Manuscript Hawk07-B@1 · Hawk07-B.tex · theorem · b9785538

Theorem: Complementary moment at arbitrary payload dimension

Scaled preparation and a rederived recovery threshold › The dimension-dependent moment · lines 515–579 · proof included or linked

Hawk07-B@1#thm:moment, Hawk07-B@1#eq:moment, Hawk07-B@1#eq:momentbound

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At a cut $D = r b$ ,

E_{U} X_{QB}^{U}_{2}^{2} E_{U} z (U)^{2} = \frac{r ( b ^{2} - 1 ) ( d ^{2} - 1 )}{d ^{2} h ( D ^{2} - 1 )}, \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 two-copy Haar twirl of $I_{P P^{'}} \otimes S_{B B^{'}}$ is $α I + β S$ , where

α = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1}, β = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1} .

Indeed, the twirl commutes with all $U \otimes U$ , and its identity and swap coefficients are fixed by the traces $r^{2} b$ and $r b^{2}$ .

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

∣ 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 therefore gives

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

Subtracting the second expression divided by $d$ proves (eq:moment).

For an input dimension $d$ , normalized Choi operators obey

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

To see the factor, write any pure referenced input as $(T \otimes I) ∣ Φ_{d} ⟩$ , with $Tr (T^{†} T) = d$ and $∥ T ∥_{\infty}^{2} \leq d$ . Apply trace-norm multiplication bounds and then convexity 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} .

Taking expectations gives $K$ . Finally use

\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

Hawk07-B@1#eq:moment · line 518
$E_{U} X_{QB}^{U}_{2}^{2} E_{U} z (U)^{2} = \frac{r ( b ^{2} - 1 ) ( d ^{2} - 1 )}{d ^{2} h ( D ^{2} - 1 )}, \leq K (r, b) := \frac{d D ( b ^{2} - 1 ) ( d ^{2} - 1 )}{4 h ( D ^{2} - 1 )} \leq \frac{d ^{4} f}{4 r ^{2}} .$
Hawk07-B@1#eq:momentbound · line 518
$E_{U} X_{QB}^{U}_{2}^{2} E_{U} z (U)^{2} = \frac{r ( b ^{2} - 1 ) ( d ^{2} - 1 )}{d ^{2} h ( D ^{2} - 1 )}, \leq K (r, b) := \frac{d D ( b ^{2} - 1 ) ( d ^{2} - 1 )}{4 h ( D ^{2} - 1 )} \leq \frac{d ^{4} f}{4 r ^{2}} .$
unlabeled · line 536
$α = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1}, β = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1} .$
unlabeled · line 545
$∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{Q L} \otimes I_{H} / h \otimes ∣ 0 ⟩ ⟨ 0 ∣_{F} .$
unlabeled · line 551
$E Tr [(ω_{QB}^{U})^{2}] = α / d + β / h, E Tr [(σ_{B}^{U})^{2}] = α + β / (d h) .$
unlabeled · line 559
$∥ Φ - Ψ ∥_{⋄} \leq d ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1} .$
unlabeled · line 569
$z (U)^{2} \leq \frac{d ^{2}}{4} X_{QB}^{U}_{1}^{2} \leq \frac{d ^{3} b}{4} X_{QB}^{U}_{2}^{2} .$
unlabeled · line 575
$\frac{b ^{2} - 1}{D ^{2} - 1} \leq \frac{b ^{2}}{D ^{2}}, d^{2} - 1 \leq d^{2}, D = d h f, b = D / r .$

Theorem: Complementary moment at arbitrary payload dimension

Displayed formulas in this passage

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