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We study the recovery of Hermitian low rank matrices $X \in \mathbb{C}^{n\times n}$ from undersampled measurements via nuclear norm minimization. Weconsider the particular scenario where the measurements are Frobenius innerproducts with random rank-one matrices of the form $a_j a_j^*$ for somemeasurement vectors $a_1,...,a_m$, i.e., the measurements are given by $y_j =\mathrm{tr}(X a_j a_j^*)$. The case where the matrix $X=x x^*$ to be recoveredis of rank one reduces to the problem of phaseless estimation (frommeasurements, $y_j = |\langle x,a_j\rangle|^2$ via the PhaseLift approach,which has been introduced recently. We derive bounds for the number $m$ ofmeasurements that guarantee successful uniform recovery of Hermitian rank $r$matrices, either for the vectors $a_j$, $j=1,...,m$, being chosen independentlyat random according to a standard Gaussian distribution, or $a_j$ being sampledindependently from an (approximate) complex projective $t$-design with $t=4$.In the Gaussian case, we require $m \geq C r n$ measurements, while in the caseof $4$-designs we need $m \geq Cr n \log(n)$. Our results are uniform in thesense that one random choice of the measurement vectors $a_j$ guaranteesrecovery of all rank $r$-matrices simultaneously with high probability.Moreover, we prove robustness of recovery under perturbation of themeasurements by noise. The result for approximate $4$-designs generalizes andimproves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. Inaddition, it has applications in quantum state tomography. Our proofs employthe so-called bowling scheme which is based on recent ideas by Mendelson andKoltchinskii.

Low rank matrix recovery from rank one measurements