Manuscript Hawk05-B@1 · Hawk05-B.tex · section · f2e5d4f5

Evaluation of the exact Page average

Evaluation of the exact Page average · lines 1669–1701

Hawk05-B@1#eq:pagegamma

This appendix evaluates the standard random-state average. The result is due to Page and Sen; no priority is claimed PageAverage,Sen. Natural logarithms are used until the final conversion to bits.

Let $1 \leq m \leq n$ , $D = mn$ , and let $G$ be an $m \times n$ matrix of independent standard complex Gaussians. Set

W = G G^{†}, T = Tr W, ρ = W / T .

The normalized Gaussian vector is Haar on the unit sphere and has reduction $ρ$ . Gaussian radius and direction are independent, so $T$ and $ρ$ are independent. Moreover, $T$ has Gamma density with shape $D$ :

E T = D, E (T ln T) = D ψ_{0} (D + 1) .

Since

Tr (W ln W) = T ln T + T Tr (ρ ln ρ),

the desired natural-log entropy satisfies

E S_{nat} (ρ) = ψ_{0} (D + 1) - \frac{1}{D} E Tr (W ln W) .

Displayed formulas in this passage

Hawk05-B@1#eq:pagegamma · line 1695
$E S_{nat} (ρ) = ψ_{0} (D + 1) - \frac{1}{D} E Tr (W ln W) .$
unlabeled · line 1679
$W = G G^{†}, T = Tr W, ρ = W / T .$
unlabeled · line 1686
$E T = D, E (T ln T) = D ψ_{0} (D + 1) .$
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$Tr (W ln W) = T ln T + T Tr (ρ ln ρ),$