Manuscript Hawk06-A@1 · Hawk06-A.tex · lemma · c59fe67c

Lemma: Normalized Choi-to-diamond conversion

Exact Haar decoupling and the emission threshold · lines 659–704 · proof included or linked

Hawk06-A@1#lem:Choi, Hawk06-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, with

z (U) = d_{⋄} (C^{U}, T_{σ_{B}^{U}}),

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}} .

Proof. The lower Choi inequality uses the Bell input. A pure referenced input is $(T \otimes I) ∣ Φ_{d} ⟩$ , with $Tr (T^{†} T) = d$ and $∥ T ∥_{\infty}^{2} \leq d$ . Multiplying the Choi difference by $T \otimes I$ on both sides bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs WatrousBook.

Since $X_{QB}^{U}$ acts on dimension $d b$ ,

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:gamma). For the last bound,

\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 .

Displayed formulas in this passage

Hawk06-A@1#eq:K · line 671
$E_{U} z (U)^{2} \leq K (r, b) := \frac{d D ( b ^{2} - 1 ) ( d ^{2} - 1 )}{4 h ( D ^{2} - 1 )} \leq \frac{d ^{4} f}{4 r ^{2}} .$
unlabeled · line 662
$\frac{1}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} \leq d_{⋄} (Φ, Ψ) \leq \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$
unlabeled · line 668
$z (U) = d_{⋄} (C^{U}, T_{σ_{B}^{U}}),$
unlabeled · line 693
$z (U)^{2} \leq \frac{d ^{2}}{4} X_{QB}^{U}_{1}^{2} \leq \frac{d ^{3} b}{4} X_{QB}^{U}_{2}^{2} .$
unlabeled · line 700
$\frac{b ^{2} - 1}{D ^{2} - 1} \leq \frac{b ^{2}}{D ^{2}}, d^{2} - 1 \leq d^{2}, D = d h f, b = D / r .$

Lemma: Normalized Choi-to-diamond conversion

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