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

Physica Scripta

On the interpretation of quantum mechanical weak values

B. E. Y. Svensson

Published Dec 1, 2014• 10 citations
Publisher

Abstract

After a brief review of how quantum mechanical weak measurements plus post-selection give rise to the weak value concept, I criticize an interpretation of a weak value as any sort of straightforward, bona fide property of the system under investigation.
DOI
10.1088/0031-8949/2014/t163/014025

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

Understanding weak values without new physics

Riuji Mochizuki

The physical meaning of weak values and measurements can be completely understood within the framework of a conventional quantum mechanical approach. We should not expect the weak value of an observable $\hat A$ with post-selection $\langle F|$ within the eigenvalue range of $\hat A$ because it is not an expectation value of $\hat A$.

•Apr 7, 2016•0 citations

Theory of “Weak Value" and Quantum Mechanical Measurements

Yutaka Shikano

We review the definition and the concepts of the weak values and some measurement model to extract the weak value. This material is based on the author Ph.D. thesis "Time in Weak Values and Discrete Time Quantum Walk" at Tokyo Institute of Technology (2011).

arXiv:1110.5055•InTech eBooks•Feb 22, 2012•20 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

Understanding Weak Values Without New Probability Theory

Riuji Mochizuki

The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable $\hat {A}$ with post-selection 〈F| may be out of the eigenvalue range of $\hat {A}$ . This is because the weak value of $\hat {A}$ with the post-selection is, in general, not the expectation value of $\hat {A}$ , but the expectation value of $\hat {A}| F\rangle \langle F|$ boosted by the post-selection.

arXiv:1604.02197•International Journal of Theoretical Physics•Jan 16, 2018•1 citation

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
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.
The physical meaning of weak values and measurements can be completely understood within the framework of a conventional quantum mechanical approach. We should not expect the weak value of an observable A^\hat AA^ with post-selection ⟨F∣\langle F|⟨F∣ within the eigenvalue range of A^\hat AA^ because it is not an expectation value of A^\hat AA^.
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 physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable A^\hat {A}A^ with post-selection 〈F| may be out of the eigenvalue range of A^\hat {A}A^ . This is because the weak value of A^\hat {A}A^ with the post-selection is, in general, not the expectation value of A^\hat {A}A^ , but the expectation value of A^∣F⟩⟨F∣\hat {A}| F\rangle \langle F|A^∣F⟩⟨F∣ boosted by the post-selection.
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.