Manuscript En07-B@1 · En07-B.tex · lemma · 45dff4a1

Lemma: Normalized Choi versus channel distance

Approximate recovery and finite certificates › Dimension-explicit Choi certificates · lines 1100–1144 · proof included or linked

En07-B@1#lem:choi

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

c (Φ, Ψ) = \frac{1}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .

Then

c (Φ, Ψ) \leq d_{⋄} (Φ, Ψ) \leq d c (Φ, Ψ) .

Proof. The lower bound is the Bell input.

For the upper bound, use the standard stabilization fact stated in Section sec:interface. Any normalized pure state on $Q \otimes C$ , with $dim Q = d$ , has the form

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

Thus $∥ S ∥^{2} \leq d$ , and

∥ (id_{Q} \otimes (Φ - Ψ)) (∣ ψ ⟩ ⟨ ψ ∣) ∥_{1} = (S \otimes I) (J_{n} (Φ) - J_{n} (Ψ)) (S^{†} \otimes I)_{1} \leq d ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .

Convexity handles mixed states. Divide by two and take the supremum.

Displayed formulas in this passage

unlabeled · line 1103
$c (Φ, Ψ) = \frac{1}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$
unlabeled · line 1109
$c (Φ, Ψ) \leq d_{⋄} (Φ, Ψ) \leq d c (Φ, Ψ) .$
unlabeled · line 1124
$∣ ψ ⟩ = (S \otimes I_{C}) ∣ Φ_{d} ⟩, Tr (S^{†} S) = d .$
unlabeled · line 1130
$∥ (id_{Q} \otimes (Φ - Ψ)) (∣ ψ ⟩ ⟨ ψ ∣) ∥_{1} = (S \otimes I) (J_{n} (Φ) - J_{n} (Ψ)) (S^{†} \otimes I)_{1} \leq d ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$

Lemma: Normalized Choi versus channel distance

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