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

Let three parties share the normalized quantum stateFor $n$ copies, group each party's $n$ qubits together. Across any nontrivial division of these three parties, the Schmidt decomposition has one nonzero coefficient for every binary string of length $n$, so its Schmidt rank is $2^n$. Similarly, $m$ copies of the standard GHZ state have Schmidt rank $2^m$ across the same division.

Every recorded LOCC branch acts by a product of local Kraus operators. Schmidt-rank contraction under product operators therefore rules out an exact successful conversion when $m<n$, even if success need only have nonzero probability. This is a rank obstruction for the full target, not merely a bound on its entanglement entropy.

Approximation changes the question. In the expansion of $|{\Psi }_p{⟩}^{\otimes n}$, squared coefficients have the binary-product distribution with parameter $p$. A quantum typical subspace retains weight tending to one while using only $2^{n{h}_2(p)+o(n)}$ strings, where $h_2$ is binary entropy. The normalized truncation approaches the full target in quantum fidelity, although its Schmidt rank can be far smaller than $2^n$. Consequently, the exact finite-copy rank argument does not forbid an asymptotic entanglement dilution protocol with vanishing approximation error.