= Local Kraus rank bound from an entangled resource
{title2=$\operatorname{OSR}(K_{\mu\nu})\leq R$}
If two parties share a pure resource of <Schmidt rank> $R$, a fully refined branch of their local operations has <Kraus operator> $K_{\mu\nu}=\sum_{j=1}^R\sqrt{p_j}A_{\mu j}\otimes B_{\nu j}$. Hence its <operator Schmidt rank> is at most $R$. Local ancillas, locally adaptive readouts and refined discarded environments are included in the branch operators. Shared classical randomness only mixes such branches. Classical comparison of records does not increase their <operator Schmidt rank>.
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