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arXiv:1409.3538 · Physical Review A

Disturbance in weak measurements and the difference between quantum and classical weak values

Asger C. Ipsen

Published Jun 16, 2015• 19 citations
arXivHTMLPDF

Abstract

The role of measurement-induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three different notions and get three different results: (1) For a ``strong'' definition of disturbance, we find that weak measurements are disturbing. (2) For a weaker definition we find that a general class of weak measurements is nondisturbing, but that one gets weak values which depend on the measurement process. (3) Finally, with respect to an operational definition of the ``degree of disturbance,'' we find that the AAV weak measurements are the least disturbing, but that the disturbance is always nonzero.

DOI
10.1103/physreva.91.062120

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The role of measurement-induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three different notions and get three different results: (1) For a strong'' definition of disturbance, we find that weak measurements are disturbing. (2) For a weaker definition we find that a general class of weak measurements is nondisturbing, but that one gets weak values which depend on the measurement process. (3) Finally, with respect to an operational definition of the degree of disturbance,'' we find that the AAV weak measurements are the least disturbing, but that the disturbance is always nonzero.
Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.
A recent Letter in Physical Review Letters, "Contextual Values of Observables in Quantum Measurements", by J. Dressel, S. Agarwal, and A. N. Jordan (abbreviated DAJ below), introduces the concept of "contextual values" and claims that they lead to "a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit". However, they do not define "minimal disturbance limit". The present paper is in part the saga of my search for a definition of "minimal disturbance limit" under which this claim could be proved. The search finally ended in what is probably a definitive counterexample to the claim.
Weak values arise experimentally as conditioned averages of weak (noisy) observable measurements that minimally disturb an initial quantum state, and also as dynamical variables for reduced quantum state evolution even in the absence of measurement. These averages can exceed the eigenvalue range of the observable ostensibly being estimated, which has prompted considerable debate regarding their interpretation. Classical conditioned averages of noisy signals only show such anomalies if the quantity being measured is also disturbed prior to conditioning. This fact has recently been rediscovered, along with the question whether anomalous weak values are merely classical disturbance effects. Here we carefully review the role of the weak value as both a conditioned observable estimation and a dynamical variable, and clarify why classical disturbance models will be insufficient to explain the weak value unless they can also simulate other quantum interference phenomena.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
The main goal of this thesis is to introdude to readers the concept of weak measure- ments in quantum mechanics. Firstly a model of indirect measuremnts is established as an alternative to Von Neumann measurement postulate. This model leads to a possible generalization in a form of weak measurements. Then the concept of postselection is introduced and involved into the issue of weak measurements which leads to the weak values of physical quantities. All these theoretical concepts are presented using simple model examples. Finally, a discussion in provided on the possible interpretation and of weak values and their experimental verification.
Weak values are the fundamental values for observables in a pre- and postselected system. Weak values are typically measured by weak measurement, in which weak values appear in the change of not the pre- and postselected system but the probe system. This indirect characteristic of weak measurement obscures the meaning of weak values for the pre- and postselected system, in contrast to conventional physical quantities, which have a clear operational meaning. In this study, we operationally formulate weak values as the sensitivity of postselection probability amplitude to small transformation in a pre- and postselected system. This formulation of weak values, which is free from the concept of probe shift assumed in weak measurement, gives a direct interpretation of strange weak values for the pre- and postselected system. Furthermore, this formulation can simplify weak-value measurement experiments because the associated measurements appearing in the modulus change of the postselection probability amplitude do not require external degrees of freedom.