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

TL;DR

This paper investigates the minimal number of rank-1 POVM elements needed to uniquely determine pure states in a d-dimensional quantum system, providing bounds and constructions for various dimensions.

Contribution

It establishes new bounds on the minimal number of rank-1 POVMs for pure state determination and constructs adaptive measurement schemes across dimensions.

Findings

Lower bound of 3d-2 elements is not tight for most dimensions.

Minimal number for d=3 is exactly 8 elements.

Constructs adaptive measurement schemes with d+2k-2 elements.

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 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 $1\le k \le d$.