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The techniques of low-rank matrix recovery were adapted for Quantum StateTomography (QST) previously by D. Gross et al. [Phys. Rev. Lett. 105, 150401(2010)], where they consider the tomography of $n$ spin-$1/2$ systems. For thedensity matrix of dimension $d = 2^n$ and rank $r$ with $r \ll 2^n$, it wasshown that randomly chosen Pauli measurements of the order $O(dr \log(d)^2)$are enough to fully reconstruct the density matrix by running a specific convexoptimization algorithm. The result utilized the low operator-norm of the Paulioperator basis, which makes it `incoherent' to low-rank matrices. For quantumsystems of dimension $d$ not a power of two, Pauli measurements are notavailable, and one may consider using SU($d$) measurements. Here, we point outthat the SU($d$) operators, owing to their high operator norm, do not provide asignificant savings in the number of measurement settings required forsuccessful recovery of all rank-$r$ states. We propose an alternative strategy,in which the quantum information is swapped into the subspace of a power-twosystem using only $\textrm{poly}(\log(d)^2)$ gates at most, with QST beingimplemented subsequently by performing $O(dr \log(d)^2)$ Pauli measurements. Weshow that, despite the increased dimensionality, this method is more efficientthan the one using SU($d$) measurements.

Compressed Sensing Tomography for qudits in Hilbert spaces of non-power-of-two dimensions