# Are almost perfectly distinguishable ensembles almost orthogonal?

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!