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arXiv:2309.00809 · AVS Quantum Science

Testing precision and accuracy of weak value measurements in an IBM quantum system

David R. A. Ruelas Paredes, Mariano Uria, E. Massoni, F. De Zela

Published Jan 19, 2024• 2 citations
arXivHTMLPDF

Abstract

Historically, weak values have been associated with weak measurements performed on quantum systems. Over the past two decades, a series of works have shown that weak values can be determined via measurements of arbitrary strength. One such proposal by Denkmayr et al. [Phys. Rev. Lett. 118, 010402 (2017)], carried out in neutron interferometry experiments, yielded better outcomes for strong than for weak measurements. We extend this scheme and explain how to implement it in an optical setting as well as in a quantum computational context. Our implementation in a quantum computing system provided by IBM confirms that weak values can be measured, with varying degrees of performance, over a range of measurement strengths. However, at least for this model, strong measurements do not always perform better than weak ones.

DOI
10.1116/5.0184965

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Historically, weak values have been associated with weak measurements performed on quantum systems. Over the past two decades, a series of works have shown that weak values can be determined via measurements of arbitrary strength. One such proposal by Denkmayr et al. [Phys. Rev. Lett. 118, 010402 (2017)], carried out in neutron interferometry experiments, yielded better outcomes for strong than for weak measurements. We extend this scheme and explain how to implement it in an optical setting as well as in a quantum computational context. Our implementation in a quantum computing system provided by IBM confirms that weak values can be measured, with varying degrees of performance, over a range of measurement strengths. However, at least for this model, strong measurements do not always perform better than weak ones.
Weak measurements have thus far been considered instrumental in the so-called direct measurement of the quantum wave function [4J. S. Lundeen, Nature (London) 474, 188 (2011).]. Here we show that a direct measurement of the wave function can be obtained by using measurements of arbitrary strength. In particular, in the case of strong measurements, i.e., those in which the coupling between the system and the measuring apparatus is maximum, we compared the precision and the accuracy of the two methods, by showing that strong measurements outperform weak measurements in both for arbitrary quantum states in most cases. We also give the exact expression of the difference between the original and reconstructed wave function obtained by the weak measurement approach; this will allow one to define the range of applicability of such a method.
The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.
In their original framework weak values must be measured by weak measurements that are minimally disturbing, meaning that the coupling between an investigated quantum system and a measurement device has no influence on the evolution of the system. However, under certain circumstances this weakness of the interaction is not necessary. In that case weak values can still be exactly determined from the statistics of the outcomes of arbitrary-strength generalized measurements. Here, we report an experimental procedure for neutron matter-waves that extends the notion of generalized eigenvalues for the neutron's path system to allow the exact determination of weak values using both strong and weak interactions . Experimental evidence is given that strong interactions outperform weak ones both for precision and accuracy.
Weak values, introduced more than 25 years ago, underwent a metamorphosis from a theoretical curiosity to a powerful resource in photonics for exploring foundations of quantum mechanics, as well as a practical laboratory tool. Due to the tiny coherence volume of particles used in matter-wave optics, a straightforward implementation of weak measurements is not feasible. We have overcome this hurdle by developing a method to weakly measure a massive particle's spin component. A neutron optical approach is realized by utilizing neutron interferometry, where the neutron's spin is coupled weakly to its spatial degree of freedom. Here, we present how one can fully characterize the weak value of the Pauli spin operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}$ of neutrons by extracting its real and imaginary components, as well as its modulus. The results show good agreement with theoretical predictions and demonstrate that the (spin) weak value is actually accessible for a purely quantum system of massive particles.
Weak values and measurements have been proposed as a means to achieve dramatic enhancements in metrology based on the greatly increased range of possible measurement outcomes. Unfortunately, the very large values of measurement outcomes occur with highly suppressed probabilities. This raises three vital questions in weak-measurement-based metrology. Namely, (Q1) Does postselection enhance the measurement precision? (Q2) Does weak measurement offer better precision than strong measurement? (Q3) Is it possible to beat the standard quantum limit or to achieve the Heisenberg limit with weak measurement using only classical resources? We analyze these questions for two prototypical, and generic, measurement protocols and show that while the answers to the first two questions are negative for both protocols, the answer to the last is affirmative for measurements with phase-space interactions, and negative for configuration space interactions. Our results, particularly the ability of weak measurements to perform at par with strong measurements in some cases, are instructive for the design of weak-measurement-based protocols for quantum metrology.