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.
Categories: math-ph math.AG math.DG math.MP quant-ph