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Progress of Theoretical Physics

Probabilistic Interpretation and the Quantum Theory of Measurement

Ziro Maki

Published Jan 1, 1988• 9 citations
Publisher

Abstract

The logical foundation of the probabilistic intrepretation of quantum-mechanical states is re-examined in view of Fukuda's new theory of measurement. We suggest that the probabilistic interpretation could be viewed as a natural consequence of the reduction of states upon meaurement, rather than an a priori ansatz contained in the “external observation” framework of quantum mechanics.

DOI
10.1143/ptp.79.313

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The logical foundation of the probabilistic intrepretation of quantum-mechanical states is re-examined in view of Fukuda's new theory of measurement. We suggest that the probabilistic interpretation could be viewed as a natural consequence of the reduction of states upon meaurement, rather than an a priori ansatz contained in the “external observation” framework of quantum mechanics.
A general theory of quantum-mechanical measurement is presented. It is shown that some general conditions should be satisfied for any apparatus to carry out the ideal measurement, and the reduction of state is an inevitable consequence of the action of such an apparatus. The statistical operator (in weak sense) is defined uniquely for ensembles prepared with respect to the measured observables, without recourse to the expectation value hypothesis. The probabilistic interpretation (the Born ansatz) comes out as a natural explanation of physical contents of the statistical operator after the measurement, rather than an axiomatic presumption of quantum mechanics.
The problem of measurement in quantum mechanics is reanalyzed within a general, strictly probabilistic framework (without reduction postulate). Based on a novel comprehensive definition of measurement the natural emergence of objective events is demonstrated and their formal representation within quantum mechanics is obtained. In order to be objective an event is required to be observable or readable in at least two independent, mutually non-interfering ways with necessarily agreeing results. Consistency in spite of unrestricted validity of reversibility of the evolution or the superposition principle is demonstrated and the role played by state reduction, in a properly defined restricted sense, is discussed. Some general consequences are pointed out.
Where is the problem? by J.M. Levy-Leblond Experiment and quantum measurement theory by A. Zeilinger A consistent interpretation of quantum mechanics. The measurement process in the individual interpretation of quantum mechanics by H. Primas Principle of stationarity in the action functional and the theory of measurement by R. Fukuda Quantum (statistical) mechanics, measurement and information by R. Balian State vector collapse as a classical statistical effect of measurement by M. Cini and M. Serva Consecutive quantum measurements by A. Peres No-collapse versions of quantum mechanics by Y. Ben-Dov An attempt to understand the many-worlds interpretation of quantum theory by E.J. Squires Uncertainty and measurement by P.T. Landsberg Author Index
Endeavoring to formulate an exhaustive solution to the measurement problem in view of the theory of decoherence leads to a better understanding of the status of the collapse and of the emergence of classicality, thanks to a precise definition of the measurement and some new vocabulary to speak about quantum mechanics. Considering the latter as a probabilistic theory all along allows us to avoid the usual probability problem of the many-worlds interpretations. A thorough verification of the consistency of quantum mechanics at all scales is proposed, as well as a discussion of what can be deemed an observer.
This paper explores a thought experiment: what would physics look like if quantum mechanics had not originated in the early 20th century from the exploration of the physical properties of microscopic particles (such as blackbody radiation and the particle and wave nature of electron orbits), but had instead been established first as a dynamics and probability theory about "physical measurement" itself? Through extrapolation from this historical turning point, this paper reconstructs a theoretical framework centered on "measurement dynamics," not on "particles," "wave functions," or "quantum fields" as fundamental ontology, but on a trinity of "system-measuring instrument-observer." Research shows that under this paradigm, the "measurement problem" is no longer a philosophical dilemma added after the theory's establishment, but rather the logical starting point for its construction; wave functions and quantum fields are relegated to derivative mathematical tools for calculating measurement probability distributions. This paper elaborates on the dynamics of measurement operators, the law of information evolution, and a systematic critique of counterfactual definiteness under this paradigm, providing a new and unified perspective for re-examining the fundamental interpretation of modern quantum mechanics.
This book is devoted to aspects of the foundations of quantum mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of positive operator-valued measures. During the past years there has been substantial progress in this direction, stimulated to a great extent by new applications such as Quantum Optics, Quantum Communication and high-precision experiments. The questions of statistical interpretation, quantum symmetries, theory of canonical commutation re