  Measuring the closest distance between two states is an alternative and
significant approach in the resource quantification, which is the core task in
the resource theory. Quite limited progress has been made for this approach
even in simple systems due to the various potential complexities. Here we
analytically solve the optimal scheme to find out the closest distance between
the objective qubit state and all the possible states convexly mixed by some
limited states, namely, to optimally construct the objective qubit state using
the quantum states within any given state set. In particular, we find the least
number of (not more than four) states within a given set to optimally construct
the objective state and also find that any state can be optimally established
by at most four quantum states of the set. The examples in various cases are
presented to verify our analytic solutions further.
