Manuscript En07-C@1 · En07-C.tex · lemma · 3f7e9ebb

Lemma: Normalized Choi and matrix-unit conversions

Approximate complement certificates and finite tests › Finite matrix certificates · lines 1062–1102 · proof included or linked

En07-C@1#lem:choi, En07-C@1#eq:choi-conversion, En07-C@1#eq:matrix-unit-conversion

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

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

If $∥ Φ (E_{ij}) - Ψ (E_{ij}) ∥_{1} \leq r$ for every matrix unit, then

d_{⋄} (Φ, Ψ) \leq min {1, d r /2} .

Proof. The Bell state is an admissible referenced input, proving the first inequality. A pure referenced input can be written, after a reference isometry, as $(A \otimes I_{C}) ∣ Φ_{d} ⟩$ with $∥ A ∥_{\infty}^{2} \leq d$ . Applying $Φ - Ψ$ and multiplying its Choi difference on the reference side gives a trace norm at most $d ∥ J_{Φ} - J_{Ψ} ∥_{1}$ . Convexity and the channel-difference state optimization prove the upper bound.

Alternatively, write a pure referenced input vector as $\sum_{i} ∣ v_{i} ⟩ \otimes ∣ i ⟩$ , with $\sum_{i} ∥ v_{i} ∥^{2} = 1$ . The matrix-unit hypothesis bounds the output difference by

r i, j \sum ∥ v_{i} ∥ ∥ v_{j} ∥ \leq d r .

Divide by two and use convexity.

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En07-C@1#eq:choi-conversion · line 1065
$d_{tr} (J_{Φ}, J_{Ψ}) \leq d_{⋄} (Φ, Ψ) \leq min {1, d d_{tr} (J_{Φ}, J_{Ψ})} .$
En07-C@1#eq:matrix-unit-conversion · line 1074
$d_{⋄} (Φ, Ψ) \leq min {1, d r /2} .$
unlabeled · line 1097
$r i, j \sum ∥ v_{i} ∥ ∥ v_{j} ∥ \leq d r .$

Lemma: Normalized Choi and matrix-unit conversions

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