  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.
