= Entropy of a binary diagonal qubit mixture
{title2=$S((1-u)|0\rangle\langle0|+u|1\rangle\langle1|)=h_2(u)$}
A mixture of two orthogonal pure qubit states has <Von Neumann entropy> equal to the <binary entropy> of its weights. Its entropy difference from the first <pure state> is exactly $h_2(u)$ and its <trace distance> from that state is $u$. The elementary bound $-(1-u)\ln(1-u)\leq u$ gives $h_2(u)\leq u\log_2(e/u)$; entropy also never exceeds one bit. For small $u$, the leading term is $u\log_2(1/u)$.
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