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arXiv:1410.6252 · Quantum Measurements and Quantum Metrology

Weak-value amplification: state of play

George C. Knee, Joshua Combes, Christopher Ferrie, Erik M. Gauger

Published Jan 1, 2016• 28 citations
arXivHTMLPDF

Abstract

Abstract Weak values arise in quantum theory when the result of a weak measurement is conditioned on a subsequent strong measurement. The majority of the trials are discarded, leaving only very few successful events. Intriguingly those can display a substantial signal amplification. This raises the question of whether weak values carry potential to improve the performance of quantum sensors, and indeed a number of impressive experimental results suggested this may be the case. By contrast, recent theoretical studies have found the opposite: using weak-values to obtain an amplification generally worsens metrological performance. This survey summarises the implications of those studies, which call for a reappraisal of weak values’ utility and for further work to reconcile theory and experiment.

DOI
10.1515/qmetro-2016-0006

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Abstract Weak values arise in quantum theory when the result of a weak measurement is conditioned on a subsequent strong measurement. The majority of the trials are discarded, leaving only very few successful events. Intriguingly those can display a substantial signal amplification. This raises the question of whether weak values carry potential to improve the performance of quantum sensors, and indeed a number of impressive experimental results suggested this may be the case. By contrast, recent theoretical studies have found the opposite: using weak-values to obtain an amplification generally worsens metrological performance. This survey summarises the implications of those studies, which call for a reappraisal of weak values’ utility and for further work to reconcile theory and experiment.
Abstract Quantum measurement is one of the most fascinating and discussed phenomena in quantum physics, due to the impact on the system of the measurement action and the resulting interpretation issues. Scholars proposed weak measurements to amplify measured signals by exploiting a quantity called a weak value, but also to overcome philosophical difficulties related to the system perturbation induced by the measurement process. The method finds many applications and raises many philosophical questions as well, especially about the proper interpretation of the observations. In this paper, we show that any weak value can be expressed as the expectation value of a suitable non-normal operator. We propose a preliminary explanation of their anomalous and amplification behavior based on the theory of non-normal matrices and their link with non-normality: the weak value is different from an eigenvalue when the operator involved in the expectation value is non-normal. Our study paves the way for a deeper understanding of the measurement phenomenon, helps the design of experiments, and it is a call for collaboration to researchers in both fields to unravel new quantum phenomena induced by non-normality.
The concept of weak measurement and associated weak value amplification has sharpened our understanding of the measurement process in quantum mechanics. Recent experiments show that elastic scattering events in resonance fluorescence experiments can exhibit weak value amplification effect, by post-selecting a particular measurement outcome. In this article, we theoretically analyse the physics behind this amplification process. We show that, in general, weak interaction and the associated weak value amplification in resonance fluorescence can be derived from the wellknown theory of spontaneous emission put forth by Wigner and Weiskopff. Using this theory we show that in the elastic scattering regime of resonance fluorescence, weak value amplification helps in segregating rare events. To our knowledge, weak value amplification has not been applied earlier in the time domain as a potential tool to study rare events.
The ability to manipulate light has allowed scientists to verify fundamental theories of physics and to develop a new generation of technologies that use photons as a primary resource. Recent developments in quantum measurement theory have offered new alternatives to approach some of the most remarkable problems in quantum physics. Consequently, the principles of quantum mechanics have been exploited in the development of quantum technologies such as optical metrology, quantum communication, and quantum information. In recent years, weak measurements and two of its most remarkable variants: weak value amplification and direct measurement, have been developed and are considered important resources for quantum applications. In this paper, we discuss weak measurements and some significant applications of weak values. We elaborate on how distinct forms of weak values are used to observe and amplify small effects or to directly measure the quantum wave function of photons, a crucial task for schemes for quantum communication and quantum information. We also review some of the most recent methods for weak value amplification and direct measurement that our group has developed.
Despite their important applications in metrology and in spite of numerous experimental demonstrations, weak measurements are still confusing for part of the community. This sometimes leads to unjustified criticism. Recent papers have experimentally clarified the meaning and practical significance of weak measurements, yet in Kastner (Found Phys 47:697-707, 2017), Kastner seems to take us many years backwards in the the debate, casting doubt on the very term "weak value" and the meaning of weak measurements. Kastner appears to ignore both the basics and frontiers of weak measurements and misinterprets the weak measurement process and its outcomes. In addition, she accuses the authors of Aharonov et al. (Ann Phys 355:258-268, 2015) in statements completely opposite to the ones they have actually made. There are many points of disagreement between Kastner and us, but in this short reply I will leave aside the ontology (which is indeed interpretational and far more complex than that described by Kastner) and focus mainly on the injustice in her criticism. I shall add some general comments regarding the broader theory of weak measurements and the two-state-vector formalism, as well as supporting experimental results. Finally, I will point out some recent promising results, which can be proven by (strong) projective measurements, without the need of employing weak measurements.
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.