= Trace-norm variational principle for Hermitian operators
{title2=$\|X\|_1=\max_{-I\leq T\leq I}\operatorname{Tr}(XT)$}
A <Hermitian operator> satisfies $\|X\|_1=\max_{-I\leq T\leq I}\operatorname{Tr}(XT)$. Each diagonal entry of $T$ in an eigenbasis of $X$ lies between minus one and one, giving the upper bound $\sum_i|\lambda_i|$. The sign operator of $X$ attains it. The substitution $T=2E-I$ converts this formula to optimization over binary <POVM> effects and proves the <Holevo–Helstrom theorem>.
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