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For real vectors of equal length and equal sum, write $x\prec y$ when ${\sum }_{j=1}^k{x}_j^{\downarrow }\le {\sum }_{j=1}^k{y}_j^{\downarrow }$ for every proper initial segment; the downward arrow means decreasing rearrangement. Majorization compares how unevenly a fixed total is distributed. The uniform probability vector is majorized by every probability vector, while $(1,0,\dots ,0)$ majorizes every probability vector. The order controls exact pure-state entanglement conversion in Nielsen's pure-state conversion theorem.

The Schmidt rank of a bipartite pure state cannot increase under local operations, even in a nonzero selected branch. A coefficient matrix $C$ changes under a local product operator to $AC{B}^T$, whose rank is at most that of $C$. For deterministic exact conversions the same conclusion follows from Nielsen's pure-state conversion theorem: at the input rank $r$, the majorization inequality forces all output coefficients after $r$ to vanish.

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.