Pure-State Quantum Tomography with Minimal Rank-One POVMs

Quantum state tomography seeks to reconstruct an unknown state from measurement statistics. A finite measurement (POVM) is \emph{pure-state informationally complete} (PSI-Complete) if the outcome probabilities determine any pure state up to a global phase. We study \emph{rank-one} POVMs that are minimally sufficient for this task. We call such a POVM \emph{vital} if it is PSI-Complete but every proper subcollection is not PSI-Complete. We prove sharp upper bounds on the size of vital rank-one POVMs in dimension $n$: the size is at most $\binom{n+1}{2}$ over $\mathbb{R}$ and at most $n^{2}$ over $\mathbb{C}$, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever $w \mid n(n-1)$, an $(n,w,w-1)$ design produces a vital rank-one POVM with $n + n(n-1)/w$ outcomes. We provide explicit constructions for $w=2,n-1$, and $n$.