Weak Value v2

---

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,: and is related to the two-state vector formalism. The first experimental realization came from researchers at Rice University in 1991. The physical interpretation and significance of weak values remains a subject of ongoing discussion in the quantum foundations and metrology literature.

Definition and Derivation

Definition

The initial state of the system is denoted as $|{\psi }_i⟩$ and the final state as $|{\psi }_f⟩$. These states are often referred to as the pre- and post-selected quantum states. Also consider an observable $A$ with minimal and maximal eigenvalues $a_{\mathrm{m}\mathrm{i}\mathrm{n}},a_{\mathrm{m}\mathrm{a}\mathrm{x}}$. With respect to these states, the weak value of $A$ is defined as $$A_w=\frac{⟨{\psi }_f|A|{\psi }_i⟩}{⟨{\psi }_f|{\psi }_i⟩}.$$

If $|{\psi }_f⟩=|{\psi }_i⟩$, then the weak value reduces to the usual expectation value in either the initial state $⟨{\psi }_i|A|{\psi }_i⟩$ or the final state $⟨{\psi }_f|A|{\psi }_f⟩$. These expectation values are necessarily bounded by the eigenvalue range of $A$, for example, $a_{\mathrm{m}\mathrm{i}\mathrm{n}}\le ⟨{\psi }_i|A|{\psi }_i⟩\le a_{\mathrm{m}\mathrm{a}\mathrm{x}}$. 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, $|{\psi }_f⟩$, approaches being orthogonal to the pre-selected state, $|{\psi }_i⟩$, i.e. $|⟨{\psi }_f|{\psi }_i⟩|\ll 1$. If $A_w$ is larger than the largest eigenvalue of $A$, $a_{\mathrm{m}\mathrm{a}\mathrm{x}}$, or smaller than its smallest eigenvalue, $a_{\mathrm{m}\mathrm{i}\mathrm{n}}$, 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.

Example

As an example consider a spin 1/2 particle. Take $A$ to be the Pauli Z operator $A={\sigma }_z$ with eigenvalues $\pm 1$. Using the initial state $$|{\psi }_i⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}\cos \frac{\alpha }{2}+\sin \frac{\alpha }{2}\\ \cos \frac{\alpha }{2}-\sin \frac{\alpha }{2}\end{array}\right)$$ and the final state $$|{\psi }_f⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$$ one can calculate the weak value to be $$A_w=\frac{⟨{\psi }_f\mid {\sigma }_z\mid {\psi }_i⟩}{⟨{\psi }_f\mid {\psi }_i⟩}=\frac{\sin (\alpha /2)}{\cos (\alpha /2)}=\tan \frac{\alpha }{2}.$$

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

Derivation

The derivation below follows the presentation given References.

Derivation (click to show)

A quantum system is measured by coupling it to an ancillary quantum system that acts as the measuring device. The joint Hilbert space is $H_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}\otimes H_{\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}}$. The observable to be measured on the system is $A$. The system and ancilla interact through the Hamiltonian $$H=\gamma A\otimes p,$$ where the coupling constant is integrated over an interaction time $\gamma ={\int }_{t_i}^{t_f}g(t)dt\ll 1$ and $[q,p]=i$ is the canonical commutator. The Hamiltonian generates the unitary $$U=\exp [-i\gamma A\otimes p].$$ Let the initial state of the ancilla be a Gaussian wavepacket in position space, $$|\mathrm{\Phi }⟩=\frac{1}{(2\pi {\sigma }^2{)}^{1/4}}\int d{q}^{\prime }\exp [-q{\text{'}}^2/4{\sigma }^2]|q^{\prime }⟩.$$ Its position wavefunction is $$\mathrm{\Phi }(q)=⟨q|\mathrm{\Phi }⟩=\frac{1}{(2\pi {\sigma }^2{)}^{1/4}}\exp [-q^2/4{\sigma }^2],$$ where $\sigma$ characterizes the initial uncertainty in the pointer position of the measuring device and $|q⟩$ denotes an eigenstate of the position operator $q$. The system begins in the state $|{\psi }_i⟩$. Thus, the combined initial state of the system and ancilla is $|\mathrm{\Psi }⟩$, jointly describing the initial state of the system and ancilla, is given then by: $$|\mathrm{\Psi }⟩=|{\psi }_i⟩\otimes |\mathrm{\Phi }⟩.$$ Next the system and ancilla interact via the unitary $U|\mathrm{\Psi }⟩$. After this one performs a projective measurement of the projectors $\{|{\psi }_f⟩⟨{\psi }_f|,I-|{\psi }_f⟩⟨{\psi }_f|\}$ on the system. Postselecting (or condition) on getting the outcome $|{\psi }_f⟩⟨{\psi }_f|$, then the (unnormalized) final state of the meter is $$\begin{array}{rl}|{\mathrm{\Phi }}_f⟩& =⟨{\psi }_f|U|{\psi }_i⟩\otimes |\mathrm{\Phi }⟩\\ & \approx ⟨{\psi }_f|(I\otimes I-i\gamma A\otimes p)|{\psi }_i⟩\otimes |\mathrm{\Phi }⟩\text{(I)}\\ & =⟨{\psi }_f|{\psi }_i⟩(1-i\gamma A_{w}p)|\mathrm{\Phi }⟩\\ & \approx ⟨{\psi }_f|{\psi }_i⟩\exp (-i\gamma A_{w}p)|\mathrm{\Phi }⟩.\text{(II)}\end{array}$$ Here it looks like the ancilla state will be shifted by $\gamma A_w$ due to the momentum operator in the exponential. It should be noted that there is a way to obtain weak values without postselection.. To arrive at this conclusion, the first order series expansion of $U$ on line (I) is used, and one requires that $$\begin{array}{r}\frac{|\gamma |}{\sigma }{\left|\frac{⟨{\psi }_f|A^n|{\psi }_i⟩}{⟨{\psi }_f|A|{\psi }_i⟩}\right|}^{1/(n-1)}\ll 1,(n=2,3,\dots )\end{array}$$ On line (II) the approximation that $e^{-x}\approx 1-x$ for small $x$ was used. This final approximation is only valid when $$|\gamma A_w|/\sigma \ll 1.$$ As $p$ is the generator of translations, the ancilla's wavefunction is now given by $${\mathrm{\Phi }}_f(q)=\mathrm{\Phi }(q-\gamma A_w).$$ This is the original wavefunction, shifted by an amount $\gamma A_w$. By Busch's theorem the system and meter wavefunctions are necessarily disturbed by the measurement. Although the protocol used to access the weak value is minimally disturbing in a specific sense, it still induces disturbance.

Applications

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.

Quantum metrology

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

Another striking aspect of this experiment becomes evident when we consider it as a device for measuring a small gradient of the magnetic field ... yields a tremendous amplification.

Aharonov, Albert, Vaidman

In modern language, when the weak value $A_w$ lies outside the eigenvalue range of the observable $A$, 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. 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..

Quantum Tomography

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, 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 , aims to improve upon standard tomography by exploiting the minimal disturbance from weak measurements, allowing the same system to be reused for additional measurements.

Quantum foundations

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.

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. 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.

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.’ Experiments in 2010 reported photon trajectories in a double-slit interferometer that qualitatively matched earlier predictions for photons in the de Broglie-Bohm interpretation. 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. A shorter overview of these arguments appears in Ref.

Criticisms

Criticisms of weak values include philosophical and practical criticisms. Some noted researchers such as Asher Peres, Tony Leggett, David Mermin, and Charles H. Bennett are critical of weak values. Below criticism of weak values are grouped by the kind of criticism.

Interpretation of the Weak Value

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

Asher Peres in his comment , 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 ($p=\pm 1$), 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 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 $A_w$ 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 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 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 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.

Foundational Significance

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.

Weak values resolving paradoxes

In Vaidman’s 2013 paper, 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) 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.

Classical vs quantum weak values

Dressel and Jordan 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 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 and a reply by the authors

Ipsen 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 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.

Metrological Significance

There has been extensive debate in the primary literature regarding the role of weak values in quantum metrology, including critiques and rebuttals. According to a review article, 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 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.

Tomographic Significance

Gross et al argued that weak-measurement-based quantum state tomography (proposed in Refs) 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.

Further reading

• 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.
• "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.

References

This article "Weak Value v2" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Weak Value v2. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

---