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Understanding weak values without new physics

Riuji Mochizuki

Published Apr 7, 2016• 0 citations
OpenAlex

Abstract

The physical meaning of weak values and measurements can be completely understood within the framework of a conventional quantum mechanical approach. We should not expect the weak value of an observable $\hat A$ with post-selection $\langle F|$ within the eigenvalue range of $\hat A$ because it is not an expectation value of $\hat A$.

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Understanding Weak Values Without New Probability Theory

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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 $\hat {A}$ with post-selection 〈F| may be out of the eigenvalue range of $\hat {A}$ . This is because the weak value of $\hat {A}$ with the post-selection is, in general, not the expectation value of $\hat {A}$ , but the expectation value of $\hat {A}| F\rangle \langle F|$ boosted by the post-selection.

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Physical Review A•Apr 4, 2022•12 citations
The physical meaning of weak values and measurements can be completely understood within the framework of a conventional quantum mechanical approach. We should not expect the weak value of an observable A^\hat AA^ with post-selection ⟨F∣\langle F|⟨F∣ within the eigenvalue range of A^\hat AA^ because it is not an expectation value of A^\hat AA^.
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.
A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.