Optimal convex approximations of quantum states

We consider the problem of optimally approximating an unavailable quantum state p by the convex mixing of states drawn from a set of available states {v(i)}. The problem is recast to look for the least distinguishable state from p among the convex set Sigma(i) p(i)v(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.