Draft:Weak Value v2

Submission declined on 1 December 2025 by Shocksingularity (talk). This draft appears to be a duplicate of an existing article. Wikipedia does not permit multiple articles on the same topic. To improve the existing article, please edit it directly at Weak value. If the article is protected and you cannot edit it, propose your changes on the article's talk page. If you are writing about a different subject with the same name, ask the reviewer to rename your draft to distinguish it. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by Shocksingularity 6 months ago. Last edited by Johnjbarton 5 months ago. Reviewer: Inform author. Resubmit Please note that if the issues are not fixed, the draft will be declined again.

• Comment: Please make edits to the Weak value article one subsection at a time from the notes you have here. Add an edit summary with each one explaining the reason for the change. If we disagree we may revert your change, then we can discuss that item in in Talk:Weak value.
  I will say that presenting conflicts in Wikipedia is very challenging. Your current section presents the viewpoints of primary sources but Wikipedia generally wants to present secondary sources. So the bit where you have "A comprehensive survey of individual contributions is beyond the scope of this article. For overviews of the broader literature, see the review articles in Refs." is pretty much the opposite of how we work. In general the article should be a summary of those reviews, not an alternative review. Johnjbarton (talk) 03:18, 8 December 2025 (UTC)

• Comment: I am just going to give my point of view, other may not agree.
  * "There are many excellent review articles ..."
  ** Remove. Move the sources to the sentences that they verify. Don't apologize about brevity or basics. Avoid meta-comments; focus on the topic. ✅
  * Definition
  ** Needs inline sources
  ** Be direct and succinct. Avoid "consider", "notice"✅
  ** Avoid storylines. "...it helpful to consider...". An encyclopedia will only provide helpful info, so we don't need to say it  ;-) ✅
  * Derivation
  ** Just give the refs not a story about giving the refs.✅
  * Applications
  ** Avoid stories: "Some researchers find weak values intriguing..." The applications are notable for an encyclopedia or not, that's all..✅
  ** Avoid mixing history and "facts". If the history is notable, have a History section. .✅
  ** "As shown by Matthew Pusey in 2014..." If this amounts to a notable discovery it should be given a secondary ref in the history. Otherwise the date and name are not notable for an "application" ✅
  * Criticisms
  ** Per WP:NPOV this kind of section requires care in citations. ✅ -- this took the most time. Thanks for pointing this out — I hadn’t noticed the non-neutral phrasing.
  ** "Below three recent criticisms are summarized." This kind of statement implies an editor making choices on a controversial topic. Instead a review should be cited and the criticisms noted in the review should be discussed. ✅
  ** The source and content by Stephen Parrott should not be used based on an arxiv preprint alone, WP:SELFPUB. ✅
  ** "To avoid adding to this ongoing controversy, ...", "The summary of the criticisms below is based on the review article..." no stories. Just cut this stuff and give the summary of the review. ✅
  I hope this is helpful. Johnjbarton (talk) 00:05, 29 November 2025 (UTC)

• Comment: This looks like Weak value. What's the status? Johnjbarton (talk) 04:03, 25 November 2025 (UTC)

User:Johnjbarton I think I've made all the revisions. Let me know what you think.

If I understand User:Shocksingularity’s decline correctly — “Thank you for your submission, but the subject of this article already exists in Wikipedia. You can find it and improve it at Weak value instead.” — then I should be editing the main Weak value page. If you’re happy with my edits and that interpretation, I can add them directly to the main article. Yama jlac (talk) 05:15, 7 December 2025 (UTC)

User:Johnjbarton Thank you for the review! I appreciate the time you took, and I agree with all your suggestions and comments. I’ll make the revisions and, once they’re done, I’ll resubmit it to you and User:Shocksingularity.

User:Johnjbarton I am trying to get this page moved to the make Weak Value page on wikipedia. I previously re wrote the page about 10 years ago. I have tried to make improvements all around to the article.

In quantum mechanics (and computation), a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection. It should not be confused with a weak measurement, which is often defined in conjunction. The weak value was first defined by Yakir Aharonov, David Albert, and Lev Vaidman, published in Physical Review Letters 1988,:^[1] and is related to the two-state vector formalism. The first experimental realization came from researchers at Rice University in 1991.^[2] The physical interpretation and significance of weak values remains a subject of ongoing discussion in the quantum foundations and metrology literature.

The initial state of the system is denoted as ${\displaystyle |\psi _{i}\rangle }$ and the final state as ${\displaystyle |\psi _{f}\rangle }$. These states are often referred to as the pre- and post-selected quantum states. Also consider an observable ${\displaystyle A}$ with minimal and maximal eigenvalues ${\displaystyle {a_{\rm {min}},a_{\rm {max}}}}$. With respect to these states, the weak value of ${\displaystyle A}$ is defined as^[3][4][5][6] $${\displaystyle A_{w}={\frac {\langle \psi _{f}|A|\psi _{i}\rangle }{\langle \psi _{f}|\psi _{i}\rangle }}.}$$

If ${\displaystyle |\psi _{f}\rangle =|\psi _{i}\rangle }$, then the weak value reduces to the usual expectation value in either the initial state ${\displaystyle \langle \psi _{i}|A|\psi _{i}\rangle }$ or the final state ${\displaystyle \langle \psi _{f}|A|\psi _{f}\rangle }$. These expectation values are necessarily bounded by the eigenvalue range of ${\displaystyle A}$, for example, ${\displaystyle a_{\rm {min}}\leq \langle \psi _{i}|A|\psi _{i}\rangle \leq a_{\rm {max}}}$. These expectations are always real numbers.

In general the weak value quantity is a complex number. The weak value of the observable becomes large when the post-selected state, ${\displaystyle |\psi _{f}\rangle }$, approaches being orthogonal to the pre-selected state, ${\displaystyle |\psi _{i}\rangle }$, i.e. ${\displaystyle |\langle \psi _{f}|\psi _{i}\rangle |\ll 1}$. If ${\displaystyle A_{w}}$ is larger than the largest eigenvalue of ${\displaystyle A}$, ${\displaystyle a_{\rm {max}}}$, or smaller than its smallest eigenvalue, ${\displaystyle a_{\rm {min}}}$, the weak value is said to be anomalous. Such anomalous weak values are interesting because they can be complex and fall outside the usual eigenvalue range, both features absent in standard expectation values.

The example and derivation below show how such expectations arise in practice.

As an example consider a spin 1/2 particle.^[7] Take ${\displaystyle A}$ to be the Pauli Z operator ${\displaystyle A=\sigma _{z}}$ with eigenvalues ${\displaystyle \pm 1}$. Using the initial state $${\displaystyle |\psi _{i}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\\\cos {\frac {\alpha }{2}}-\sin {\frac {\alpha }{2}}\end{pmatrix}}}$$ and the final state $${\displaystyle |\psi _{f}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}}$$ one can calculate the weak value to be $${\displaystyle A_{w}={\frac {\langle \psi _{f}\mid \sigma _{z}\mid \psi _{i}\rangle }{\langle \psi _{f}\mid \psi _{i}\rangle }}={\frac {\sin(\alpha /2)}{\cos(\alpha /2)}}=\tan {\frac {\alpha }{2}}.}$$

For ${\displaystyle |\alpha |>{\frac {\pi }{2}}}$ the weak value is anomalous.

The derivation below follows the presentation given References.^[7][3]

Weak values have been proposed as potentially useful for quantum metrology and for clarifying aspects of quantum foundations. The sections below briefly outline these applications.

At the end of the original weak value paper^[1] the authors suggested weak values could be used in quantum metrology:

In modern language, when the weak value ${\displaystyle A_{w}}$ lies outside the eigenvalue range of the observable ${\displaystyle A}$, the effect is known as weak value amplification. In this regime, the shift of the measuring device’s pointer can appear much larger than expected, for example a component of spin may seem 100 times greater than its largest eigenvalue. This amplification effect has been viewed as potentially beneficial for metrological applications where small physical signals need to be detected with high sensitivity.

This weak value amplification subsequently demonstrated experimentally^[12][13]. This area has developed into an active field of research investigating the use of weak values in quantum sensing and precision measurement. A comprehensive survey of individual contributions is beyond the scope of this article. For overviews of the broader literature, see the review articles in Refs.^[6][14][15].

Weak values have also been explored in the context of quantum state tomography. Two main approaches have emerged. The first approach is called, "direct state tomography", and the second "weak-measurement tomography".

Direct state tomography^[16][17], uses weak measurements and post-selection, motivated by weak-value protocols, to reconstruct the quantum state. It also provides an operational interpretation of wavefunction amplitudes.

Weak-measurement tomography ^[18], aims to improve upon standard tomography by exploiting the minimal disturbance from weak measurements, allowing the same system to be reused for additional measurements.

Weak values appear to have many applications in quantum foundations. Broadly, they are used as indicators of nonclassicality, as tools for explaining quantum paradoxes, and as links between different interpretations of quantum mechanics.

Anomalous weak values, those lying outside the eigenvalue range of an observable, are considered indicators of nonclassicality. They serve as proofs of quantum contextuality, showing that measurement outcomes cannot be reproduced by any noncontextual hidden-variable model^[19].

Weak values have been used to create (see e.g. Quantum Cheshire cat) and explain some of the paradoxes in the foundations of quantum theory^[20]. They have also been used in experimental studies of Hardy's paradox, where joint weak measurements of entangled pairs of photons reproduced the paradoxical predictions.^[21][22][23]

Weak values have been proposed as a way to define a particle’s velocity at a given position, referred to as the ‘naively observable velocity.’^[24] Experiments in 2010 reported photon trajectories in a double-slit interferometer that qualitatively matched earlier predictions^[25] for photons in the de Broglie-Bohm interpretation.^[26][27] Subsequent analyses have argued that weak velocity measurements do not provide new evidence for or against de Broglie-Bohm theory and cannot directly reveal the form of particle trajectories, even under deterministic assumptions^[28]. A shorter overview of these arguments appears in Ref. ^[29]

Criticisms of weak values include philosophical and practical criticisms. Some noted researchers such as Asher Peres, Tony Leggett, David Mermin^{[citation needed]}, and Charles H. Bennett ^[30][31] are critical of weak values. Below criticism of weak values are grouped by the kind of criticism.

After Aharonov, Albert, and Vaidman published their paper, two critical comments and a reply were subsequently published.

Asher Peres in his comment ^[32], argued that even if a weak measurement yields a precisely determined average value, this does not imply the observable itself actually took that value. He stated that a detailed analysis of the measurement data would reveal two peaks at (${\displaystyle p=\pm 1}$), corresponding to the true eigenvalues, with their widths reflecting the initial measurement uncertainty. According to Peres, such results are fully consistent with standard quantum mechanics, and the apparently "anomalous" weak values arise from overlap and interference effects rather than any violation of quantum principles.

Tony Leggett's^[33] comment raised two main objections. First, he argued that Aharonov, Albert, and Vaidman’s result is largely irrelevant to standard quantum measurement theory because it relies on a nonstandard definition of measurement, treating a specific weak interaction Hamiltonian as fundamental rather than as one component of a broader process. Second, he maintained that the weak value ${\displaystyle A_{w}}$ is not a true value of the observable—it is neither an individual outcome nor an ensemble average, but instead reflects how the system affects the measuring device under weak coupling and postselection, and is valid only to lowest order in the interaction strength.

The reply by Aharonov and Vaidman^[34] to the comments by Peres and Leggett addressed several of the technical points raised in the critiques, although later discussions in the literature have expressed differing views on how fully these issues were resolved.

Subsequent analyses have continued to question aspects of the weak-value framework.

Svensson^[35] argued that weak values should not be interpreted as ordinary physical properties. He noted that, as ratios of quantum amplitudes, weak values lack justification in the axioms of quantum mechanics for interpretation as real, measurable quantities such as probabilities or expectation values. In his analysis, their dependence on both pre- and postselection makes them context-dependent and adjustable rather than objective system features. He further contended that applying realistic interpretations of weak values to cases such as the Three-Box Paradox or Hardy’s Paradox leads to results he regards as unphysical, such as assigning “minus one particle” to a box or path. Svensson concluded that weak values cannot meaningfully represent physical quantities and should not be taken as indicators of underlying quantum realities.

Kastner^[36] argues that weak values are not new physical quantities and do not provide evidence of retrocausality. She maintains that weak measurements are “weak” only in their coupling to the system, while still involving a strong, projective measurement on the pointer that disturbs the system. In her analysis, weak values are normalized transition amplitudes derivable within standard quantum mechanics, with the observed correlations accounted for by ordinary unitary evolution and postselection. She further contends that the Two-State Vector Formalism adds no additional explanatory power, since conventional quantum theory already describes the phenomena associated with weak values.

Weak Values are often used to explain paradoxical quantum phenomena. There is a lot of work on this topic so below only a few key examples are explored.

In Vaidman’s 2013 paper^[37], the “weak trace” is defined through the weak value of a projection operator. In this account, a nonzero weak value is taken to indicate where the particle left a measurable influence, and such regions are interpreted as places the particle was.

Hance, Rarity, and Ladyman (2023)^[38] offer a critique of this interpretation, focusing on its use in describing the "past" of a quantum particle. They argue that weak measurements disturb the system, that nonzero weak values should not be regarded as evidence of a particle’s presence, and that weak values describe ensemble averages rather than properties of individual systems. Related objections to this interpretation appear in other sources.^[39]

Dressel and Jordan^[40] presented classical measurement models in which ambiguous or noisy detectors can yield amplified values that lie outside the observable’s eigenvalue range. Their analysis indicates that classical systems can reproduce certain anomalous averages, although they noted that quantum weak values display additional structure not captured by classical disturbance alone.

Ferrie and Combes^[41] introduced a simple classical model that produces anomalous weak values. Their paper title, “How the Result of a Single Coin Toss Can Turn Out to be 100 Heads,” refers to the original Aharonov–Albert–Vaidman paper. They argued that anomalous weak values can arise in purely classical systems, such as a noisy coin-toss model with pre- and postselection, and interpreted weak values as statistical effects of disturbance rather than inherently quantum phenomena. Their analysis prompted a published comment by Brodutch^[42] and a reply by the authors^[43]

Ipsen^[44] compared classical and quantum weak values within a single operational framework. In this account, anomalous weak values in classical systems originate from measurement disturbance, whereas in quantum systems they can also arise from interference effects in the weak-value denominator. A later analysis by Ipsen^[45] argued that anomalous weak values arise not from new physics but from the small, unavoidable disturbance caused by weak measurements. In this view, even infinitesimal interactions alter the post-selection probability enough to shift the measurement outcomes beyond an observable’s eigenvalue range. Thus weak measurements are never truly non-invasive and that such “anomalous” results are statistical effects of measurement back-action, not evidence of deeper quantum paradoxes.

There has been extensive debate in the primary literature regarding the role of weak values in quantum metrology^[46], including critiques and rebuttals. According to a review article^[47], analyses based on Fisher information and parameter estimation indicate that postselected weak-value amplification does not generally improve precision in metrological tasks. Although the technique can increase signal size, postselection reduces data efficiency because most trials are discarded. The review concludes that weak-value methods do not provide a fundamental quantum advantage in metrology, and that large amplification factors alone do not enhance estimation performance.

Rostom^[48] argued that the amplification observed in postselected weak-value experiments can be understood as a phase-dependent interference effect within the measurement apparatus, rather than as a direct physical manifestation of the weak value. In this account, postselection recovers an interference pattern that would otherwise remain hidden, and the resulting sensitivity enhancement is attributed to entanglement and interference effects rather than to anomalous weak values considered as standalone physical quantities.

Gross et al^[49] argued that weak-measurement-based quantum state tomography (proposed in Refs^[16][17][18]) offers no fundamental advantage over standard methods. They contended that weak measurements provide no information beyond that available from conventional generalized measurements, and typically perform less efficiently because of postselection and weak coupling. Gross and colleagues also questioned claims that weak measurements yield a more “direct” or less disturbing procedure for reconstructing the wave function. In their assessment, weak-value-inspired tomography does not circumvent disturbance constraints and does not provide a deeper operational interpretation of quantum states than standard tomographic techniques.

• Zeeya Merali (April 2010). "Back From the Future". Discover. A series of quantum experiments shows that measurements performed in the future can influence the present.{{cite journal}}: CS1 maint: postscript (link)
• "Quantum physics first: Researchers observe single photons in two-slit interferometer experiment". phys.org. June 2, 2011.
• Adrian Cho (5 August 2011). "Furtive Approach Rolls Back the Limits of Quantum Uncertainty". Science. 333 (6043): 690–693. Bibcode:2011Sci...333..690C. doi:10.1126/science.333.6043.690. PMID 21817029.