# Approximating an ensemble with an orthogonal ensemble

Consider an arbitrary ensemble $$\{p_x\rho_x\}_{x\in X}$$ and define the state $$\rho = \sum_{x\in X} \vert x \rangle\langle x \vert \otimes p_x\rho_x.$$ I am interested in understanding the quantity $$\inf_{\xi, \hspace{2mm}\xi_x\perp\xi_{x'}} \Vert \rho - \xi \Vert_1, \quad\quad \text{with } \Vert\cdot\Vert_1 = \operatorname{Tr}\vert\cdot\vert,$$ where the infimum is taken over all orthogonal ensembles, that is for the ensemble $$\{q_x\xi_x\}_{x\in X}$$ we define $$\xi = \sum_{x\in X}\vert x \rangle\langle x \vert \otimes q_x\xi_x$$ and we assume $$\xi_x\perp \xi_{x'}$$ for all distinct $$x,x'\in X$$ in the sense $$\operatorname{Tr}\xi_x\xi_{x'} = 0$$. In particular, my curiosity is directed at having some intuition as to what a good choice of $$\xi$$ depending on $$\rho$$ is when trying to minimize the expression above. For instance, one could let $$\{M_x\}_{x\in X}$$ denote a set of orthogonal projections and define $$\xi_x = \frac{M_x\rho_x M_x}{\operatorname{Tr} M_x\rho_x M_x}$$ and $$q_x = p_x$$. Then \begin{align*} \Vert \rho - \xi\Vert_1 &= \sum_{x\in X} \Vert p_x\rho_x - p_x\xi_x\Vert_1 \\ &= \sum_{x\in X} p_x\left\Vert \left(1 - \frac{1}{\operatorname{Tr} M_x\rho_xM_x}\right)M_x\rho_xM_x - (Id-M_x)\rho_x(Id-M_x)\right\Vert_1 \\ &= \sum_{x\in X} p_x\left(\left(\frac{1}{\operatorname{Tr} M_x\rho_xM_x}-1\right)\operatorname{Tr} M_x\rho_xM_x + \operatorname{Tr} (Id-M_x)\rho_x(Id-M_x)\right) \\ &= 1 - \sum_{x\in X}\operatorname{Tr}(2M_x-Id)p_x\rho_x \\ &= 2\left(1 - \sum_{x\in X}\operatorname{Tr}M_xp_x\rho_x\right) \end{align*} So choosing $$\{M_x\}_{x\in X}$$ such that $$\sum_{x\in X}\operatorname{Tr}M_xp_x\rho_x$$ is maximized is a decent choice! However, is this optimal in general? Or are there any other natural constructions of $$\xi$$, which give a pretty good upper bound on the quantity in question?

Any help is appreciated!