J_{n} (F) - J_{n} (Q)] (T^{†} \otimes I) .$

Lemma: Normalized Choi comparison

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