 Variational characterization of trace distance (source code)

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= Variational characterization of trace distance

For density operators $\rho,\sigma$,
$$
D(\rho,\sigma)=\max_{0\leq P\leq I}\operatorname{Tr}[P(\rho-\sigma)].
$$
The maximum is attained by the projector onto the positive spectral subspace of $\rho-\sigma$.

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