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Kluwer Academic Publishers eBooks

Dual Classical and Quantum Descriptions of a Measuring Apparatus

Ori Hay, Asher Peres

Published Dec 19, 2005• 0 citations
Publisher

Abstract

The measuring process is the interface of classical and quantum phenomena. A measuring apparatus must be described by quantum mechanics while it interacts with the quantum system under observation. Then, it has to be be given a classical description so that the result of the measurement appears as objective reality. Alternatively, that apparatus may be treated at all times by quantum mechanics. In that case, it never achieves the “objective” status, but it can be measured by a second apparatus which has such a dual description. The question is whether these two different descriptions of the same apparatus are mutually consistent. A detailed analysis leads to the conclusion that not every physical system is a legitimate measuring apparatus. The quantum measurement problem has a satisfactory solution only if the measuring apparatus is properly designed.

DOI
10.1007/0-306-47097-7_16

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The measuring process is the interface of classical and quantum phenomena. A measuring apparatus must be described by quantum mechanics while it interacts with the quantum system under observation. Then, it has to be be given a classical description so that the result of the measurement appears as objective reality. Alternatively, that apparatus may be treated at all times by quantum mechanics. In that case, it never achieves the “objective” status, but it can be measured by a second apparatus which has such a dual description. The question is whether these two different descriptions of the same apparatus are mutually consistent. A detailed analysis leads to the conclusion that not every physical system is a legitimate measuring apparatus. The quantum measurement problem has a satisfactory solution only if the measuring apparatus is properly designed.
A measuring apparatus is described by quantum mechanics while it interacts with the quantum system under observation and then it must be given a classical description so that the result of the measurement appears as objective reality. Alternatively, the apparatus may always be treated by quantum mechanics and be measured by a second apparatus, which has such a dual description. This article examines whether these two different descriptions are mutually consistent. It is shown that if the dynamical variable used in the first apparatus is represented by an operator of the Weyl-Wigner type (for example, if it is a linear coordinate), then the conversion from quantum to classical terminology does not affect the final result. However, if the first apparatus encodes the measurement in a different type of operator (e.g., the phase operator), the two methods of calculation may give different results.
The quantum measurement problem has to do with the compatibility of two different prescriptions for the end state of a quantum measurement process: (1) a superposition representing an entangled state of object system and apparatus; (2) a mixture of correlated states of object system and apparatus. The first prescription is a consequence of the linearity of time evolution in quantum mechanics; the second one (corresponding to what is usually called the 'collapse of the wave function') is a basic feature of the axiomatic structure of the theory. This paper proves that the two prescriptions are exactly equivalent whenever information about the results of the measurement persists, i.e. whenever there is some kind of record or trace of the results (all the measurements we are ordinarily concerned with fulfill this requirement). Experimental evidence is cited suggesting that indeed we should expect the equivalence of prescriptions (1) and (2) to hold when and only when information about measurement results persists. Finally the notion of 'element of reality' introduced by Einstein, Podolsky and Rosen is discussed in the context of this description of the measurement process, and a brief remark is offered about the 'classical' nature of the world we experience.
Abstract The measurement problem in quantum mechanics still appears to be an unresolved issue. Here we present a new quantum theory of measurement that overcomes many of the difficulties previously found. It is based on a consistent use of the linear superposition principle and distinguishes two aspects: recording and observation. A recording elicits the full interaction of the object quantum system with the quantum measuring apparatus. No wavefunction collapse is introduced. Statistics may appear at the observation of the recording only and depends on filtering processes. The theory presented here uses the existing mathematical structure of quantum mechanics but requires no ad hoc measurement postulates. Well‐known paradoxical aspects in standard quantum mechanics, for instance, wave–particle duality, Schrödinger's cat, and Zeno effects do not appear in the current formulation. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
We introduce an unconventional approach to the measurement problem in quantum mechanics: we treat the apparatus as a classical system belonging to the macroworld. To describe the quantum measurement process we must couple the classical apparatus to the quantum system. In this paper we explain how this is to be done: we embed the classical apparatus into a larger quantum-mechanical structure, making use of a superselection principle. The apparatus can now be coupled to the quantum system in a straightforward manner. We discuss what constraints should be placed on the coupling so that an interpretation of the interaction as a measurement results. We require that unambiguous information of the values of a quantum observable should be transferred to the variables of the apparatus. We also require that the apparatus should retain its classical identity. This latter requirement is formulated as a principle of integrity, in both weak and strong forms.
The so-called measurement problem of quantum theory (QT) is still lacking a satisfactory, or at least widely agreed upon, solution. A number of theories, known as interpretations of quantum theory, have been proposed and found differing acceptance among physicists. Most of the proposed theories try to explain what happens during a QT measurement using a modification of the declarative equations that define the possible results of a measurement of QT observables or by making assumptions outside the scope of falsifiable physics. This paper proposes a solution to the QT measurement problem in terms of a model of the process for the evolution of two QT systems that interact in a way that represents a measurement. The model assumes that the interactions between the measured QT object and the measurement apparatus are ''normal" interactions which adhere to the laws of quantum field theory.
Measurement in quantum mechanics is generally described as an irreversible process that perturbs the wavefunction describing a quantum system. In this work we establish a formal connection between the measurement description within the Copenhagen interpretation (i.e., through the collapse of the wavefunction) compared versus a picture in which the system and the measurement apparatus are considered as a whole. We first consider a projective measurement. In this limiting case, the natural requirements of consistency and equivalence between the two pictures lead to the rigorous definition of consistent measuring apparatus: the orthonormal wavefunctions from the Schmidt decomposition of the system plus apparatus must have non-overlapping supports. This result arises from the comparison of the two pictures (otherwise hidden), and while it seems to be an obvious conclusion in the limit of projective measurements, it has some nontrivial implications as one extends its validity to the domain of weak measurements. In this respect, we argue on the existence of two alternative approaches to mathematically constructing a weak measurement protocol. While the two approaches are equivalent from the system's perspective, they do strongly differ from the apparatus point of view, and hence can be only distinguished one from each other in the picture where system and apparatus are considered as a whole. We show that only one of the two mathematical formulations of the weak measurement fulfills the consistent apparatus condition, while the combination of the two gives rise to a generalized weak measurements framework.