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arXiv:quant-ph/0402134

Consistent measurement of a quantum dynamical variable using classical apparatus

E. C. G. Sudarshan

Published Feb 18, 2004• 4 citations
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Abstract

The claim that there is an inconsistency of quantum-classical dynamics [1] is investigated. We point out that a consistent formulation of quantum and classical dynamics which can be used to describe quantum measurement processes is already available in the literature [2]. An example in which a quantum system is interacting with a classical system is worked out using this formulation.

DOI
10.48550/arxiv.quant-ph/0402134

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The claim that there is an inconsistency of quantum-classical dynamics [1] is investigated. We point out that a consistent formulation of quantum and classical dynamics which can be used to describe quantum measurement processes is already available in the literature [2]. An example in which a quantum system is interacting with a classical system is worked out using this formulation.
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 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.
MEASUREMENT INTERACTIONS In the present interpretation terms such as ‘measurement,’ ‘observation,’ ‘find,’ ‘classical,’ and ‘apparatus’ do not appear in the formulation of quantum mechanics. Yet it must of course be true that quantum mechanics can be used to predict features of the results of measurements – in particular their statistics. What is required, then, is an account in quantum mechanical terms of those physical interactions capable of being used to perform measurements of quantum dynamical variables, from which it follows that the predicted probabilities for specified behavior of certain quantum systems in these interactions coincide with what, on a more conventional interpretation, are simply postulated as the quantum mechanical probabilities for various measurement results on a system with a given quantum state. It is necessary to see how this consequence may be derived, not only in order to explicate the Born rules, but also to explain the role of quantum states within the present interpretation. The first step toward the required account is to analyze the nature of an interaction capable of being used to perform a measurement of the value of a quantum dynamical variable in a quantum system. Any such interaction will be called an M -type interaction, whether or not it is in fact used to perform a measurement. The basic characteristic of an M -type interaction is that it establishes a correlation between the initial dynamical state of one quantum system σ (the object system) and the final dynamical state of another quantum system α (the apparatus system).
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.
By considering the measurement of a quantum spin by two different models of the measuring apparatus, we try to throw some light on the nature of the apparatus that can perform quantum measurement. Our scheme is to consider a quantum spin in interaction with another quantum system playing the role of an apparatus, which in turn is dissipatively coupled to an environment. The coupling with the environment drives the density matrix of the apparatus to a diagonal form. However, measurement is performed only when the reduced density matrix of the apparatus contains correlations with the states of the spin. The two examples considered here show that this happens when the quantum apparatus has a well-defined classical limit. If the apparatus does not have a well-defined classical limit, the reduced density matrix of the apparatus is not correlated with the states of the measured system.