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No, this is not possible. Consider $\rho_1 = \sigma_2 = \vert 0\rangle\langle 0 \vert$ and $\rho_2 = \sigma_1 = \vert 1\rangle\langle 1 \vert$. Then, $$D_{\max}(\rho_i\|\sigma_i) = \infty\quad \text{for } i = 1,2.$$ Let $p_i = (1/2, 1/2)$ and you see that $D_{\max}(\rho\|\sigma) = 0$.


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