Theorem: Two exact means for one cell-release proxy

For every fixed pure input independent of $U$, coherent radiation and immediate destructive cell readout have the exact means

The classical mean increases strictly at each release. The coherent mean increases on cuts with $r_k\le b_k$ that remain on that side, and decreases on cuts with $r_k\ge b_k$ that remain on that side. If the capacity crossing is traversed, a maximal cut is among the two cuts bracketing it.

If $b_{\ast }=1$, the coherent endpoint mean is zero while the record endpoint mean is

If the schedule stops before a crossing, no turnover is guaranteed.

Proof. Haar invariance makes $UJ|\psi ⟩$ a Haar unit vector, and Lemma lem:pagesen applies at every cut.

Write its product-basis components as normalized independent complex Gaussians. A radiation prefix has $b_k$ squared Gaussian components. The $r_k$ prefix probabilities therefore have a Dirichlet distribution with all parameters $b_k$. Its moment identity is

Differentiate at $t=1$. Using ${\psi }_0(\ell +1)={\mathsf{H}}_{\ell }-{\gamma }_{\mathrm{E}}$ gives the record mean. Later SWAPs act on unmeasured factors, so immediate readout has precisely these prefix probabilities.

For a same-side coherent step, write the smaller and larger dimensions as $m,n$, and transfer an integer factor $s\ge 2$ so that the next early-side dimensions are $sm,n/s$. Remaining on that side implies $n\ge s^{2}m$. The Page-mean increment in natural units is

Exchange the two factors for the late side. The record increment is $({\mathsf{H}}_{b_{k-1}}-{\mathsf{H}}_{b_k})/\ln 2>0$. The endpoint assertions follow by substitution.