Manuscript Hawk03-A@1 · Hawk03-A.tex · remark · 9a109c12

Remark: Diagnostic versus worst-case certificate

Complete complements and reference-preserving recovery › Approximate recovery · lines 618–636

Hawk03-A@1#eq:choisandwich

A single Bell-input test is not automatically a diamond-norm certificate. For input dimension $m$ , if

J_{Λ} = (id_{Q} \otimes Λ) (∣ Φ_{m} ⟩ ⟨ Φ_{m} ∣),

then

d_{tr} (J_{Λ}, J_{Γ}) \leq d_{⋄} (Λ, Γ) \leq m d_{tr} (J_{Λ}, J_{Γ}) .

For the upper bound, write a purified input as $(A \otimes I) ∣ Φ_{m} ⟩$ , with $∥ A ∥_{\infty} \leq m$ , and apply the trace-norm ideal property. Convexity and Schmidt compression cover arbitrary inputs and references. This dimension factor must not be silently discarded.

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Hawk03-A@1#eq:choisandwich · line 625
$d_{tr} (J_{Λ}, J_{Γ}) \leq d_{⋄} (Λ, Γ) \leq m d_{tr} (J_{Λ}, J_{Γ}) .$
unlabeled · line 621
$J_{Λ} = (id_{Q} \otimes Λ) (∣ Φ_{m} ⟩ ⟨ Φ_{m} ∣),$

Remark: Diagnostic versus worst-case certificate

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