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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.

Let $r$ be the input Schmidt rank, so ${\lambda }_j=0$ for $j>r$ and ${\sum }_{j=1}^r{\lambda }_j=1$. If $r=d$, the output rank is already at most $r$. Otherwise, Nielsen's pure-state conversion theorem and majorization at $k=r$ giveThus all output coefficients beyond $r$ vanish, provingThis is monotonicity of Schmidt rank under LOCC. It also holds separately in any nonzero postselected branch: represent the input amplitudes by a matrix $C$; a local branch maps it to $AC{B}^T$, whose rank cannot exceed the rank of $C$. The deterministic result requested here follows already from the majorization criterion.

Nielsen's pure-state conversion theorem gives the exact deterministic LOCC criterion. Let the Schmidt decompositions be $|\psi ⟩={\sum }_j\sqrt{{\lambda }_j}|a_j{b}_j⟩$ and $|\varphi ⟩={\sum }_j\sqrt{{\mu }_j}|a_j^{\prime }{b}_j^{\prime }⟩$. The vectors $\lambda ,\mu$ consist of squared Schmidt coefficients, equivalently the eigenvalues of either reduced density operator. Order each in decreasing order and pad with zeros to a common length $d$.

Then deterministic conversion is possible if and only ifThis is majorization, with the input vector majorized by the output vector. The direction matters: a maximally entangled state has a uniform vector, which is majorized by a product state's vector $(1,0,\dots )$, so entanglement can be discarded by LOCC. The criterion is for certain exact conversion, without catalysts or postselection on a successful branch.