Stable low-rank matrix recovery from 3-designs

We study the recovery of low-rank Hermitian matrices from rank-one measurements obtained by uniform sampling from complex projective 3-designs, using nuclear-norm minimization. This framework includes phase retrieval as a special case via the PhaseLift method. In general, complex projective $t$-designs provide a practical means of partially derandomizing Gaussian measurement models. While near-optimal recovery guarantees are known for $4$-designs, and it is known that $2$-designs do not permit recovery with a subquadratic number of measurements, the case of $3$-designs has remained open. In this work, we close this gap by establishing recovery guarantees for (exact and approximate) $3$-designs that parallel the best-known results for $4$-designs. In particular, we derive bounds on the number of measurements sufficient for stable and robust low-rank recovery via nuclear-norm minimization. Our results are especially relevant in practice, as explicit constructions of $4$-designs are significantly more challenging than those of $3$-designs.