Weak value

Definition

The weak value of the observable A is defined as:{{cite journal A_w = \frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}, where |\psi_i\rangle is the initial or preselection state and |\psi_f\rangle is the final or postselection state. The nth order weak value, A^n_w is defined using the nth power of the operator in this expression.

Weak values arise in small perturbations of quantum measurements. Representing a small perturbation with the operator \exp(-i\epsilon \hat{A}), the probability of detecting a system in a final state given the initial state is P_\epsilon = |{\langle\psi_f|\exp(-i\epsilon \hat{A})|\psi_i\rangle}|^2, For small perturbations, \epsilon is small and the exponential can be expanded in a Taylor series P_\epsilon = |{\langle\psi_f|1-i\epsilon \hat{A} +\dots|\psi_i\rangle}|^2, The first term is the unperturbed probability of detection, P =|{\langle\psi_f|\psi_i\rangle}|^2, and the first order correction involves the first order weak value: \frac{P_\epsilon}{P} \approx 1 + 2 \epsilon A_w. In general the weak value quantity is a complex number. In the weak interaction regime, the ratio P_\epsilon/P is close to one and \epsilon Im A_w is significantly larger than higher order terms.

For example, two Stern-Gerlach analyzers can be arranged along the y axis, with the field of the first one along the z axis set at low magnetic field and second on along the x axis with sufficient field to separate the spin 1/2 particle beams. Going into the second analyzer is the initial state |\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 will be |\psi_f\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1 \ 1 \end{pmatrix}. The perturbing action of the first analyzer is described with Pauli z-axis operator as A= \sigma_z giving the weak value{{cite journal A_w = (\sigma_z)_w = \tan\frac{\alpha}{2}.

Real part

The real part of the weak value provides a quantitative way to discuss non-classical aspects of quantum systems. When the real part of a weak value is falls outside the range of the eigenvalues of the operator, it is called an "anomalous weak value". In addition to being important in discussions of quantum paradoxes, anomalous weak values are the basis of quantum sensor applications.

Derivation

The derivation below follows the presentation given References.{{cite journal 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_{\rm system} \otimes H_{\rm ancilla}. 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, |\Phi\rangle = \frac{1}{(2\pi \sigma^2)^{1/4}}\int dq' \exp[-q'^2/4\sigma^2]|q'\rangle. Its position wavefunction is \Phi(q) =\langle q|\Phi\rangle = \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\rangle denotes an eigenstate of the position operator q.

The system begins in the state |\psi_i\rangle. Thus, the combined initial state of the system and ancilla is |\Psi\rangle, jointly describing the initial state of the system and ancilla, is given then by: |\Psi\rangle =|\psi_i\rangle \otimes |\Phi\rangle.

Next the system and ancilla interact via the unitary U |\Psi\rangle. After this one performs a projective measurement of the projectors { |\psi_f\rangle\langle \psi_f |, I- |\psi_f\rangle\langle \psi_f |} on the system. Postselecting (or condition) on getting the outcome |\psi_f\rangle\langle \psi_f |, then the (unnormalized) final state of the meter is

\begin{align} |\Phi_f \rangle &= \langle \psi_f |U |\psi_i\rangle \otimes |\Phi\rangle\ &\approx \langle \psi_f |(I\otimes I -i \gamma A\otimes p ) |\psi_i\rangle \otimes|\Phi\rangle \quad \text{(I)}\ &= \langle \psi_f|\psi_i\rangle (1 -i \gamma A_w p ) |\Phi\rangle\ &\approx \langle \psi_f|\psi_i\rangle \exp(-i \gamma A_w p) |\Phi\rangle. \quad \text{(II)} \end{align} Here it looks like the ancilla state will be shifted by \gamma A_w due to the momentum operator in the exponential. 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{align} \frac{|\gamma|}{\sigma} \left|\frac{\langle \psi_f |A^n |\psi_i \rangle}{ \langle \psi_f| A |\psi_i \rangle }\right|^{1/(n-1)} \ll 1, \quad (n = 2, 3, \dots) \end{align}

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 \Phi_f(q) = \Phi(q-\gamma A_w).

This is the original wavefunction, shifted by an amount \gamma A_w . By Busch's theorem{{cite book| author = Paul Busch | article-number = 062120

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:

a small gradient of the magnetic field ... yields a tremendous amplification.

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.{{cite journal | article-number = 173601 |article-number = 100518 .

Quantum Tomography

Weak values have also been explored in the context of quantum state tomography. Direct state tomography{{cite journal |article-number = 070402 |article-number = 062121

Quantum foundations

Weak values 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. |article-number = 200401 .

Weak values have been used to create and explain some of the paradoxes in the foundations of quantum theory, for example, the Quantum Cheshire cat. 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.{{cite journal | article-number = 020404 | access-date = June 8, 2013 | archive-date = May 30, 2013 | archive-url = https://web.archive.org/web/20130530051055/http://www.perimeterinstitute.ca/news/hardys-paradox-confirmed-experimentally

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.'{{cite journal | chapter-url = https://doi.org/10.1093/oso/9780198901853.003.0010

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 |access-date = November 5, 2025 |access-date = October 31, 2025

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

|url-access = subscription

|url-access = subscription

The reply by Aharonov and Vaidman |url-access = subscription

Metrological Significance

There has been extensive debate in the primary literature regarding the role of weak values in quantum metrology |article-number = 042116 According to a review article

References

References

1. (2019). "Anomalous Weak Values Without Post-Selection". Quantum.
2. (2019-02-13). "Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer". Physical Review A.
4. Lundeen Jeff S., Sutherland Brandon, Patel Aabid, Stewart Corey, Bamber Charles. (2011). "Direct measurement of the quantum wavefunction". Nature.
5. (2008-09-26). "Quantum Paradoxes: Quantum Theory for the Perplexed". John Wiley & Sons.
6. Yokota K., Yamamoto T., Koashi M., Imoto N.. (2009). "Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair". New J. Phys..
7. Ghose Partha, Majumdar A.S., Guhab S., Sau J.. (2001). "Bohmian trajectories for photons". Physics Letters A.
8. Sacha Kocsis, Sylvain Ravets, Boris Braverman, Krister Shalm, Aephraim M. Steinberg: Observing the trajectories of a single photon using weak measurement, 19th Australian Institute of Physics (AIP) Congress, 2010 [https://web.archive.org/web/20120324224949/http://www.aip.org.au/Congress2010/Abstracts/Monday%206%20Dec%20-%20Orals/Session_3E/Kocsis_Observing_the_Trajectories.pdf]
9. Kocsis Sacha, Braverman Boris, Ravets Sylvain, Stevens Martin J., Mirin Richard P., Shalm L. Krister, Steinberg Aephraim M.. (2011). "''Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer''". Science.
10. Fankhauser Johannes, Dürr Patrick. (2021). "How (not) to understand weak measurements of velocity". Studies in History and Philosophy of Science Part A.