Manuscript Oai280-C@1 · Oai280-C.tex · corollary · 7d44b8c0

Corollary: Identity-target overlap conversion

Dimension-correct channel conversion · lines 1308–1332 · proof included or linked

Oai280-C@1#cor:pure, Oai280-C@1#eq:pureconversion

If $Λ : B (C^{d}) \to B (C^{d})$ and $f = Tr (Φ_{d} J (Λ))$ , then

1 - f \leq d_{⋄} (Λ, id_{d}) \leq min {1, d 1 - f} .

Proof. The Bell-projector measurement proves the lower bound. For completeness, the pure-state trace-distance inequality

\frac{1}{2} ∥ σ - ∣ ψ ⟩ ⟨ ψ ∣ ∥_{1} \leq 1 - ⟨ ψ ∣ σ ∣ ψ ⟩

follows by purifying $σ$ with overlap $⟨ ψ ∣ σ ∣ ψ ⟩$ with a purification of $∣ ψ ⟩ ⟨ ψ ∣$ , evaluating the trace distance of the two pure states, and tracing the purifier. Apply this inequality to $J (Λ)$ and use Lemma lem:choi.

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Oai280-C@1#eq:pureconversion · line 1312
$1 - f \leq d_{⋄} (Λ, id_{d}) \leq min {1, d 1 - f} .$
unlabeled · line 1322
$\frac{1}{2} ∥ σ - ∣ ψ ⟩ ⟨ ψ ∣ ∥_{1} \leq 1 - ⟨ ψ ∣ σ ∣ ψ ⟩$

Corollary: Identity-target overlap conversion

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