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arXiv:2103.05884 · Chinese Physics B

Increasing the efficiency of post-selection in direct measurement of the quantum wave function

Yong-Li Wen, Shanchao Zhang, Hui Yan, Shi-Liang Zhu

Published Aug 19, 2021• 3 citations
arXivHTMLPDF

Abstract

Direct weak or strong measurement of quantum wave function based on post-selections has been widely explored; however, the efficiency of the measurement is heavily limited by the success probability of post-selection. Here we propose a modified scheme to directly measure photon–s wave function by simply inserting a liquid crystal plate before the post-selection stage. Numerical simulations demonstrate that our modified method can significantly increase the efficiency of post selection. Our proposal would speed up the quantum wave function measurement with high resolution and high fidelity.

DOI
10.1088/1674-1056/ac1f04

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•Jan 1, 2006•6 citations
Direct weak or strong measurement of quantum wave function based on post-selections has been widely explored; however, the efficiency of the measurement is heavily limited by the success probability of post-selection. Here we propose a modified scheme to directly measure photon–s wave function by simply inserting a liquid crystal plate before the post-selection stage. Numerical simulations demonstrate that our modified method can significantly increase the efficiency of post selection. Our proposal would speed up the quantum wave function measurement with high resolution and high fidelity.
Weak measurements have been under intensive investigation in both experiment and theory. Numerous experiments have indicated that the amplified meter shift is produced by the post-selection, yielding an improved precision compared to conventional methods. However, this amplification effect comes at the cost of a reduced rate of acquiring data, which leads to an increasing uncertainty to determine the level of meter shift. From this point of view, a number of theoretical works have suggested that weak measurements cannot improve the precision, or even damage the metrology information due to the post-selection. In this review, we give a comprehensive analysis of the weak measurements to justify their positive effect on prompting measurement precision. As a further step, we introduce two modified weak measurement protocols to boost the precision beyond the standard quantum limit. Compared to previous works beating the standard quantum limit, these protocols are free of using entangled or squeezed states. The achieved precision outperforms that of the conventional method by two orders of magnitude and attains a practical Heisenberg scaling up to n=106 photons.
The measurement of a quantum state wave function not only acts as a fundamental part in quantum physics but also plays an important role in developing practical quantum technologies. Conventional quantum state tomography has been widely used to estimate quantum wave functions, which usually requires complicated measurement techniques. The recent weak-value-based quantum measurement circumvents this resource issue but relies on an extra pointer space. Here, we theoretically propose and then experimentally demonstrate a direct and efficient measurement strategy based on a δ-quench probe: by quenching its complex probability amplitude one by one (δ quench) in the given basis, we can directly obtain the quantum wave function of a pure ensemble by projecting the quenched state onto a postselection state. We confirm its power by experimentally measuring photonic complex temporal wave functions. This new method is versatile and can find applications in quantum information science and engineering.
The wave function is at the foundation of quantum theory, which is assumed to give a complete description of a quantum system. From early inception, the wave function was introduced as an abstract element of the theory and lacked any way to measure it directly. The situation, however, has somewhat changed with the reporting of the direct measurement of the quantum wave function via weak measurements, giving the wave function a clearly operational definition. The weak measurement method requires sequential measurements of conjugate observables position and momentum with the position measurement sufficiently weak. Surprisingly, recent research showed that performing sequential strong measurements realized the same object, in which case no approximation in weak measurements has to be made. Here, we report on experiments of direct measurements of the two-dimensional transverse wave function of photons via strong measurement, implying that an accurate and clear operational definition may be given to the wave function. We have measured the Gaussian and Laguerre--Gaussian of l=1l=1l=1 spatial wave functions of photons with fidelity of 0.93. As a potential important application, we show that this direct measurement of the wave function provides an alternative way to realize digital holography of three-dimensional objects. This realization of direct measurement of the wave function has significant consequences in quantum information processing and quantum imaging.
Recent work has revealed that the wave function of a pure state can be measured directly and that complementary knowledge of a quantum system can be obtained simultaneously by weak measurements. However, the original scheme applies only to pure states, and it is not efficient because most of the data are discarded by post-selection. Here, we propose tomography schemes for pure states and for mixed states via weak measurements, and our schemes are more efficient because we do not discard any data. Furthermore, we demonstrate that any matrix element of a general state can be directly read from an appropriate weak measurement. The density matrix (with all of its elements) represents all that is directly accessible from a general measurement.
Post-selection is the selection of a subset of an ensemble based on a measurement on that ensemble. In the last few years, post-selection has taken on new importance as a mechanism to induce the optical nonlinearities required for quantum logic gates. I present experiments and theory that explore post-selection in photonic systems. I then go on to describe an experimental implementation of Hardy's Paradox, which is one of the three great thought-experiments about entanglement and correlations in quantum mechanics (the others being the EPR and GHZ "paradoxes"). In the place of the electron--positron annihilation in the original proposal, I apply, for the first time, a post-selection-based absorptive photon switch that coworkers and I had previously developed. Weak measurement promises to be an important technique for characterizing post-selected quantum systems during their evolution. The technique minimizes disturbance to the system and therefore can be used in situ. In a third experiment, I attempt to resolve a recent debate: Which is more fundamental--complementarity or the uncertainty principle? Specifically, can interference be destroyed by a which-path measurement without transferring momentum to the particle? This question is difficult to investigate as any measurement of the momentum transfer causes collapse, severely disturbing the system. I circumvent this problem with the technique of weak measurement and show that which-path measurements do indeed induce a momentum transfer as required by the uncertainty principle. First, I experimentally demonstrate effective nonlinear absorption between two single photons. I show that this photon-exchange effect can be understood as a post-selection induced nonlinearity similar to the one that occurs in Hong-Ou-Mandel interference. In a fourth experiment, I use the latter theory and return to Hardy's Paradox to perform the first weak measurement of an entangled system. I post-select on the paradoxical measurement result while weakly measuring a set of observables. The results are surprising in that they no longer conflict; they now satisfy a classical-logic truth-table. I then proceed to develop theoretically a new method for performing multiparticle weak measurements that is unique because it does not require single-particle-level interactions. This theory provides new insight into the importance of the imaginary component of the weak measurement result.