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Metrologia

Theory of Quantum Mechanical Measurements

Günther Ludwig

Published Jan 1, 1986• 9 citations
Publisher

Abstract

No measurement is possible without any theory. With respect to a theory under consideration one has to distinguish between direct and indirect measurements. Direct measurements are defined by pre-theories, i.e. without the theory under consideration. The direct measurements for quantum theory are given by measurable processes on macroscopic devices. It is demonstrated how indirect measurements in quantum mechanics can be reduced to direct measurements. A theory of the measuring device is in this context not possible by quantum mechanics alone, but by a more comprehensive theory of macrosystems.

DOI
10.1088/0026-1394/22/3/002

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No measurement is possible without any theory. With respect to a theory under consideration one has to distinguish between direct and indirect measurements. Direct measurements are defined by pre-theories, i.e. without the theory under consideration. The direct measurements for quantum theory are given by measurable processes on macroscopic devices. It is demonstrated how indirect measurements in quantum mechanics can be reduced to direct measurements. A theory of the measuring device is in this context not possible by quantum mechanics alone, but by a more comprehensive theory of macrosystems.
This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr) of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.
The two main types of quantum measurements von Neumann's postulate of the reduction of the wave function answers the question of what happens to the quantum object during the measurement. However, the reduction postulate does not answer the question of how the measuring device must be designed to realize the desired measurement. To answer this second question, one must understand the connection between the measuring device as an ordinary physical system, and the nature of the desired measurement (the quantity to be measured, the desired precision, and the choice of how the measurement is to perturb the quantum object). The Schrödinger equation cannot tell us the connection between the design of the measuring device and the nature of the measurement, because the Schrödinger equation neither governs nor describes the process of measurement. More specifically: The evolution of the wave function as described by the Schrödinger equation has two key features: it is a reversible evolution (from the final state in principle one can always evolve back to the initial one), and it is a deterministic evolution (the final state is determined uniquely by the initial one). By contrast, the reduction of the wave function in a measurement is irreversible (once the measurement is finished and the information has been extracted from it, it is impossible to return to the pre-measurement state), and it is nondeterministic (one cannot predict, before the measurement, the post-measurement state of the measured object).
The aim of this paper is to consider the consequences of an information-theoretic interpretation of quantum mechanics for the measurement problem. The motivating idea of the interpretation is that the relation between quantum mechanics and the structure of information is analogous to the relation between special relativity and the structure of space-time. Insofar as quantum mechanics deals with a class of probabilistic correlations that includes correlations structurally different from classical correlations, the theory is about the structure of information: the possibilities for representing, manipulating, and communicating information in a genuinely indeterministic quantum world in which measurement outcomes are intrinsically random are different than we thought. Part of the measurement problem is deflated as a pseudo-problem on this view, and the theory has the resources to deal with the remaining part, given certain idealizations in the treatment of macrosystems.
The standard theory of measurements in quantum mechanics is reviewed with special emphasis on the conceptual and epistemological implications. It is concluded that the standard theory remains the only one which is compatible with present quantum mechanics. Hence, if one wants to avoid the conclusion that quantum mechanics only gives probability connections between subsequent observations, the quantum-mechanical equations would have to be modified. Particular attention is paid to the case that the measuring apparatus is macroscopic and its state vector not accurately known before the measurement.