On the regularization and optimization in quantum detector tomography

TL;DR

This paper investigates how regularization techniques can enhance the accuracy of quantum detector tomography in both complete and incomplete informational scenarios, providing theoretical insights and experimental validation.

Contribution

It introduces regularization methods tailored for QDT, analyzes their impact on error scaling, and identifies optimal regularization strategies for different informational scenarios.

Findings

Regularization improves QDT accuracy in both scenarios.

Optimal regularization minimizes mean squared error.

Experimental results confirm theoretical predictions.

Abstract

Quantum detector tomography (QDT) is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we utilize regularization to improve the QDT accuracy whenever the probe states are informationally complete or informationally incomplete. In the informationally complete scenario, without regularization, we optimize the resource (probe state) distribution by converting it to a semidefinite programming problem. Then in both the informationally complete and informationally incomplete scenarios, we discuss different regularization forms and prove the mean squared error scales as $ O(\frac{1}{N}) $ or tends to a constant with $ N $ state copies under the static assumption. We also characterize the ideal best regularization for the identifiable parameters, accounting for both the informationally complete and informationally incomplete scenarios. Numerical examples demonstrate the effectiveness of different regularization forms and a quantum optical experiment test shows that a suitable regularization form can reach a reduced mean squared error.