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What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in -dimensional Hilbert space ? The known result is that the number is no less than . We show that this lower bound is not tight except for  or 4. Then we give an upper bound . For , many rank-1 POVMs with four elements can determine any pure states in . For , we show eight is the minimal number by construction. For , the minimal number is in the set of . We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in . For any dimension , we construct  adaptive rank-1 positive operators for the reconstruction of any unknown pure state in , where .

Pure state ‘really’ informationally complete with rank-1 POVM