Determination of all pure quantum states from a minimal number of observables

TL;DR

This paper demonstrates that four generic full-rank observables are sufficient to uniquely determine any pure quantum state in complex Hilbert space, providing an optimal measurement scheme for quantum state tomography.

Contribution

It proves that four generic unitary-based measurements are enough for complete pure state reconstruction, improving understanding of minimal measurement requirements.

Findings

Four measurements suffice for pure state determination in complex space.

The measurement maps are generically injective modulo phase.

This result is sharp for dimensions n ≥ 6.

Abstract

We show that for any positive integer $n$, the maps $x \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of $\mathbb{C}P^{n-1}$ into $\mathbb{R}^{4n-4}$. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for $n \geq 6$. To complement this information-theoretic study, we establish in a companion paper that the PhaseLift algorithm yields efficient phase retrieval from quadratic measurements with $O(1)$ unitary matrices, with high probability, where the unitaries are iid according to Haar measure.