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Mathematics 2016, 4(3), 56;

Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”

Departamento de Química-Física, Universidad del País Vasco, UPV/EHU, Leioa 48940, Bizkaia, Spain
IKERBASQUE, Basque Foundation for Science, Maria Diaz Haroko Kalea, 3, Bilbao 48013, Bizkaia, Spain
Academic Editors: Paul Busch, Takayuki Miyadera and Teiko Heinosaari
Received: 6 May 2016 / Revised: 28 July 2016 / Accepted: 23 August 2016 / Published: 15 September 2016
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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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. View Full-Text
Keywords: quantum probabilities; uncertainty principle; transition amplitudes; “weak values” quantum probabilities; uncertainty principle; transition amplitudes; “weak values”

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Sokolovski, D. Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”. Mathematics 2016, 4, 56.

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