# Weak value amplification

Weak value amplification (WVA) is a quantum measurement technique that enlarges the detector signal produced by a small physical parameter, such as a beam deflection, phase shift, or frequency shift, by coupling the system only weakly to a meter and then keeping only the rare outcomes of a post-selection measurement. It exploits weak values, quantities introduced by [Yakir Aharonov](https://www.edgepedia.org/yakir-aharonov), David Z. Albert, and Lev Vaidman in 1988 that can lie far outside the range of the measured observable's eigenvalues.<sup>[1](https://doi.org/10.1103/physrevlett.60.1351)</sup> In optical experiments the amplified quantity is usually a tiny spatial displacement or phase tilt of a light beam, read out as a shift of the beam's centroid on a detector.<sup>[2](https://arxiv.org/html/2407.10087)</sup>

| Key fact | Value |
|---|---|
| Weak value definition | \( \langle \hat{A} \rangle_{w} = \langle \psi_{f} | \hat{A} | \psi_{i} \rangle / \langle \psi_{f} | \psi_{i} \rangle \)<sup>[2](https://arxiv.org/html/2407.10087)</sup> |
| Sagnac-interferometer amplification | Over 100; 560 frad mirror deflection measured (20 fm piezo travel)<sup>[3](https://arxiv.org/html/0906.4828)</sup> |
| Spin Hall effect of light | Amplified by four orders of magnitude; 1 angstrom sensitivity<sup>[2](https://arxiv.org/html/2407.10087)</sup> |
| Post-selection cost | Depends on parameterization; in the Sagnac scheme, \( P_{\mathrm{ps}} \approx \phi^{2}/4 \) for weak value \( -2i/\phi \)<sup>[2](https://arxiv.org/html/2407.10087)</sup> |
| On-chip enhancement | 7 dB (2021) and 30 dB phase-response enhancement over a standard Mach-Zehnder interferometer<sup>[4](https://www.nature.com/articles/s41467-021-26522-2)</sup><sup> • </sup><sup>[5](https://par.nsf.gov/servlets/purl/10711180)</sup> |
| Shot-noise-limited SNR | Remains constant as amplification increases<sup>[6](https://justindressel.com/papers/RevModPhys.86.307.pdf)</sup> |

## How it works

A weak value is assigned to an observable \( \hat{A} \) of a system prepared in a pre-selection state \( | \psi_{i} \rangle \) and later found in a post-selection state \( | \psi_{f} \rangle \). It is defined as the ratio of the post-selected matrix element to the post-selection overlap:<sup>[2](https://arxiv.org/html/2407.10087)</sup>

\[ \langle \hat{A} \rangle_{w} = \frac{ \langle \psi_{f} | \hat{A} | \psi_{i} \rangle }{ \langle \psi_{f} | \psi_{i} \rangle }. \]

When the two states approach orthogonality, \( | \langle \psi_{f} | \psi_{i} \rangle | \rightarrow 0 \), the denominator becomes small and the modulus of the weak value greatly exceeds the maximum eigenvalue of \( \hat{A} \); the value can also be complex. Such values are called anomalous.<sup>[2](https://arxiv.org/html/2407.10087)</sup>

The real and imaginary parts of a large weak value shift different meter variables. Both can be read from mean values of the incompatible observables \( Q \) and \( P \) of the meter, so experiments typically adopt a purely real or a purely imaginary weak value.<sup>[2](https://arxiv.org/html/2407.10087)</sup> In practice the real part amplifies transverse beam deflection (a position shift), while the imaginary part amplifies phase-like couplings that appear as momentum shifts of the pointer.<sup>[2](https://arxiv.org/html/2407.10087)</sup><sup> • </sup><sup>[6](https://justindressel.com/papers/RevModPhys.86.307.pdf)</sup>

## How it is done

A WVA experiment combines two key ingredients, weak measurement and post-selection, followed by signal readout.<sup>[7](https://www.mdpi.com/2304-6732/11/4/291)</sup>

1. **Weak interaction.** The system couples to a meter so gently that the meter is barely disturbed. In typical optical experiments the polarization of a light beam plays the role of the quantum system and the beam deflection replaces the meter needle.<sup>[8](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)</sup>
2. **Post-selection.** Most photons are discarded; only those found in \( | \psi_{f} \rangle \) are kept. Choosing a small parameter \( \epsilon \) or \( \phi \) yields a large real weak value \( 2/\epsilon \) or a large imaginary weak value \( -2i/\phi \), at the cost of a post-selection success probability \( p_{f} \approx \epsilon^{2} \).<sup>[2](https://arxiv.org/html/2407.10087)</sup>
3. **Signal readout.** The small change in the parameter of interest now produces a large shift of the pointer, which is read out, often with a lock-in amplifier.<sup>[3](https://arxiv.org/html/0906.4828)</sup>

In the Sagnac-interferometer implementation, the post-selection is simply a photon emerging from the dark port, with probability \( P_{\mathrm{ps}} = | \langle \psi_{f} | \psi_{i} \rangle |^{2} = \sin^{2}(\phi/2) \); the weak value is purely imaginary, \( A_{w} = -i \cot(\phi/2) \approx -2i/\phi \) for small \( \phi \), and the amplified position expectation is \( \langle x \rangle = 2 k \cdot a^{2} | A_{w} | \approx 4 k \cdot a^{2} / \phi \).<sup>[3](https://arxiv.org/html/0906.4828)</sup>

## Origin

Weak values were introduced by Yakir Aharonov, David Z. Albert, and Lev Vaidman in their 1988 [Physical Review Letters](https://www.edgepedia.org/physical-review-letters) paper, whose title poses the question of how the result of a spin component measurement of a spin-1/2 particle can turn out to be 100.<sup>[1](https://doi.org/10.1103/physrevlett.60.1351)</sup> In the original example, the weak value inferred from the pointer shift of the pre- and post-selected ensemble in a weak measurement of a spin component of a spin-1/2 particle was 100, far outside the eigenvalue range of \( \pm 1/2 \).<sup>[1](https://doi.org/10.1103/physrevlett.60.1351)</sup>

The first laboratory realization came from N. W. M. Ritchie, J. G. Story, and [Randall G. Hulet](https://www.edgepedia.org/randall-g-hulet) in 1991: a birefringent crystal separated the two linear-polarization components of a laser beam by a distance small compared to the beam waist, and a subsequent strong measurement translated the beam centroid by a distance far larger than that separation; the authors noted the effect might have application in amplifying and detecting weak effects.<sup>[9](https://doi.org/10.1103/physrevlett.66.1107)</sup> In that experiment the birefringence-induced transverse displacement was amplified up to 20 times its actual size.<sup>[2](https://arxiv.org/html/2407.10087)</sup> A 1998 study by A. D. Parks, D. W. Cullin, and D. C. Stoudt observed and measured an optical Aharonov-Albert-Vaidman effect, identifying an intrinsic "weak energy" in the equation of motion for a weak value that is non-vanishing only for pre- and post-selected systems.<sup>[10](https://doi.org/10.1098/rspa.1998.0288)</sup> P. Ben Dixon and colleagues reported ultrasensitive beam-deflection measurement via interferometric weak value amplification in 2009.<sup>[11](https://doi.org/10.1103/physrevlett.102.173601)</sup>

## Variants

**Real and imaginary schemes.** [Experiments](https://www.edgepedia.org/experiments) choose a weak value that is purely real, to amplify a position (deflection) signal, or purely imaginary, to amplify a phase-type signal, because reading both requires measuring two incompatible meter observables.<sup>[2](https://arxiv.org/html/2407.10087)</sup>

**Interferometric WVA.** The Sagnac-interferometer scheme uses which-path information and dark-port post-selection to reach amplification factors over 100.<sup>[3](https://arxiv.org/html/0906.4828)</sup>

**Inverse WVA.** Ordinary WVA measures a spatial phase front tilt using a known phase shift; inverse WVA reverses the roles, measuring the phase shift itself with the signal amplified by a known tilt. An integrated-photonic inverse-WVA device showed a 7 dB signal enhancement over a standard Mach-Zehnder interferometer at equal detected optical power.<sup>[4](https://www.nature.com/articles/s41467-021-26522-2)</sup> An on-chip weak value interferometer later demonstrated 30 dB signal enhancement in phase response over a standard Mach-Zehnder interferometer, and its authors position on-chip WVA as an alternative to squeezing, which is difficult to implement and susceptible to loss.<sup>[5](https://par.nsf.gov/servlets/purl/10711180)</sup>

**Photon recycling.** Instead of discarding the post-selected-away photons, a recycling scheme returns them for another pass; for beam-deflection measurement it boosts the [Fisher information](https://www.edgepedia.org/fisher-information) (A measure of how much data reveals a parameter) by the inverse post-selection probability relative to the standard case.<sup>[12](https://link.aps.org/doi/10.1103/PhysRevX.4.011031)</sup>

**Single-photon nonlinear amplification.** With proper post-selection, a single photon calibrated to write a nonlinear phase shift \( \phi_{o} \) on a probe beam produced measured phase shifts as large as \( 8 \phi_{o} \), an effect equal to that of eight photons.<sup>[13](https://www.nature.com/articles/nphys4040)</sup> A 2024 review extended the WVA framework to optical nonlinear, opto-mechanical, and spin-mechanical systems, where amplification of number operators occurs.<sup>[7](https://www.mdpi.com/2304-6732/11/4/291)</sup>

## Applications

A postselected weak-measurement amplification study applied to a small physical effect detected the spin [Hall effect](https://www.edgepedia.org/hall-effect) of light, a coupling between polarization and transverse momentum at an interface between media of different refractive index; the effect was amplified by four orders of magnitude, with 1 angstrom sensitivity achieved without isolating air disturbance or mechanical vibration.<sup>[8](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)</sup><sup> • </sup><sup>[2](https://arxiv.org/html/2407.10087)</sup> In the Sagnac-interferometer implementation, amplification factors over 100 allowed measurement of 560 frad of mirror deflection caused by 20 fm of piezo actuator travel.<sup>[3](https://arxiv.org/html/0906.4828)</sup> The 2021 inverse-WVA device reached a minimum resolvable phase of 0.2 µrad, versus 0.44 µrad for the standard Mach-Zehnder interferometer at 0.1 Hz resolution bandwidth.<sup>[4](https://www.nature.com/articles/s41467-021-26522-2)</sup>

## Limitations and alternatives

**No free photons.** The amplification factor and the post-selection probability trade against each other: a weak value of \( 2/\epsilon \) costs \( p_{f} \approx \epsilon^{2} \), so most photons are discarded.<sup>[2](https://arxiv.org/html/2407.10087)</sup> In the shot-noise limit this cancels the gain: the signal-to-noise ratio for determining a small parameter within a fixed time remains constant as amplification increases, because the amplified signal is exactly offset by the shot noise from the reduced detection rate.<sup>[6](https://justindressel.com/papers/RevModPhys.86.307.pdf)</sup>

**Where it does help.** WVA significantly improves SNR where technical noise such as alignment noise dominates; one analysis concluded it could outperform standard interferometry by 3 orders of magnitude in the presence of technical noise.<sup>[14](https://ar5iv.labs.arxiv.org/html/1201.6039)</sup> The weakness of the measurement also makes the amplification robust against certain technical noise such as 1/f noise, and the detector handles only a fraction of the beam power while retaining sensitivity comparable to optimal estimation methods.<sup>[6](https://justindressel.com/papers/RevModPhys.86.307.pdf)</sup>

**The Fisher-information debate.** Published analyses disagree on whether WVA offers a fundamental sensitivity advantage. One position holds that weak-value techniques concentrate all Fisher information about the detected parameter into a small portion of events, giving technical advantages over standard techniques that use all photons.<sup>[12](https://link.aps.org/doi/10.1103/PhysRevX.4.011031)</sup> The other holds that "given the same number of input resources, a weak-value strategy will generally not outperform the standard metrology strategy", and that a large amplification factor by itself is not sufficient for an advantage in parameter estimation.<sup>[8](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)</sup>

**Failure modes.** When the weak value becomes large, nonlinear effects of the von-Neumann measurement cap the pointer shift, which vanishes when the pre- and post-selected states are exactly orthogonal, so there is an optimal interaction strength rather than an arbitrarily large amplification.<sup>[14](https://ar5iv.labs.arxiv.org/html/1201.6039)</sup> In quantum-noise-limited systems with ideal detectors, post-selected WVA normally does not beat standard methods because the increased weak value is compensated by reduced post-selected power.<sup>[15](https://ar5iv.labs.arxiv.org/html/2104.14393)</sup> Decoherence imposes a fundamental limitation on the quantum Fisher information achievable via WVA, from which an optimal post-selection state can be derived for different noise types.<sup>[16](https://beta.iopscience.iop.org/article/10.1088/1674-1056/ae29fe)</sup>

## References

1. [Yakir Aharonov, David Z. Albert, Lev Vaidman (1988). How the result of a measurement of a component of the spin of a spin- 1/2 particle can turn out to be 100. Physical Review Letters.](https://doi.org/10.1103/physrevlett.60.1351)
2. [Progress and Perspectives on Weak-value Amplification](https://arxiv.org/html/2407.10087)
3. [Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification (Dixon/Starling et al.)](https://arxiv.org/html/0906.4828)
4. [Enhanced on-chip phase measurement by inverse weak value amplification](https://www.nature.com/articles/s41467-021-26522-2)
5. [30dB Signal Enhancement by Integrated Weak Value Amplification](https://par.nsf.gov/servlets/purl/10711180)
6. [Colloquium: Understanding quantum weak values: Basics and applications (Reviews of Modern Physics 86, 307)](https://justindressel.com/papers/RevModPhys.86.307.pdf)
7. [Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems](https://www.mdpi.com/2304-6732/11/4/291)
8. [Weak-value amplification (debate paper on metrological advantage, arXiv:1410.6252)](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)
9. [N. W. M. Ritchie, J. G. Story, Randall G. Hulet (1991). Realization of a measurement of a ‘‘weak value’’. Physical Review Letters.](https://doi.org/10.1103/physrevlett.66.1107)
10. [A. D. Parks, D. W. Cullin, D. C. Stoudt (1998). Observation and measurement of an optical Aharonov–Albert–Vaidman effect. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences.](https://doi.org/10.1098/rspa.1998.0288)
11. [P. Ben Dixon and colleagues (2009). Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification. Physical Review Letters.](https://doi.org/10.1103/physrevlett.102.173601)
12. [Technical Advantages for Weak-Value Amplification: When Less Is More (Phys. Rev. X 4, 011031)](https://link.aps.org/doi/10.1103/PhysRevX.4.011031)
13. [Weak-value amplification of the nonlinear effect of a single photon | Nature Physics](https://www.nature.com/articles/nphys4040)
14. [Weak value amplification in a shot-noise limited interferometer](https://ar5iv.labs.arxiv.org/html/1201.6039)
15. [Enhanced weak-value amplification via photon recycling](https://ar5iv.labs.arxiv.org/html/2104.14393)
16. [Impact of decoherence on the metrological advantage of weak-value amplification](https://beta.iopscience.iop.org/article/10.1088/1674-1056/ae29fe)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Measurement and decoherence*

*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*

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