Draft:Weak Value v2

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

There are many excellent review articles on weak values (see e.g. ) here we briefly cover the basics.

Definition

We denote the initial state of the system as $|\psi _{i}\rangle$ and the final state as $|\psi _{f}\rangle$. These states are often referred to as the pre- and post-selected quantum states. We also consider an observable $A$ with minimal and maximal eigenvalues ${a_{\rm {min}},a_{\rm {max}}}$. With respect to these states, the weak value of $A$ is defined as: $$A_{w}={\frac {\langle \psi _{f}|A|\psi _{i}\rangle }{\langle \psi _{f}|\psi _{i}\rangle }}.$$

Notice that if $|\psi _{f}\rangle =|\psi _{i}\rangle$, then the weak value reduces to the usual expectation value in either the initial state $\langle \psi _{i}|A|\psi _{i}\rangle$ or the final state $\langle \psi _{f}|A|\psi _{f}\rangle$. These expectation values are necessarily bounded by the eigenvalue range of $A$, for example, $a_{\rm {min}}\leq \langle \psi _{i}|A|\psi _{i}\rangle \leq a_{\rm {max}}$. 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, $|\psi _{f}\rangle$, approaches being orthogonal to the pre-selected state, $|\psi _{i}\rangle$, i.e. $|\langle \psi _{f}|\psi _{i}\rangle |\ll 1$. If $A_{w}$ is larger than the largest eigenvalue of $A$, $a_{\rm {max}}$, or smaller than its smallest eigenvalue, $a_{\rm {min}}$, the weak value is said to be anomalous. Such anomalous weak values are especially interesting because they can be complex and fall outside the usual eigenvalue range, both features absent in standard expectation values.

To understand how such strange expectations could arise in practice it is helpful to consider the example and derivation below.

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 $$|\psi _{i}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\\\cos {\frac {\alpha }{2}}-\sin {\frac {\alpha }{2}}\end{pmatrix}}$$ and the final state $$|\psi _{f}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}$$ one can calculate the weak value to be $$A_{w}={\frac {\langle \psi _{f}\mid \sigma _{z}\mid \psi _{i}\rangle }{\langle \psi _{f}\mid \psi _{i}\rangle }}={\frac {\sin(\alpha /2)}{\cos(\alpha /2)}}=\tan {\frac {\alpha }{2}}.$$

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

Derivation

Here we follow the presentation given by Duck, Stevenson, and Sudarshan, (with some notational updates from Kofman et al.) which makes explicit when the approximations used to derive the weak value are valid.

Derivation (click to show)

Consider a quantum system that is to be measured by coupling it to an ancillary (also quantum) 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 ${\textstyle \gamma =\int _{t_{i}}^{t_{f}}g(t)dt\ll 1}$ 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

$${\begin{aligned}|\Phi _{f}\rangle &=\langle \psi _{f}|U|\psi _{i}\rangle \otimes |\Phi \rangle \\&\approx \langle \psi _{f}|(I\otimes I-i\gamma A\otimes p)|\psi _{i}\rangle \otimes |\Phi \rangle \quad {\text{(I)}}\\&=\langle \psi _{f}|\psi _{i}\rangle (1-i\gamma A_{w}p)|\Phi \rangle \\&\approx \langle \psi _{f}|\psi _{i}\rangle \exp(-i\gamma A_{w}p)|\Phi \rangle .\quad {\text{(II)}}\end{aligned}}$$ 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

$${\begin{aligned}{\frac {|\gamma |}{\sigma }}\left|{\frac {\langle \psi _{f}|A^{n}|\psi _{i}\rangle }{\langle \psi _{f}|A|\psi _{i}\rangle }}\right|^{1/(n-1)}\ll 1,\quad (n=2,3,\dots )\end{aligned}}$$

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 the system and meter wavefunctions are necessarily disturbed by the measurement. There is a certain sense in which the protocol that allows one to measure the weak value is minimally disturbing, but there is still disturbance.

Applications

Some researchers find weak values intriguing because they may advance quantum technologies such as metrology, while also deepening our understanding of quantum foundations. Below we briefly explore these possibilities.

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 suggestion was subsequently demonstrated experimentally by Hosten and Kwiat and later by Dixon et al. This area has since developed into an active field of research exploring the applications of weak values in quantum sensing and precision measurement. Given the large number of theoretical and experimental studies, detailing individual contributions is not appropriate here. Readers interested in the broader literature are referred to review articles on weak value amplification and its applications.

Quantum Tomography

Weak values have also been explored in the context of quantum state tomography. Two main approaches have emerged. We will call the first approach, "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. As shown by Matthew Pusey in 2014, such values serve as direct proofs of quantum contextuality, demonstrating that measurement outcomes cannot be explained by any noncontextual hidden-variable theory.

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. For example, the research group of Aephraim M. Steinberg at the University of Toronto confirmed Hardy's paradox experimentally using joint weak measurement of the locations of entangled pairs of photons. (also see)

Howard M. Wiseman proposed using weak values to define a particle's velocity at a position, a quantity he called its “naïvely observable velocity.” In 2010, a first experimental observation of trajectories of a photon in a double-slit interferometer was reported, which displayed the qualitative features predicted in 2001 by Partha Ghose for photons in the de Broglie-Bohm interpretation. Following up on Wiseman's weak velocity measurement, Johannes Fankhauser and Patrick Dürr suggest in a paper that weak velocity measurements constitute no new arguments, let alone empirical evidence, in favor of or against standard de Broglie-Bohm theory. According to the authors such measurements could not provide direct experimental evidence displaying the shape of particle trajectories, even if it is assumed that some deterministic particle trajectories exist. A shorter and less technical presentation of the main arguments appears in: .

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 we break down criticism of weak values based on 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 mean the observable itself actually took that value. A careful 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. Properly interpreted, such results are fully consistent with standard quantum mechanics, and the apparently "anomalous" weak values arise from overlap and interference effects rather than any real violation of quantum principles.

Tony Leggett's comment had 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 noted 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 clarified several technical points raised by their critics, but their response has been viewed by some commentators as only partially satisfactory.

Weak values continue to be questioned by several authors. Below we summarize three more recent criticisms.

Parrott argues that weak values are not unique and depend on details of the measurement interaction rather than reflecting an intrinsic property of the observable. He shows that by varying the choice of meter system and coupling, one can obtain arbitrarily many different weak values for the same pre- and postselected states. Consequently, a weak value by itself does not unambiguously characterize the observable and should not be interpreted as a property of the system alone but as dependent on the specific measurement procedure used.

Svensson argued that weak values should not be interpreted as ordinary physical properties. As ratios of quantum amplitudes, they lack justification in quantum mechanics' axioms for interpretation as real, measurable properties like probabilities or expectation values. Their dependence on both pre- and postselection makes them context-dependent and tunable rather than objective system features. Moreover, realistic interpretations of weak values in cases like the Three-Box Paradox or Hardy's Paradox produce nonsensical results such as "minus one particle" in a box or path. Svensson concludes these values cannot meaningfully represent physical quantities and should not be treated as revealing hidden quantum realities.

Kastner argues that weak values are not new physical quantities or evidence of retrocausality. She contends that weak measurements are only “weak” in their coupling to the system but involve a strong, projective measurement on the pointer that disturbs the system. Weak values, she argues, are simply normalized transition amplitudes derivable from standard quantum mechanics, with all observed correlations explained by ordinary evolution and post-selection. The Two-State Vector Formalism, she concludes, adds no explanatory power, as conventional quantum theory already accounts for weak-value phenomena.

Foundational Significance

Weak Values are often used to explain paradoxical quantum phenomena. There is a lot of work and this could quickly spiral out so here we will just select a few key examples.

Weak values resolving paradoxes

For example in Vaidman’s 2013 paper , the weak trace is defined through the weak value of a projection operator. According to Vaidman (2013), the weak trace indicates where the particle left a measurable influence. In this view, regions where the weak value (and hence the weak trace) is nonzero are interpreted as places the particle “was”.

In response, Hance, Rarity, and Ladyman (2023) critique the foundational interpretation of weak values, particularly their use in describing the “past” of a quantum particle. They argue that weak measurements disturb the system, that nonzero weak values should not be taken as evidence of a particle’s presence, and that weak values represent ensemble averages rather than properties of individual particles. Many authors have objected to Vaidman's idea.

Classical vs quantum weak values

In 2012 Dressel and Jordan showed that even in a purely classical measurement scenario, ambiguous or noisy detectors can yield amplified values, which lie outside the observable’s eigenvalue range. That is classical models can reproduce some anomalous averages. However Dressel and Jordan emphasized that quantum weak values exhibit a richer structure that cannot be fully captured by classical disturbance alone.

In 2014, Ferrie and Combes presented a simple and striking example of classical weak values. Their provocatively titled paper “How the Result of a Single Coin Toss Can Turn Out to be 100 Heads,” is a reference to the original Aharonov–Albert–Vaidman paper. They showed that anomalous weak values can arise in purely classical systems, such as a noisy coin toss with pre- and postselection, arguing that weak values are statistical effects of disturbance rather than inherently quantum. The paper was controversial and prompted a published comment from Brodutch and a reply by the authors.

Ipsen compares classical and quantum weak values within a single operational framework. In classical systems, anomalous weak values arise only from measurement disturbance, whereas in quantum systems, interference in the weak-value denominator allows such anomalies even under minimal disturbance. More recently Ipsen argues that anomalous weak values arise not from new physics but from the small, unavoidable disturbance caused by weak measurements. He shows that even infinitesimal interactions alter the post-selection probability enough to shift the measurement outcomes beyond an observable’s eigenvalue range. He concludes that 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 over the role of weak values in quantum metrology, with many papers offering critiques and rebuttals. To avoid adding to this ongoing controversy, readers are directed to the paper that initiated much of the discussion. and the review article by Knee et al..

The summary of the criticisms below is based on the review article which provides a balanced summary of theoretical and experimental work on the use of weak values in precision measurement. Analyses based on Fisher information and parameter estimation show that postselected weak-value amplification does not generally improve precision over standard methods: while it increases signal size, it reduces data efficiency because most trials are discarded. The authors conclude that weak-value techniques offer no fundamental quantum advantage in metrology, and large amplification factors alone do not enhance estimation performance.

Rostom (2022) 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 itself. On this view, 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 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 showed that weak measurements provide no new information beyond conventional generalized measurements. Typically perform less efficiently due to postselection and weak coupling. Gross and colleagues also questioned the claims that weak measurements provide a more “direct” or less disturbing way to reconstruct the wave function. Weak-value-inspired tomography neither avoids disturbance constraints nor offers a deeper operational interpretation of quantum states beyond standard tomography.

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.{{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.