= Schmidt-rank contraction under product operators
{c}
{title2=$\operatorname{rank}(AMB^{\mathsf T})\leq\operatorname{rank}M$}
The coefficient matrix of a bipartite <pure state> has rank equal to its <Schmidt rank>, by <singular value decomposition>. A local product operation $A\otimes B$ sends it to $AMB^{\mathsf T}$, whose rank cannot increase. A complete classical transcript of an <LOCC> protocol still selects one product <Kraus operator>, so every nonzero branch obeys the bound. Discarding the transcript gives a mixed output with <Schmidt number> at most the initial rank.
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