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arXiv:0905.3810 · Physical Review A

Weak values, “negative probability,” and the uncertainty principle

D. Sokolovski

Published Oct 25, 2007• 45 citations
arXivHTMLPDF

Abstract

A quantum transition can be seen as a result of interference between various pathways (e.g., Feynman paths), which can be labeled by a variable $f$. An attempt to determine the value of $f$ without destroying the coherence between the pathways produces a weak value of $\overline{f}$. We show $\overline{f}$ to be an average obtained with an amplitude distribution which can, in general, take negative values, which, in accordance with the uncertainty principle, need not contain information about the actual range of $f$ which contributes to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to a misinterpretation of the weak-measurement results. We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double-slit diffraction experiment, weak von Neumann and von Neumann--like measurements, traversal time for an elastic collision, phase time, and local angular momentum.

DOI
10.1103/physreva.76.042125

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A quantum transition can be seen as a result of interference between various pathways (e.g., Feynman paths), which can be labeled by a variable fff. An attempt to determine the value of fff without destroying the coherence between the pathways produces a weak value of f‾\overline{f}f​. We show f‾\overline{f}f​ to be an average obtained with an amplitude distribution which can, in general, take negative values, which, in accordance with the uncertainty principle, need not contain information about the actual range of fff which contributes to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to a misinterpretation of the weak-measurement results. We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double-slit diffraction experiment, weak von Neumann and von Neumann--like measurements, traversal time for an elastic collision, phase time, and local angular momentum.
The outcomes of a series of measurements, made on a quantum system, form a sequence of random events which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become "real", and the mean values of a quantity (a functional) is directly related to the frequencies with which the paths are travelled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one to distinguish between interfering alternatives, while leaving the interference between them intact.
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
In quantum mechanics, it is well known that counterfactual arguments sometimes lead to paradoxes, which implies that our classical intuition is not always reliable.However, it has been pointed out by several authors that the weak value, which was introduced by Aharonov et al. in 1988, may be useful to comprehend quantum mechanics intuitively.Another extraordinary aspect of the weak value is its amplification, which has been exploited for precision measurement.Nevertheless, the physical meaning of the weak value is still obscure due to its complexity of the value and the very fact of the amplification.In this thesis, by considering various situations of quantum interference, we attempt to provide a possible interpretation of the weak value.We find, on one hand, that the imaginary part of the weak value is relevant to the wave nature.On the other hand, in the double-slit experiment the particle nature can be seen in the real part of the position weak value.This property for the real part does not hold for more general cases such as the multiple (more than two) slit experiment.The position weak value, nevertheless, can be interpreted as the average of the classical trajectories based on a complex probability.We also investigate the possibility of knowing which path the particle takes in the multiple-slit experiment without destroying the interference fringes.We show that, because of the statistical nature of the weak value, obtaining the which-path information and the interference pattern is compatible with complementarity.Our study suggests that the weak value is a reasonable physical quantity.
Using the quantum transition path time probability distribution we show that time averaging of weak values leads to unexpected results. We prove a weak value time-energy uncertainty principle and time-energy commutation relation. We also find that time averaging allows one to predict in advance the momentum of a particle at a post selected point in space with accuracy greater than the limit of ℏ/2 as dictated by the uncertainty principle. This comes at a cost—it is not possible at the same time to predict when the particle will arrive at the post selected point. A specific example is provided for one dimensional scattering from a square barrier.
Weak measurements can be seen as an attempt at answering the Which way? question without destroying interference between the pathways involved. Unusual mean values obtained in such measurements represent the response of a quantum system to this forbidden question, in which the true composition of virtual pathways is hidden from the observer. Such values indicate a failure of a measurement where the uncertainty principle says it must fail, rather than provide an additional insight into physical reality. Quanta 2013; 2: 50–57.
Weak measurements can be seen as an attemptat answering the Which way? question with-out destroying interference between the path-ways involved. Unusual mean values obtained in such measurements represent the response of a quantum system to this forbidden question, in which the true composition of virtual pathways is hidden from the observer. Such values indicate a failure of a measure-ment where the uncertainty principle says it must fail, rather than provide an additional insight into physi-cal reality. Quanta 2013; 2: 50–57. 1