{ "id": "epfp00bbmp", "slug": "weak-value-amplification", "title": "Weak value amplification", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.quantum", "label": "Quantum physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.quantum" }, { "id": "physical.physics.quantum.mechanics", "label": "Quantum mechanics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.quantum.mechanics" }, { "id": "physical.physics.quantum.mechanics.phenomena-and-measurement", "label": "Quantum phenomena and measurement", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.quantum.mechanics.phenomena-and-measurement" }, { "id": "physical.physics.quantum.mechanics.phenomena-and-measurement.qpm-measurement-decoherence", "label": "Measurement and decoherence", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.quantum.mechanics.phenomena-and-measurement.qpm-measurement-decoherence" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.physics", "label": "Non-geographic · 1946 to 2000: Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" } ] } ], "excerpt": "Weak value amplification is a quantum measurement technique that enlarges tiny detector signals by weak coupling and post-selection, introduced by Aharonov, Albert, and Vaidman in 1988.", "snippet": "Weak value amplification is a quantum measurement technique that enlarges tiny detector signals by weak coupling and post-selection, introduced by Aharonov, Albert, and Vaidman in 1988.", "node": "physical.physics.quantum.mechanics.phenomena-and-measurement.qpm-measurement-decoherence", "markdown": "# Weak value amplification\n\nWeak 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.[1](https://doi.org/10.1103/physrevlett.60.1351) 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.[2](https://arxiv.org/html/2407.10087)\n\n| Key fact | Value |\n|---|---|\n| Weak value definition | \\( \\langle \\hat{A} \\rangle_{w} = \\langle \\psi_{f} | \\hat{A} | \\psi_{i} \\rangle / \\langle \\psi_{f} | \\psi_{i} \\rangle \\)[2](https://arxiv.org/html/2407.10087) |\n| Sagnac-interferometer amplification | Over 100; 560 frad mirror deflection measured (20 fm piezo travel)[3](https://arxiv.org/html/0906.4828) |\n| Spin Hall effect of light | Amplified by four orders of magnitude; 1 angstrom sensitivity[2](https://arxiv.org/html/2407.10087) |\n| Post-selection cost | Depends on parameterization; in the Sagnac scheme, \\( P_{\\mathrm{ps}} \\approx \\phi^{2}/4 \\) for weak value \\( -2i/\\phi \\)[2](https://arxiv.org/html/2407.10087) |\n| On-chip enhancement | 7 dB (2021) and 30 dB phase-response enhancement over a standard Mach-Zehnder interferometer[4](https://www.nature.com/articles/s41467-021-26522-2) • [5](https://par.nsf.gov/servlets/purl/10711180) |\n| Shot-noise-limited SNR | Remains constant as amplification increases[6](https://justindressel.com/papers/RevModPhys.86.307.pdf) |\n\n## How it works\n\nA 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:[2](https://arxiv.org/html/2407.10087)\n\n\\[ \\langle \\hat{A} \\rangle_{w} = \\frac{ \\langle \\psi_{f} | \\hat{A} | \\psi_{i} \\rangle }{ \\langle \\psi_{f} | \\psi_{i} \\rangle }. \\]\n\nWhen 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.[2](https://arxiv.org/html/2407.10087)\n\nThe 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.[2](https://arxiv.org/html/2407.10087) 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.[2](https://arxiv.org/html/2407.10087) • [6](https://justindressel.com/papers/RevModPhys.86.307.pdf)\n\n## How it is done\n\nA WVA experiment combines two key ingredients, weak measurement and post-selection, followed by signal readout.[7](https://www.mdpi.com/2304-6732/11/4/291)\n\n1. **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.[8](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)\n2. **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} \\).[2](https://arxiv.org/html/2407.10087)\n3. **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.[3](https://arxiv.org/html/0906.4828)\n\nIn 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 \\).[3](https://arxiv.org/html/0906.4828)\n\n## Origin\n\nWeak 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.[1](https://doi.org/10.1103/physrevlett.60.1351) 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 \\).[1](https://doi.org/10.1103/physrevlett.60.1351)\n\nThe 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.[9](https://doi.org/10.1103/physrevlett.66.1107) In that experiment the birefringence-induced transverse displacement was amplified up to 20 times its actual size.[2](https://arxiv.org/html/2407.10087) 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.[10](https://doi.org/10.1098/rspa.1998.0288) P. Ben Dixon and colleagues reported ultrasensitive beam-deflection measurement via interferometric weak value amplification in 2009.[11](https://doi.org/10.1103/physrevlett.102.173601)\n\n## Variants\n\n**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.[2](https://arxiv.org/html/2407.10087)\n\n**Interferometric WVA.** The Sagnac-interferometer scheme uses which-path information and dark-port post-selection to reach amplification factors over 100.[3](https://arxiv.org/html/0906.4828)\n\n**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.[4](https://www.nature.com/articles/s41467-021-26522-2) 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.[5](https://par.nsf.gov/servlets/purl/10711180)\n\n**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.[12](https://link.aps.org/doi/10.1103/PhysRevX.4.011031)\n\n**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.[13](https://www.nature.com/articles/nphys4040) A 2024 review extended the WVA framework to optical nonlinear, opto-mechanical, and spin-mechanical systems, where amplification of number operators occurs.[7](https://www.mdpi.com/2304-6732/11/4/291)\n\n## Applications\n\nA 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.[8](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf) • [2](https://arxiv.org/html/2407.10087) 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.[3](https://arxiv.org/html/0906.4828) 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.[4](https://www.nature.com/articles/s41467-021-26522-2)\n\n## Limitations and alternatives\n\n**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.[2](https://arxiv.org/html/2407.10087) 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.[6](https://justindressel.com/papers/RevModPhys.86.307.pdf)\n\n**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.[14](https://ar5iv.labs.arxiv.org/html/1201.6039) 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.[6](https://justindressel.com/papers/RevModPhys.86.307.pdf)\n\n**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.[12](https://link.aps.org/doi/10.1103/PhysRevX.4.011031) 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.[8](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)\n\n**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.[14](https://ar5iv.labs.arxiv.org/html/1201.6039) 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.[15](https://ar5iv.labs.arxiv.org/html/2104.14393) 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.[16](https://beta.iopscience.iop.org/article/10.1088/1674-1056/ae29fe)\n\n## References\n\n1. [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)\n2. [Progress and Perspectives on Weak-value Amplification](https://arxiv.org/html/2407.10087)\n3. [Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification (Dixon/Starling et al.)](https://arxiv.org/html/0906.4828)\n4. [Enhanced on-chip phase measurement by inverse weak value amplification](https://www.nature.com/articles/s41467-021-26522-2)\n5. [30dB Signal Enhancement by Integrated Weak Value Amplification](https://par.nsf.gov/servlets/purl/10711180)\n6. [Colloquium: Understanding quantum weak values: Basics and applications (Reviews of Modern Physics 86, 307)](https://justindressel.com/papers/RevModPhys.86.307.pdf)\n7. [Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems](https://www.mdpi.com/2304-6732/11/4/291)\n8. [Weak-value amplification (debate paper on metrological advantage, arXiv:1410.6252)](https://pure.hw.ac.uk/ws/files/9632915/1410.6252v3.pdf)\n9. [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)\n10. [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)\n11. [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)\n12. [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)\n13. [Weak-value amplification of the nonlinear effect of a single photon | Nature Physics](https://www.nature.com/articles/nphys4040)\n14. [Weak value amplification in a shot-noise limited interferometer](https://ar5iv.labs.arxiv.org/html/1201.6039)\n15. [Enhanced weak-value amplification via photon recycling](https://ar5iv.labs.arxiv.org/html/2104.14393)\n16. [Impact of decoherence on the metrological advantage of weak-value amplification](https://beta.iopscience.iop.org/article/10.1088/1674-1056/ae29fe)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Measurement and decoherence*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/weak-value-amplification", "markdown_url": "https://www.edgepedia.org/weak-value-amplification.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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