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

Lemma: Two-copy Haar twirl

Exact Haar decoupling and the emission threshold · lines 577–608 · proof included or linked

Hawk06-A@1#lem:twirl, Hawk06-A@1#eq:twirlcoeff

On a $D$ -dimensional space with $D \geq 2$ ,

E_{U} U^{†\otimes 2} O U^{\otimes 2} = α I + β F,

where

α = \frac{Tr O - Tr ( O F ) / D}{D ^{2} - 1}, β = \frac{Tr ( O F ) - Tr O / D}{D ^{2} - 1} .

For $O = I_{R R^{'}} \otimes F_{B B^{'}}$ ,

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

Proof. The averaged operator commutes with every $U \otimes U$ . The symmetric and antisymmetric subspaces give its two scalar blocks, equivalently its expression as $α I + β F$ . Preservation of $Tr O$ and $Tr (O F)$ solves the two linear equations. For the specified $O$ , those traces are $r^{2} b$ and $r b^{2}$ .

Displayed formulas in this passage

Hawk06-A@1#eq:twirlcoeff · line 593
$α = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1}, β = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1} .$
unlabeled · line 580
$E_{U} U^{†\otimes 2} O U^{\otimes 2} = α I + β F,$
unlabeled · line 585
$α = \frac{Tr O - Tr ( O F ) / D}{D ^{2} - 1}, β = \frac{Tr ( O F ) - Tr O / D}{D ^{2} - 1} .$

Lemma: Two-copy Haar twirl

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