2020/06/22 by Revanth Badveli, Vinayak Jagadish, R. Srikanth +1
Engineering · Mathematics · Medicine · Physics and Astronomy · #Advanced MRI Techniques and Applications #Combinatorics #Density matrix #Dimension (graph theory) #Electrical and Bioimpedance Tomography #Hilbert space #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Norm (philosophy) #Operator (biology) #Pauli exclusion principle #Pauli matrices #Physics #Pure mathematics #Quantum #Quantum mechanics #Quantum operation #Quantum state #Quantum tomography #Rank (graph theory) #SIC-POVM #Sparse and Compressive Sensing Techniques #Subspace topology #quant-ph
paper · pdf · doi:10.1103/physreva.101.062328
published as Phys. Rev. A 101, 062328 (2020) · 6 pages, 2 figures
openalex publication_date 2020/06/22 · arxiv created 2020/07/06 · arxiv updated 2020/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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=2n and rank r with r\ensuremath≪2n, it was shown that randomly chosen Pauli measurements of the order O[dr\phantom\rule0.16em0exlog(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 poly[log(d)2] gates at most with QST being implemented, subsequently, by performing O[dr\phantom\rule0.16em0exlog(d)2] Pauli measurements. We show that, despite the increased dimensionality, this method is more efficient than the one using SU(d) measurements.