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

Theorem: Exact complementary Choi moment

The Haar decoupling calculation · lines 762–807 · proof included or linked

Hawk07-A@1#thm:moment, Hawk07-A@1#eq:moment

For the complete entangled preparation,

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

Proof. After tracing $E$ , the initial reference–vault state is

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

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

The swap identity and Lemma lem:twirl give

E Tr [(ω_{QB}^{U})^{2}] = α / d + β / h,

E Tr [(σ_{B}^{U})^{2}] = α + 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:twirl).

Displayed formulas in this passage

Hawk07-A@1#eq:moment · line 765
$E_{U} X_{QB}^{U}_{2}^{2} = \frac{r ( b ^{2} - 1 ) ( d ^{2} - 1 )}{d ^{2} h ( D ^{2} - 1 )} .$
unlabeled · line 778
$ρ_{Q, vault} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{Q L} \otimes I_{H} / h \otimes ∣ 0 ⟩ ⟨ 0 ∣_{F} .$
unlabeled · line 788
$E Tr [(ω_{QB}^{U})^{2}] = α / d + β / h,$
unlabeled · line 793
$E Tr [(σ_{B}^{U})^{2}] = α + β / (d h) .$
unlabeled · line 799
$X_{QB}^{U}_{2}^{2} = Tr [(ω_{QB}^{U})^{2}] - \frac{1}{d} Tr [(σ_{B}^{U})^{2}] .$