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arXiv:0909.0295

Quantum weak values are not unique; what do they actually measure?

Stephen Parrott

Published Sep 2, 2009• 10 citations
arXivHTMLPDF

Abstract

Precise definitions of "weak [quantum] measurements" and "weak value" [of a quantum observable] are offered, which seem to capture the meaning of the often vague ways that these terms are used in the literature. Simple finite dimensional examples are given showing that weak values of an observable are not unique, and in fact arbitrary weak values can be obtained by appropriate weak measurements. This implies that a "weak value" of an observable A, *by itself*, can furnish no unambiguous information about A; any information in a weak value is inextricably connected with the particular measurement procedure used to obtain that weak value. Moreover, arbitrary weak values can be obtained using a "meter space" of dimension as small as 2. A "Remarks" section questions the utility of "weak measurement".

DOI
10.48550/arxiv.0909.0295

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Precise definitions of "weak [quantum] measurements" and "weak value" [of a quantum observable] are offered, which seem to capture the meaning of the often vague ways that these terms are used in the literature. Simple finite dimensional examples are given showing that weak values of an observable are not unique, and in fact arbitrary weak values can be obtained by appropriate weak measurements. This implies that a "weak value" of an observable A, by itself, can furnish no unambiguous information about A; any information in a weak value is inextricably connected with the particular measurement procedure used to obtain that weak value. Moreover, arbitrary weak values can be obtained using a "meter space" of dimension as small as 2. A "Remarks" section questions the utility of "weak measurement".
A precise definition of "weak [quantum] measurements" and "weak value" (of a quantum observable) is offered, and simple finite dimensional examples are given showing that weak values are not unique and therefore probably do not correspond to any physical attribute of the system being "weakly" measured, contrary to impressions given by most of the literature on weak measurements. A possible mathematical error in the seminal paper introducing "weak values" is explicitly identified. A mathematically rigorous argument obtains results similar to, and more general than, the main result of that paper and concludes that even in the infinite-dimensional context of that paper, weak values are not unique. This implies that the "usual" formula for weak values is not universal, but can apply only to specific physical situations. The paper is written in a more pedagogical and informal style than is usual in the research literature in the hope that it might serve as an introduction to weak values.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.
We generalize the concept of the weak value of a quantum observable to cover arbitrary real positive operator measures. We show that the definition is operationally meaningful in the sense that it can be understood within the quantum theory of sequential measurements. In particular, we show that the weak value can be obtained from a single measurement scheme. We then present a detailed analysis of the recent experiment [J. S. Lundeen et al., Nature (London) 474, 188 (2011)] concerning the reconstruction of the state of a photon using weak measurements. We compare their method with the reconstruction method through informationally complete phase space measurements. In particular, we show that unlike with phase space measurements, the reconstruction of a completely unknown state is not always possible using the method of weak measurements.
'Weak measurements'—involving a weak unitary interaction between a quantum system and a meter followed by a projective measurement—are investigated when the system has a non-Hermitian Hamiltonian. We show in particular how the standard definition of the 'weak value' of an observable must be modified. These studies are undertaken in the context of bound-state scattering theory, a non-Hermitian formalism for which the involved Hilbert spaces are unambiguously defined and the metric operators can be explicitly computed. Numerical examples are given for a model system. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
Since weak measurements are known to produce measurement values that can be much larger than the maximal eigenvalues of the measured observable, it is an interesting question how this enhancement of the measurement signal relates to the sensitivity of quantum measurements as investigated in the field of quantum metrology. In this presentation, it is pointed out that the estimation of a small interaction parameter using weak measurements actually corresponds to standard quantum metrology, where the logarithmic derivatives of the final measurement probabilities are proportional to the corresponding weak values. The analysis of the general weak measurement formalism then shows that extreme weak values do not improve the sensitivity. Instead, all final measurements with real weak values have the same sensitivity as a final measurement of the eigenvalues. This result supports the view that real weak values can be interpreted as precise, zero uncertainty estimates of a quantum observable, despite their deviation from the eigenvalues of the corresponding operator.
The ability to manipulate light has allowed scientists to verify fundamental theories of physics and to develop a new generation of technologies that use photons as a primary resource. Recent developments in quantum measurement theory have offered new alternatives to approach some of the most remarkable problems in quantum physics. Consequently, the principles of quantum mechanics have been exploited in the development of quantum technologies such as optical metrology, quantum communication, and quantum information. In recent years, weak measurements and two of its most remarkable variants: weak value amplification and direct measurement, have been developed and are considered important resources for quantum applications. In this paper, we discuss weak measurements and some significant applications of weak values. We elaborate on how distinct forms of weak values are used to observe and amplify small effects or to directly measure the quantum wave function of photons, a crucial task for schemes for quantum communication and quantum information. We also review some of the most recent methods for weak value amplification and direct measurement that our group has developed.