Nielsen's pure-state conversion theorem ($|\psi ⟩\stackrel{\mathrm{LOCC}}{\to }|\varphi ⟩⟺\lambda (\psi )\prec \lambda (\varphi )$)

An exact deterministic LOCC conversion of a bipartite pure state to another is possible exactly when the input vector of squared Schmidt coefficients is majorized by the output vector. Sort the two probability vectors decreasingly and pad with zeros to a common length before applying majorization. A maximally entangled state may therefore be converted to a product state, while the reverse conversion is forbidden. The criterion does not assert conversion by postselection or with a catalyst.