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 in 1988, published in Physical Review Letters 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

The weak value is defined as: is the initial or preselection state is the final or postselection state is defined using the nth power of the operator in this expression.

is small and the exponential can be expanded in a Taylor series, and the first order correction involves the first order weak value: In general the weak value quantity is a complex number. In the weak interaction regime is significantly larger than higher order terms.

For example, two Stern-Gerlach The perturbing action of the first analyzer is described with Pauli giving the weak value

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.

.

. After this one performs a projective measurement on the system. Postselecting (or condition, then the (unnormalized) final state of the meter is

due to the momentum operator in the exponential. There is a way to obtain weak values without postselection..

on line (I) is used, and one requires that

was used. This final approximation is only valid when

. 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:

, 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..

Quantum Tomography

Weak values have also been explored in the context of quantum state 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 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 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.

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

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.

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

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.

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.

References

Further reading