Pure state `really' informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive-operator-valued measure (POVM) which can uniquely determine any pure state in $d$-dimensional Hilbert space $\mathcal{H}_d$? The known result is that the number is no less than $3d-2$. We show that this lower bound is not tight except for $d=2$ or 4. Then we give an upper bound of $4d-3$. For $d=2$, many rank-1 POVMs with four elements can determine any pure states in $\mathcal{H}_2$. For $d=3$, we show eight is the minimal number by construction. For $d=4$, the minimal number is in the set of $\{10,11,12,13\}$. We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases can not distinguish all pure states in $\mathcal{H}_4$. For any dimension $d$, we construct $d+2k-2$ adaptive rank-1 positive operators for the reconstruction of any unknown pure state in $\mathcal{H}_d$, where $1\le k \le d$.