Optimal convex approximations of quantum states

We consider the problem of optimally approximating an unavailable quantum state $\rho$ by the convex mixing of states drawn from a set of available states $\{{\nu }_i\}$. The problem is recast to look for the least distinguishable state from $\rho$ among the convex set ${\sum }_i{p}_i{\nu }_i$, and the corresponding optimal weights $\{p_i\}$ provide the optimal convex mixing. We present the complete solution for the optimal convex approximation of a qubit mixed state when the set of available states comprises the three bases of the Pauli matrices.