Compressed-sensing tomography for qudits in Hilbert spaces of non-power-of-two dimensions

The techniques of low-rank matrix recovery were adapted for quantum state tomography (QST) previously by Gross et al. [Phys. Rev. Lett. 105, 150401 (2010)] where they consider the tomography of $n$ spin-1/2 systems. For the density matrix of dimension $d={2}^{n}$ and rank $r$ with $r\ensuremath{\ll}{2}^{n}$, it was shown that randomly chosen Pauli measurements of the order $O[dr\phantom{\rule{0.16em}{0ex}}log{(d)}^{2}]$ are enough to fully reconstruct the density matrix by running a specific convex optimization algorithm. The result utilized the low operator norm of the Pauli operator basis, which makes it ``incoherent'' to low-rank matrices. For quantum systems of dimension $d$ not a power of two, Pauli measurements are not available, and one may consider using $\mathrm{SU}(d)$ measurements. Here, we point out that the $\mathrm{SU}(d)$ operators, owing to their high operator norm, do not provide a significant savings in the number of measurement settings required for successful recovery of all rank-$r$ states. We propose an alternative strategy in which the quantum information is swapped into the subspace of a power-two system using only $\mathrm{poly}[log{(d)}^{2}]$ gates at most with QST being implemented, subsequently, by performing $O[dr\phantom{\rule{0.16em}{0ex}}log{(d)}^{2}]$ Pauli measurements. We show that, despite the increased dimensionality, this method is more efficient than the one using $\mathrm{SU}(d)$ measurements.