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When can the max relative entropy be written as $D_{\max}(\rho\|\sigma) = \|\sigma^{-1/2}\rho\sigma^{-1/2}\|_{\infty}$?

It's worth noting that even if $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$, which you need to have $D_{\rm max}(\rho\|\sigma)<\infty$, you don't necessarily have $\sigma$ ...
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