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

Theorem: Exact complementary Hilbert–Schmidt moment

Exact Haar decoupling and the emission threshold · lines 610–657 · proof included or linked

Hawk06-A@1#thm:moment, Hawk06-A@1#eq:gamma

In the complete preparation of Section sec:model,

E_{U} X_{QB}^{U}_{2}^{2} = γ (r, b) := \frac{r ( b ^{2} - 1 ) ( d ^{2} - 1 )}{d ^{2} h ( D ^{2} - 1 )} .

The formula includes the blank dimension $f$ through $D = d h f$ .

Proof. Tracing the accessible early register before the calculation gives the initial reference–vault state

ρ_{Q, vault} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{Q L} \otimes \frac{I _{H}}{h} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{F} .

Its squared trace is $1/ h$ , whereas its vault marginal has squared trace $1/ (d h)$ , and its reference marginal has squared trace $1/ d$ .

The swap test and Lemma lem:twirl give

E Tr [(ω_{QB}^{U})^{2}] = \frac{α}{d} + \frac{β}{h}, E Tr [(σ_{B}^{U})^{2}] = α + \frac{β}{d h} .

Since $Tr_{Q} ω_{QB}^{U} = σ_{B}^{U}$ ,

X_{QB}^{U}_{2}^{2} = Tr [(ω_{QB}^{U})^{2}] - \frac{1}{d} Tr [(σ_{B}^{U})^{2}] .

The $α$ terms cancel. Insert $β$ from (eq:twirlcoeff).

Displayed formulas in this passage

Hawk06-A@1#eq:gamma · line 613
$E_{U} X_{QB}^{U}_{2}^{2} = γ (r, b) := \frac{r ( b ^{2} - 1 ) ( d ^{2} - 1 )}{d ^{2} h ( D ^{2} - 1 )} .$
unlabeled · line 630
$ρ_{Q, vault} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{Q L} \otimes \frac{I _{H}}{h} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{F} .$
unlabeled · line 641
$E Tr [(ω_{QB}^{U})^{2}] = \frac{α}{d} + \frac{β}{h}, E Tr [(σ_{B}^{U})^{2}] = α + \frac{β}{d h} .$
unlabeled · line 649
$X_{QB}^{U}_{2}^{2} = Tr [(ω_{QB}^{U})^{2}] - \frac{1}{d} Tr [(σ_{B}^{U})^{2}] .$