Complete Optimal Convex Approximations of Qubit States under B₂ Distance

We consider the optimal approximation of arbitrary qubit states with respect to an available states consisting the eigenstates of two of three Pauli matrices, the $B_2$-distance of an arbitrary target state. Both the analytical formulae of the $B_2$-distance and the corresponding complete optimal decompositions are obtained. The tradeoff relations for both the sum and the squared sum of the $B_2$-distances have been analytically and numerically investigated.