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

Abstract

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 $4d-3$. 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 cannot 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 $1\le k \le d$.