= Diagonal logarithm concavity bound
{title2=$\langle v|\log A|v\rangle\leq\log\langle v|A|v\rangle$}
For a positive-definite <Hermitian operator> $A=\sum_jt_j|u_j\rangle\langle u_j|$ and a unit vector $v$, the weights $|\langle v|u_j\rangle|^2$ form a probability distribution. Scalar logarithmic concavity bounds their average of $\log t_j$ by the logarithm of their average of $t_j$. Positive semidefinite operators follow by a limiting convention. Using the eigenbasis of a <density operator> $\rho$ gives $S(\rho\|\sigma)\geq D(r\|q)$, where $r$ is the spectrum of $\rho$ and $q$ the diagonal of $\sigma$ in that basis. This yields <nonnegativity of quantum relative entropy> without commutativity.
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