Compressed sensing quantum state tomography for qudits: A comparison of Gell-Mann and Heisenberg-Weyl observable bases

TL;DR

This paper compares Gell-Mann and Heisenberg-Weyl bases in compressed sensing quantum state tomography for qudits, showing that Heisenberg-Weyl becomes more efficient as system dimension increases.

Contribution

It provides a numerical comparison of basis choices in CS-QST for high-dimensional qudits, highlighting the efficiency of Heisenberg-Weyl basis at larger dimensions.

Findings

Heisenberg-Weyl basis is more efficient for higher qudit dimensions.

Both bases enable successful density matrix reconstruction.

Estimated measurement counts for 95% fidelity are provided.

Abstract

Quantum state tomography (QST) is an essential technique for reconstructing the density matrix of an unknown quantum state from measurement data, crucial for quantum information processing. However, conventional QST requires an exponentially growing number of measurements as the system dimension increases, posing a significant challenge for high-dimensional systems. To mitigate this issue, compressed sensing quantum state tomography (CS-QST) has been proposed, significantly reducing the required number of measurements. In this study, we investigate the impact of basis selection in CS-QST for qudit systems, which are fundamental to high-dimensional quantum information processing. Specifically, we compare the efficiency of the generalized Gell-Mann (GGM) and Heisenberg-Weyl observable (HWO) bases by numerically reconstructing density matrices and evaluating reconstruction accuracy using fidelity and trace distance metrics. Our results demonstrate that, while both bases allow for successful density matrix reconstruction, the HWO basis becomes more efficient as the qudit dimension increases. Furthermore, we find the best fitting curves that estimate the number of measurement operators required to achieve a fidelity of at least 95%. These findings highlight the significance of basis selection in CS-QST and provide valuable insights for optimizing measurement strategies in high-dimensional quantum state tomography.