The Best Approximation of an Objective State With a Given Set of Quantum States

Approximating a quantum state by the convex mixing of some given states has strong experimental significance and provides alternative understandings of quantum resource theory. It is essentially a complex optimal problem which, up to now, has only partially solved for qubit states. Here, the most general case is focused on that the approximation of a d-dimensional objective quantum state by the given state set consisting of any number of (mixed-) states. The problem is thoroughly solved with a closed solution of the minimal distance in the sense of l(2) norm between the objective state and the set. In particular, the minimal number of states in the given set is presented to achieve the optimal distance. The validity of this closed solution is further verified numerically by several randomly generated quantum states.