Let $\varepsilon>0$ and consider an ensemble of states $\{p_x\rho_x\}_{x\in X}$ and suppose there exists a measurement with POVM representation $\{M_x\}_{x\in X}$ such that $$ \sum_{x\in X} p_x\operatorname{Tr} M_x\rho_x \geq 1-\varepsilon. $$ Does this imply the existence of another ensemble $\{q_x\sigma_x\}_{X\in X}$ of orthogonal states, that is $\sigma_x \perp \sigma_{x'}$ for all distinct $x,x'\in X$, such that $$ \sum_{x\in X} \lVert p_x\rho_x - q_x\sigma_x\rVert_1 = \sum_{x\in X} \operatorname{Tr}\lvert p_x\rho_x - q_x\sigma_x\rvert_1 \leq O(\varepsilon). $$ Intuitively, I like to think states are almost perfectly distinguishable, when they are almost orthogonal. Hence the conjecture.

Any help is appreciated!



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