Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2306.01392 · Journal of Physics A Mathematical and Theoretical

Revisiting weak values through non-normality

Lorena Ballesteros Ferraz, Riccardo Muolo, Yves Caudano, Timotéo Carletti

Published Oct 18, 2023• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ad04a4

Related papers

What Is a Quantum-Mechanical “Weak Value” the Value of?

B. E. Y. Svensson

A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.

arXiv:1301.4328•Foundations of Physics•Sep 4, 2013•27 citations

Experimental realization of robust weak measurements

Enrico Rebufello, Fabrizio Piacentini, Alessio Avella +7 more

Despite being very influential on both foundations and applications of quantum mechanics, weak values are still somewhat controversial. Although there are some indications that weak values are physical properties of a single quantum system, the common way weak values are presented is statistical: it is commonly believed that for measuring weak values one has to perform many weak measurements over a large ensemble of pre- and postselected particles. Other debates surround the anomalous nature of weak value and even their quantumness. To address these issues, we present some preliminary data showing that anomalous weak values can be measured using just a single detection, i.e. with no statistics. In our experiment, a single click of a detector indicates the weak value as a single photon property, which moreover lies well beyond the range of eigenvelues of the measured operator. Importantly, the uncertainty with which the weak values is measured is smaller than the difference between the weak value and the closet eigenvalue. This is the first experimental realization of robust weak measurements.

Optical and Quantum Sensing and Precision Metrology II•Feb 25, 2020•0 citations

Colloquium : Understanding quantum weak values: Basics and applications

Justin Dressel, Mehul Malik, Filippo M. Miatto +2 more

Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation for how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration to discuss and clarify each of these applications is used.

arXiv:1305.7154•Reviews of Modern Physics•Mar 28, 2014•700 citations

Pedagogical Review of Quantum Measurement Theory with an Emphasis on Weak Measurements

B. E. Y. Svensson

The quantum theory of measurement has been with us since quantum mechanics was invented. It has recently been invigorated, partly due to the increasing interest in quantum information science. In this partly pedagogical review I attempt to give a self-contained overview of non-relativistic quantum theory of measurement expressed in density matrix formalism. I will not dwell on the applications in quantum information theory; it is well covered by several books in that field. The focus is instead on applications to the theory of weak measurement, as developed by Aharonov and collaborators. Their development of weak measurement combined with what they call post-selection - judiciously choosing not only the initial state of a system (pre-selection) but also its final state - has received much attention recently. Not the least has it opened up new, fruitful experimental vistas, like novel approaches to amplification. But the approach has also attached to it some air of mystery. I will attempt to demystify it by showing that (almost) all results can be derived in a straight-forward way from conventional quantum mechanics. Among other things, I develop the formalism not only to first order but also to second order in the weak interaction responsible for the measurement. I apply it to the so called Leggett-Garg inequalities, also known as Bell inequalities in time. I also give an outline, even if rough, of some of the ingenious experiments that the work by Aharonov and collaborators has inspired. As an application of weak measurement, not related to the approach by Aharonov and collaborators, the formalism also allows me to derive the master equation for the density matrix of an open system in interaction with an environment. An issue that remains in the weak measurement plus post-selection approach is the interpretation of the so called weak value of an observable. Is it a bona fide property of the system considered? I have no definite answer to this question; I shall only exhibit the consequences of the proposed interpretation. Quanta 2013; 2: 18–49.

arXiv:1202.5148•Quanta•May 14, 2013•103 citations

Weak-value amplification: state of play

George C. Knee, Joshua Combes, Christopher Ferrie +1 more

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.

arXiv:1410.6252•Quantum Measurements and Quantum Metrology•Jan 1, 2016•28 citations

The Trouble with Weak Values

Jacob A. Barandes

In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of state vectors. Weak values first appeared in the research literature in a pair of papers in 1987 and 1988, and were originally defined as the results of a special kind of experimental protocol involving non-disturbing measurements combined with an explicit form of post-selection. In the years since, subsequent papers on weak values have produced a number of important practical spin-offs, including new methods for signal amplification and quantum-state tomography. The present work is not concerned with those practical spin-offs, but with historical and ongoing attempts to assign weak values a transparent, single-system interpretation, as well as efforts that invoke weak values to make a number of exotic claims about the properties and behavior of individual quantum systems. This paper challenges these interpretational claims by arguing that they involve several forms of fallacious reasoning.

quant-pharXiv:2602.09380•Feb 11, 2026•0 citations
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.
A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
Despite being very influential on both foundations and applications of quantum mechanics, weak values are still somewhat controversial. Although there are some indications that weak values are physical properties of a single quantum system, the common way weak values are presented is statistical: it is commonly believed that for measuring weak values one has to perform many weak measurements over a large ensemble of pre- and postselected particles. Other debates surround the anomalous nature of weak value and even their quantumness. To address these issues, we present some preliminary data showing that anomalous weak values can be measured using just a single detection, i.e. with no statistics. In our experiment, a single click of a detector indicates the weak value as a single photon property, which moreover lies well beyond the range of eigenvelues of the measured operator. Importantly, the uncertainty with which the weak values is measured is smaller than the difference between the weak value and the closet eigenvalue. This is the first experimental realization of robust weak measurements.
Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation for how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration to discuss and clarify each of these applications is used.
The quantum theory of measurement has been with us since quantum mechanics was invented. It has recently been invigorated, partly due to the increasing interest in quantum information science. In this partly pedagogical review I attempt to give a self-contained overview of non-relativistic quantum theory of measurement expressed in density matrix formalism. I will not dwell on the applications in quantum information theory; it is well covered by several books in that field. The focus is instead on applications to the theory of weak measurement, as developed by Aharonov and collaborators. Their development of weak measurement combined with what they call post-selection - judiciously choosing not only the initial state of a system (pre-selection) but also its final state - has received much attention recently. Not the least has it opened up new, fruitful experimental vistas, like novel approaches to amplification. But the approach has also attached to it some air of mystery. I will attempt to demystify it by showing that (almost) all results can be derived in a straight-forward way from conventional quantum mechanics. Among other things, I develop the formalism not only to first order but also to second order in the weak interaction responsible for the measurement. I apply it to the so called Leggett-Garg inequalities, also known as Bell inequalities in time. I also give an outline, even if rough, of some of the ingenious experiments that the work by Aharonov and collaborators has inspired. As an application of weak measurement, not related to the approach by Aharonov and collaborators, the formalism also allows me to derive the master equation for the density matrix of an open system in interaction with an environment. An issue that remains in the weak measurement plus post-selection approach is the interpretation of the so called weak value of an observable. Is it a bona fide property of the system considered? I have no definite answer to this question; I shall only exhibit the consequences of the proposed interpretation. Quanta 2013; 2: 18–49.
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.
In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of state vectors. Weak values first appeared in the research literature in a pair of papers in 1987 and 1988, and were originally defined as the results of a special kind of experimental protocol involving non-disturbing measurements combined with an explicit form of post-selection. In the years since, subsequent papers on weak values have produced a number of important practical spin-offs, including new methods for signal amplification and quantum-state tomography. The present work is not concerned with those practical spin-offs, but with historical and ongoing attempts to assign weak values a transparent, single-system interpretation, as well as efforts that invoke weak values to make a number of exotic claims about the properties and behavior of individual quantum systems. This paper challenges these interpretational claims by arguing that they involve several forms of fallacious reasoning.