= Exact dilution obstruction from Schmidt rank
{title2=$m\geq n\quad\text{for exact }\mathrm{GHZ}^{\otimes m}\to\Psi_p^{\otimes n}$}
For $0<p<1$, every bipartition of $\Psi_p^{\otimes n}$ has <Schmidt rank> $2^n$, while $m$ copies of a <GHZ state> have <Schmidt rank> $2^m$. <Schmidt-rank contraction under product operators> therefore forbids any exact successful <LOCC> branch when $m<n$. Approximate <entanglement dilution> avoids this obstruction by retaining only a high-weight <quantum typical subspace>, whose dimension has logarithm $nh_2(p)+o(n)$. Exact finite-block conversion and asymptotically faithful conversion are different tasks.
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