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 D. 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 $r ll 2^n$, it was shown 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 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 SU($d$) measurements. Here, we point out that the 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 $textrm{poly}(log(d)^2)$ gates at most, with QST being implemented subsequently by performing $O(dr log(d)^2)$ Pauli measurements. We show that, despite the increased dimensionality, this method is more efficient than the one using SU($d$) measurements.