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AIP conference proceedings

Weak value as a fundamental concept and uncertainty relation

Akio Hosoya

Published Jan 1, 2012• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.4773134

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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.
We characterize a value of an observable by a 'sum rule' for generally non-commuting observables and a 'product rule' when restricted to a maximal commuting subalgebra of observables together with the requirement that the value is unity for the projection operator of the prepared state and the values are zero for the projection operators of the states which are orthogonal to the prepared state. The crucial requirement is that the expectation value and the variance of an observable should be independent of the way of measurement, i.e. the choice of the maximal commuting subalgebra of observables. We shall call the value a ' contextual value '. We show that the contextual value of an observable coincides with the weak value advocated by Aharonov and his colleagues by demanding the consistency of quantum mechanics with Kolmogorov's measure theory of probability. This also gives a derivation of Born's rule, which is one of the axioms of conventional quantum mechanics.
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.
Our investigation of the results of the neutron spin experiment by Ehhart et al. demonstrates that their results cannot be understood in accordance with common sense. For example, their results obtained with different measurement errors are equal to each other for a same neutron state. This is because their measurement values have a large systematic error due to adopting an incorrect measurement-value operator, which is the operator that gives the measurement values of the measured observable. They asserted that their results confirmed the validity of a universally valid uncertainty relation proposed by Ozawa. Therefore, their results demonstrate unexpectedly that Ozawa's uncertainty relation holds true for incorrect measurements. In addition, we have proved this fact theoretically. The measurement-value operator must be defined in order that the systematic error of measurement values predicted by the operator is zero. Such a measurement is called an accurate measurement. Two kind of uncertainty relations are derived for accurate measurements of a pre-measurement observable x^0\hat{x}_0x^0​. In addition, we have presented an uncertainty relation for accurate measurements of a post-measurement observable x^t\hat{x}_tx^t​. Heisenberg's original uncertainty relation for measurement processes should be interpreted as this relation between the resolution, that is, the measurement error of the post-measurement observable x^t\hat{x}_tx^t​ and the disturbance of an object observable y^0\hat{y}_0y^​0​.
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.
While there is a rigorously proven relationship about uncertainties intrinsic to any quantum system, often referred to as "Heisenberg's uncertainty principle," Heisenberg originally formulated his ideas in terms of a relationship between the precision of a measurement and the disturbance it must create. Although this latter relationship is not rigorously proven, it is commonly believed (and taught) as an aspect of the broader uncertainty principle. Here, we experimentally observe a violation of Heisenberg's "measurement-disturbance relationship", using weak measurements to characterize a quantum system before and after it interacts with a measurement apparatus. Our experiment implements a 2010 proposal of Lund and Wiseman to confirm a revised measurement-disturbance relationship derived by Ozawa in 2003. Its results have broad implications for the foundations of quantum mechanics and for practical issues in quantum measurement.