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arXiv:1509.07198

Bayesian Interpretation of Weak Values

Akio Hosoya

Published Sep 24, 2015• 0 citations
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Abstract

The real part of the weak value is identified as the conditional Bayes probability through the quantum analog of the Bayes relation. We present an explicit protocol to get the the weak values in a simple Mach-Zehnder interferometer model and derive the formulae for the weak values in terms of the experimental data consisting of the positions and momenta of detected photons on the basis of the quantum Bayes relation. The formula gives a way of tomography of the initial state almost without disturbing it in the weak coupling limit.

DOI
10.48550/arxiv.1509.07198

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The real part of the weak value is identified as the conditional Bayes probability through the quantum analog of the Bayes relation. We present an explicit protocol to get the the weak values in a simple Mach-Zehnder interferometer model and derive the formulae for the weak values in terms of the experimental data consisting of the positions and momenta of detected photons on the basis of the quantum Bayes relation. The formula gives a way of tomography of the initial state almost without disturbing it in the weak coupling limit.
Weak measurements of photon position can be used to obtain direct experimental evidence of the wavefunction of a photon between generation and ultimate detection. Significantly, these measurement results can also be understood as complex valued conditional probabilities of intermediate photon positions. It is therefore possible to interpret the quantum state as a complex valued probability distribution from which measurement probabilities can be derived according to Bayesian rules. The conventional measurement probabilities derived from squares of the wavefunction then describes the effects of measurement back-action, which originate from a non-classical relation between dynamics and statistics that is characteristic of quantum mechanics. It is pointed out that this relation can be used to derive the complete Hilbert space formalism directly from complex probabilities, without the axiomatic introduction of quantum states or operators.
Throughout the "creation" of Quantum Mechanics, a theory has been built that is based on a single way of measuring quantum systems. These measurements have always taken place so that the act of measuring disturbs the system to the point of collapsing it, and then it acquires all the properties of a single state. However, in the late 1980s, a new way of measuring a quantum system emerged, called the Weak Measurement. With it, we were able to deal with experiments with a new approach, where the configuration of the measurement is changed by a factor that disturbs the system very little. With the way of measuring before this new concept, it would be impossible to detect small changes in a quantum system, but now, the weak measurement reaches new horizons. We developed a form of weak measurement using a general coupling in terms of the superposition of an initial state and a final state, in the case of a qubit. With that, it was possible to see in more detail how the process of this way of measuring occurs, where we will see how the coupling between the two states must be done to carry out such an experiment. We have also developed a weak measurement in a quantum system using a Mach-Zehnder interferometer for such a study. In it, with the help of a value that will determine how weak the weak measure is, we can carry out a measure in which we do not need to disturb a system to obtain information.
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.
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.