Abstract
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 n+12 over R and at most n² over C, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever w 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.