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IEEE Transactions on Instrumentation and Measurement

A Fusion Measurement Approach to Improve Quantum State Tomography Efficiency and Accuracy

Sen Kuang, Benwei Wu, Shuang Cong

Published Jul 9, 2019• 5 citations
Publisher

Abstract

Quantum state tomography (QST) is an important tool for estimating an unknown quantum state, which includes a measurement process and a reconstruction process. The state estimation error involves the measurement-induced (probability estimation) error in the measurement process and the calculation error in the reconstruction process. Via the mean-square error methods, we propose two fusion measurement schemes with multiple measurement devices (MMDs) to improve the efficiency and accuracy of quantum state measurement and QST by using information fusion theory. These two schemes are founded on the parallel synchronous measurements of MMDs and, therefore, can improve the efficiency of quantum state measurement and tomography. At the same time, by fusing measurement data from different measurement devices in optimal and suboptimal manners, the proposed multiple-measurement-device fusion measurement schemes achieve the improvement of quantum state measurement and tomography accuracy. Numerical simulations are presented to demonstrate the proposed method.

DOI
10.1109/tim.2019.2927546

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The estimation of all the parameters in an unknown quantum state or measurement device, commonly known as quantum state tomography (QST) and quantum detector tomography (QDT), is crucial for comprehensively characterizing and controlling quantum systems. In this paper, we introduce a framework, in two different bases, that utilizes multiple quantum processes to simultaneously identify a quantum state and a detector. We develop a closed-form algorithm for this purpose and prove that the mean squared error (MSE) scales as $O(1/N) $ for both QST and QDT, where $N $ denotes the total number of state copies. This scaling aligns with established patterns observed in previous works that addressed QST and QDT as independent tasks. Furthermore, we formulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, where the physical constraints of the state and detector are characterized by polynomial equalities and inequalities. The effectiveness of our proposed methods is validated through numerical examples.
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Quantum state tomography (QST) is a crucial technique for characterizing the state of a quantum system, often employed in quantum metrology, quantum communication, and fundamental quantum information science. However, QST inherently suffers from significant measurement redundancy and uncertainty, demanding a large number of measurements to achieve a desired level of accuracy. This work proposes a novel framework for QST leveraging concepts from algorithmic information theory, specifically Kolmogorov complexity, to rigorously quantify this uncertainty and redundancy. We introduce a tailored complexity measure for quantum states that allows for the design of measurement strategies optimized to minimize the number of required measurements while maintaining a specified accuracy. Our approach shifts the focus from purely statistical estimation to a fundamental information-theoretic analysis, offering a deeper understanding of the reconstruction problem and potentially leading to more efficient and robust QST protocols. The core claim is that applying Kolmogorov complexity provides a quantifiable metric for the inherent uncertainty within a quantum state, directly informing the development of improved estimation techniques. The mechanism involves defining and utilizing this complexity measure to guide the selection of measurement angles, ultimately reducing redundancy and improving the precision of state reconstruction.