= Eigenvalue mixing matrix of a unital quantum channel
{title2=$A_{ji}=\langle v_j|\Lambda(|u_i\rangle\langle u_i|)|v_j\rangle$}
Choose input eigenvectors $u_i$ and output eigenvectors $v_j$ for a <unital quantum channel>. Positivity makes the displayed entries nonnegative. Trace preservation makes every column sum one; unitality makes every row sum one. Thus $A$ is a <doubly stochastic matrix> and the output spectrum satisfies $s=Ar$. The mixing matrix need not be a matrix of squared overlaps of a single unitary, even though its row and column constraints are the same.
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