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Physica Scripta

The counterfactual process in weak values

Yutaka Shikano

Published Nov 1, 2012• 2 citations
Publisher

Abstract

From the viewpoint of mathematics, quantum-mechanical probability seems to be controversial. To resolve this problem, we have reconstructed the Born rule using the weak value. We also discuss its meaning.
DOI
10.1088/0031-8949/2012/t151/014015

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In the previous meeting I demonstrated that the contextual value of an observable coincides with the weak value of Aharonov and his colleagues on the basis of consistency of quantum mechanics with Kolmogorov's measure theory of probability and that this consistency also leads to Born's rule as the unique probability measure of events to be discovered by the post-selection. I would like to point out in the meeting of this year that the error and disturbance defined by Ozawa in his work on the Heisenberg uncertainty relation can be interpreted in terms of the weak value. This observation further strengthens the viewpoint that the weak value is the "true value" of an observable before measurement.
The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable A^\hat {A}A^ with post-selection 〈F| may be out of the eigenvalue range of A^\hat {A}A^ . This is because the weak value of A^\hat {A}A^ with the post-selection is, in general, not the expectation value of A^\hat {A}A^ , but the expectation value of A^∣F⟩⟨F∣\hat {A}| F\rangle \langle F|A^∣F⟩⟨F∣ boosted by the post-selection.
In this work we revisit the important and controversial concept of quantum weak values, aiming to provide a simplified understanding to its associated physics and the origin of anomaly. Taking the Stern-Gerlach setup as a working system, we base our analysis on an exact treatment in terms of quantum Bayesian approach. We also make particular connection with a very recent work, where the anomaly of the weak values was claimed from the pure statistics in association with "disturbance" and "post-selection", rather than the unique quantum nature. Our analysis resolves the related controversies through a clear and quantitative way.
Abstract In ordinary situations involving a small part of the universe, Born's rule seems to work well for calculating probabilities of observations in quantum theory. However, there are a number of reasons for believing that it is not adequate for many cosmological purposes. Here a number of possible generalizations of Born's rule are discussed, explaining why they are consistent with the present statistical support for Born's rule in ordinary situations but can help solve various cosmological problems.
The Aharonov-Bergmann-Lebowitz rule assigns probabilities to quantum measurement results at time t on the condition that the system is prepared in a given way at t_1 < t and found in a given state at t_2 > t. The question whether the rule can also be applied counterfactually to the case where no measurement is performed at the intermediate time t has recently been the subject of controversy. I argue that the counterfactual meaning may be understood in terms of the true value of an observable at t. Such a value cannot be empirically determined for, by stipulation, the measurement that would yield it is not performed. Nevertheless, it may or may not be well-defined depending on one's proposed interpretation of quantum mechanics. Various examples are discussed illustrating what can be asserted at the intermediate time without running into contradictions.