Manuscript Hawk07-A@1 · Hawk07-A.tex · lemma · f9c0f5da

Lemma: Choi conversion with its dimension factor

The Haar decoupling calculation · lines 809–856 · proof included or linked

Hawk07-A@1#lem:choi, Hawk07-A@1#eq:K

For channels with input dimension $d$ ,

\frac{1}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} \leq d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .

Consequently,

E_{U} z (U)^{2} \leq K (r, b) := \frac{d D ( b ^{2} - 1 ) ( d ^{2} - 1 )}{4 h ( D ^{2} - 1 )} \leq \frac{d ^{4} f}{4 r ^{2}},

where $z (U) = d_{⋄} (C^{U}, T_{σ_{B}^{U}})$ .

Proof. The Bell input gives the lower inequality. Every pure referenced input with a sufficient reference dimension can be written as $(T \otimes I) ∣ Φ_{d} ⟩$ , with $Tr (T^{†} T) = d$ , hence $∥ T ∥_{\infty}^{2} \leq d$ . Multiplying the Choi difference by $T \otimes I$ on both sides gives the upper inequality; convexity handles mixed inputs. This is the usual normalized Choi-to-diamond conversion WatrousBook.

The Choi difference acts on dimension $d b$ , so

z (U)^{2} \leq \frac{d ^{2}}{4} X_{QB}^{U}_{1}^{2} \leq \frac{d ^{3} b}{4} X_{QB}^{U}_{2}^{2} .

Take expectations and use (eq:moment). Finally,

\frac{b ^{2} - 1}{D ^{2} - 1} \leq \frac{b ^{2}}{D ^{2}}, d^{2} - 1 \leq d^{2}, D = d h f, b = D / r,

which gives the last bound.

Displayed formulas in this passage

Hawk07-A@1#eq:K · line 819
$E_{U} z (U)^{2} \leq K (r, b) := \frac{d D ( b ^{2} - 1 ) ( d ^{2} - 1 )}{4 h ( D ^{2} - 1 )} \leq \frac{d ^{4} f}{4 r ^{2}},$
unlabeled · line 812
$\frac{1}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} \leq d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$
unlabeled · line 844
$z (U)^{2} \leq \frac{d ^{2}}{4} X_{QB}^{U}_{1}^{2} \leq \frac{d ^{3} b}{4} X_{QB}^{U}_{2}^{2} .$
unlabeled · line 851
$\frac{b ^{2} - 1}{D ^{2} - 1} \leq \frac{b ^{2}}{D ^{2}}, d^{2} - 1 \leq d^{2}, D = d h f, b = D / r,$

Lemma: Choi conversion with its dimension factor

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