Monotonicity of Schmidt rank under LOCC ($r_{\mathrm{out}}\le r_{\mathrm{in}}$)

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 $AC{B}^T$, 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.