Quantum tomography by regularized linear regressions. (April 2020)

• Record Type: Journal Article
• Authors: Mu, Biqiang; Qi, Hongsheng; Petersen, Ian R.; Shi, Guodong
• Is Part Of: Automatica. Volume 114(2020)
• Publication Date: 2020-04
• Subjects: Quantum state tomography -- Linear regression -- Regularization; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2020.108837
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that ℓ 2 -regularization with proper regularization gain provides even a lower mean-square error under a cost in bias. The optimal regularization parameter is defined in terms of a risk characterization for any finite sample size and a resulting implementable estimator is proposed. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed implementable tuning estimator is asymptotically optimal as the number of copies grows to infinity. Additionally, several numerical examples are provided to validate the established results.