为了描述对前、后选择的量子系统进行弱测量的测量结果,Aharonov、Albert和Vaidman在1988年提出了弱值的概念。可观测量的弱值在取值上可以突破可观测量的最大本征值,甚至可以取复数,为弱值在量子力学基本原理和发展新型量子技术的研究中提供了丰富的物理内涵。本文主要回顾了弱值在量子精密测量和量子层析中应用的进展情况。弱值在量子精密测量中的应用被称为弱值放大,弱值放大可以实现对微小物理量的放大测量,因而在精密测量领域获得了广泛关注。由于信号的放大伴随着后选择成功概率的降低,因此弱值放大技术的测量精度是否优于传统的测量方式一直是一个有争论的话题。本文先回顾了弱值放大和传统测量方式在测量精度方面的比较,阐述了弱值放大方案在特定条件下的精度优势;之后回顾了经过改进的弱值放大技术的最新进展。弱值在量子层析中的应用被称为直接量子层析,“直接”指的是该层析方案能够通过测量复数的弱值实现对量子态波函数概率幅的直接测量。本文还回顾了直接量子层析技术被推广并被应用到各种形式的量子态、量子过程、量子测量表征的理论和实验方案,分析了直接量子层析技术的准确度、精度,并回顾了提高直接层析技术效率的最新进展。最后,本文总结了基于弱值的这两项技术的发展状况,并提出了未来可能的发展方向。

To describe the results of a weak measurement of post- and preselected quantum systems, Aharonov, Albert, and Vaidman proposed the concept of weak-value in 1988. Weak-values can lie far outside the eigenvalue spectrum of the observable and even be complex, which provides rich physical insight in the investigation of quantum foundations and development of quantum technologies. This paper mainly reviews the progress of weak-value applied to quantum metrology and tomography. The former, known as weak-value amplification (WVA), can amplify ultrasmall physical effects, thus attracting extensive interest in precision metrology. Because the amplified signals are received from the successful post-selection with low probabilities, whether the WVA outperforms the conventional measurement schemes in terms of precision remains unclear. Here, we review the comparison in precision between the WVA and the conventional measurement and clarify the potential advantages of WVA under certain circumstances. Subsequently, we review the recent progress in the modified WVA. The application of weak-value in quantum tomography is known as direct quantum tomography, in which “direct” refers to the ability to directly measure the probability amplitude of the wave function based on the measurable complex weak-value. We review the generalization of the direct tomography method to characterize various quantum states, processes, and measurements. The accuracy, precision, and feasibility of the direct tomography protocol are analyzed. We also review the advances in promoting the operational efficiency of direct tomography. Finally, we draw conclusions regarding these two techniques based on weak-value and propose possible future research directions.