= Optimal stochastic conversion of a two-qubit pure state
{title2=$p_{\max}=\min\{1,s/t\}$}
Write a source and target in <Schmidt decomposition> with squared coefficients $(1-s,s)$ and $(1-t,t)$, where $0<s,t\leq1/2$. If $s\geq t$, <Nielsen's pure-state conversion theorem> allows deterministic conversion. If $s<t$, the <Schmidt-tail entanglement monotone> gives $p\leq s/t$. Alice attains this bound with a local two-outcome <quantum measurement> whose success <Kraus operator> is $K_s=\operatorname{diag}(\sqrt{s(1-t)/(t(1-s))},1)$ and whose failure operator is $K_f=\operatorname{diag}(\sqrt{1-s(1-t)/(t(1-s))},0)$. They obey $K_s^\dagger K_s+K_f^\dagger K_f=I$. The successful unnormalized state is $\sqrt{s/t}$ times the target, and failure leaves a <product state>. A classical message tells Bob which branch occurred. Known local changes of Schmidt bases allow the same protocol for arbitrary two-qubit pure states with these coefficients.
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