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We consider the optimal approximation of arbitrary qubit states with respectto an available states consisting the eigenstates of two of three Paulimatrices, the $B_2$-distance of an arbitrary target state. Both the analyticalformulae of the $B_2$-distance and the corresponding complete optimaldecompositions are obtained. The tradeoff relations for both the sum and thesquared sum of the $B_2$-distances have been analytically and numericallyinvestigated.

Complete Optimal Convex Approximations of Qubit States under $B_2$ Distance