Exact dilution obstruction from Schmidt rank ($m\ge n\text{for exact }{\mathrm{GHZ}}^{\otimes m}\to {\Psi }_p^{\otimes n}$)

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 $n{h}_2(p)+o(n)$. Exact finite-block conversion and asymptotically faithful conversion are different tasks.