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 \(n\).

翻译：量子态层析旨在通过测量统计重构未知量子态。若一个有限测量（POVM）的结果概率能确定任意纯态（忽略全局相位），则称其为纯态信息完备（PSI-Complete）的。本研究聚焦于实现该任务所需的最小秩一POVM：若一个POVM是PSI-Complete的，但其任意真子集均非PSI-Complete，则称其为关键POVM。我们证明了维度 \(n\) 下关键秩一POVM规模的严格上界：在实数域 \(\mathbb{R}\) 中规模不超过 \(\binom{n+1}{2}\)，在复数域 \(\mathbb{C}\) 中不超过 \(n^{2}\)，并给出了达到这些界限的构造。在实数情形中，我们进一步揭示了其与区组设计的关联：当 \(w \mid n(n-1)\) 时，一个 \((n,w,w-1)\) 设计可生成具有 \(n + n(n-1)/w\) 个结果的关键秩一POVM。我们针对 \(w=2,n-1\) 和 \(n\) 给出了显式构造。