Manuscript Oai265-B@2 · Oai265-B-v2.tex · lemma · 94b6f42f

Lemma: Normalized Choi conversion

Standard finite-dimensional conversions · lines 398–430 · proof included or linked

Oai265-B@2#lem:choi, Oai265-B@2#eq:choi

For channels with common input dimension $d$ and identified outputs,

d_{tr} (J_{Φ}, J_{Ψ}) \leq d_{⋄} (Φ, Ψ) \leq min {1, d d_{tr} (J_{Φ}, J_{Ψ})} .

Proof. The Bell input proves the lower bound. Put $Δ = Φ - Ψ$ . For unit vectors $u, v$ in an arbitrary reference space tensored with the input, write

u = (A \otimes I) ∣ Φ_{d} ⟩, v = (B \otimes I) ∣ Φ_{d} ⟩, Tr (A^{†} A) = Tr (B^{†} B) = d .

Hence $∥ A ∥_{\infty}, ∥ B ∥_{\infty} \leq d$ , and

∥ (id \otimes Δ) (∣ u ⟩ ⟨ v ∣) ∥_{1} = (A \otimes I) J_{Δ} (B^{†} \otimes I)_{1} \leq d ∥ J_{Δ} ∥_{1} .

A singular-value decomposition extends this bound to arbitrary trace-norm-one input operators and all reference dimensions. Divide by two and use the channel-distance cap one.

Displayed formulas in this passage

Oai265-B@2#eq:choi · line 401
$d_{tr} (J_{Φ}, J_{Ψ}) \leq d_{⋄} (Φ, Ψ) \leq min {1, d d_{tr} (J_{Φ}, J_{Ψ})} .$
unlabeled · line 414
$u = (A \otimes I) ∣ Φ_{d} ⟩, v = (B \otimes I) ∣ Φ_{d} ⟩, Tr (A^{†} A) = Tr (B^{†} B) = d .$
unlabeled · line 421
$∥ (id \otimes Δ) (∣ u ⟩ ⟨ v ∣) ∥_{1} = (A \otimes I) J_{Δ} (B^{†} \otimes I)_{1} \leq d ∥ J_{Δ} ∥_{1} .$

Lemma: Normalized Choi conversion

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