Pure-State Quantum Tomography with Minimal Rank-One POVMs
Dan Edidin, Ivan Gonzalez, Itzhak Tamo
TL;DR
This work introduces vital rank-one POVMs for pure-state quantum tomography, defining PSI-Complete measurements that are minimal in the sense that removing any outcome destroys completeness. It proves sharp upper bounds on the size of vital measurements in real and complex spaces, showing $m\le\binom{n+1}{2}$ and $m\le n^{2}$ respectively, and provides explicit frames attaining these bounds. In the real case, it develops a general construction from $(n,w,w-1)$ block designs yielding vital measurements with $m=n+n(n-1)/w$ outcomes, connecting combinatorics to quantum tomography via the Complement Property. It also demonstrates that block-design-based constructions do not exhaust all vital measurements, giving non-equivalence results and highlighting a rich landscape of vital POVMs. The results have implications for designing structured, minimal-measurement tomography schemes and motivate further study of design-inspired and non-design vital measurements in quantum state reconstruction.
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 \(\binom{n+1}{2}\) 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 \(n\).