= Operator Schmidt rank
{title2=$\operatorname{OSR}(K)=\min\{r:K=\sum_{j=1}^rA_j\otimes B_j\}$}
The operator Schmidt rank of a bipartite <linear operator> is its shortest expansion as a sum of product <linear operators>. Expanding in local orthonormal operator bases gives a coefficient <matrix>; its <matrix rank> is the <operator Schmidt rank>, by a <singular value decomposition>. It is unchanged by invertible local operator-basis changes. A rank-one projector onto a bipartite <pure state> of <Schmidt rank> $d$ has <operator Schmidt rank> $d^2$: its expansion contains all $d^2$ independent products of local matrix units.
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