Optimal convex approximations of quantum states

We consider the problem of optimally approximating an unavailable quantum state ρ by the convex mixing of states drawn from a set of available states {νi}. The problem is recast to look for the least distinguishable state from ρ among the convex set -ipiνi, and the corresponding optimal weights {pi} 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.