Progress in Weak-Value-Based Quantum Metrology and Tomography

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.