1306.1214 (Damien Mondragon et al.)

Determination of all pure quantum states from a minimal number of
observables    [PDF]

Damien Mondragon, Vladislav Voroninski

Ron Wright conjectured circa 1978 that there exist three observables $A_1, A_2, A_3$ which uniquely determine any pure state $x \in \mathbb{C}P^{n-1}$. It is now known that Wright's conjecture is false due to general obstructions to embedding $\mathbb{C}P^{n}$ into Euclidean space and it is natural to consider the minimal number of observables required for informational completeness. We prove in this paper that for any positive integer $n$, the map $x \in \mathbb{C}^n/\mathbb{S}^1 \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the rows of four generic $n\times n$ unitary matrices, is injective, yielding a family of quadratic embeddings of $\mathbb{C}P^{n-1}$ into $\mathbb{R}^{4(n-1)}$. In particular, this implies that four generic observables determine any pure state. This result is sharp for $n \geq 6$.

View original: http://arxiv.org/abs/1306.1214