= Monotonicity of Schmidt rank under LOCC
{title2=$r_{\rm out}\leq r_{\rm in}$}
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 $ACB^T$, 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.
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