The problem of whether a measurement interaction can leave the joint object-apparatus system in a mixture of states, in each state of which the apparatus's observable displays a definite value, is set within the most general quantum-theoretic framework for treating measurements. It is shown that the question posed by this problem admits only a negative answer. Some schemes for approximating the true object-apparatus state by means of such mixtures are examined. It is argued that such schemes constitute fundamental changes in the interpretation of quantum theory.
The problem of whether a measurement interaction can leave the joint object-apparatus system in a mixture of states, in each state of which the apparatus's observable displays a definite value, is set within the most general quantum-theoretic framework for treating measurements. It is shown that the question posed by this problem admits only a negative answer. Some schemes for approximating the true object-apparatus state by means of such mixtures are examined. It is argued that such schemes constitute fundamental changes in the interpretation of quantum theory.
The quantum mechanical measurement problem is the difficulty of dealing with the indefiniteness of the pointer observable at the conclusion of a measurement process governed by unitary quantum dynamics. There has been hope to solve this problem by eliminating idealizations from the characterization of measurement. We state and prove two ‘insolubility theorems’ that disappoint this hope. In both the initial state of the apparatus is taken to be mixed rather than pure, and the correlation of the object observable and the pointer observable is allowed to be imperfect. In the insolubility theorem for sharp observables, which is only a modest extension of previous results, the object observable is taken to be an arbitrary projection valued measure. In the insolubility theorem for unsharp observables, which is essentially new, the object observable is taken to be a positive operator valued measure. Both theorems show that the measurement problem is not the consequence of neglecting the ever-present imperfections of actual measurements.
This is the first of two papers showing that the quantum problem of measurement remains unsolved even when the initial state of the apparatus is described by a statistical operator and when the results of measurement have a small probability of being erroneous. A realistic treatment of the measurement of observables of microscopic objects (e.g., the position or the spin of an electron) by means of observables of macroscopic apparatus (e.g., the position of a spot on a photographic plate) requires the consideration of errors. The first paper considers measurement procedures of the following type: An initial eigenstate of the object observable leads to a final statistical operator of the object plus apparatus which describes a mixture of "approximate" eigenstates of the apparatus observable. It is proved that each of a large class of initial states leads to a final statistical operator which does not describe any mixture containing even one "approximate" eigenstate of the apparatus observable.
A transition would lead the system from its original superposition state to another state, in which the system and the measuring apparatus acquire definite properties. The problem of measurement is approached from the theory of environment-induced decoherence. This theoretical program, based on the study of the effects of the interaction between a quantum system, is considered as an open system and its environment. In turn, even the measurement problem has been reformulated, expanding its scope. As Schlosshauer points out, the conceptual difficulties of quantum measurement can now be concentrated around two cases: The definite outcome, which consists of the traditional question about quantum measurement. The problem of the preferred basis: since there is a theoretical ambiguity in the definition of the observable measured due to the mathematical possibility of a base change in the expression of the system state.
This is the first of two papers showing that the quantum problem of measurement remains unsolved even when the initial state of the apparatus is described by a statistical operator and when the results of measurement have a small probability of being erroneous. A realistic treatment of the measurement of observables of microscopic objects (e.g., the position or the spin of an electron) by means of observables of macroscopic apparatus (e.g., the position of a spot on a photographic plate) requires the consideration of errors. The first paper considers measurement procedures of the following type: An initial eigenstate of the object observable leads to a final statistical operator of the object plus apparatus which describes a mixture of “approximate” eigenstates of the apparatus observable. It is proved that each of a large class of initial states leads to a final statistical operator which does not describe any mixture containing even one “approximate” eigenstate of the apparatus observable. INTRODUCTION Several writers have tried to solve the quantum-mechanical problem of measurement through one or both of the following proposals: (a) describthe initial state of the measuring apparatus by a projection onto a subspace of the associated Hilbert space or by a statistical operator, thus taking into account the practical impossibility of knowing the exact quantum state of a macroscopic object; (b) recognizing that there may be some physical inaccuracy, such as a small error in the position of a pointer needle, in the final registration of the outcome of the measurement by the apparatus.
This thesis deals with the set of issues commonly known as the 'measurement problem' in quantum mechanics. The main thesis is that the problems are best understood as typically theoretical problems, in the sense that they are not problems directly concerned with the ability of the quantum theory to account for, or represent, actual measurements. This is contrary to the standard view that the quantum measurement problem is in fact about how to fit theory to experiment. I explain how I characterise a theoretical problem and argue against claims that quantum measurement theory is unrealistic or ineffective because it bears so little relation to actual measurement practice: I argue that the quantum theory's analysis of measurement need not be committed to doing for the experimenter what Henry Margenau and other critics think it should do. Its principal aim is to answer two questions. First, it tells us what properties are to be associated to quantum states; secondly, it tells us what, in the theory, a measurement must be if these properties are to emerge. I then discuss some of the specific aspects of the problem of measurement, in particular the results known as insolubility proofs of the quantum measurement problem and the characterisation of the quantum measurement interactions satisfying standard probabilistic constraints. I prove several results here, amongst them characterisations of all interactions jointly satisfying the conditions of unitarity and, first, objectification, then secondly, probability reproducibility conditions. These are the standard conditions which capture our intuitions about quantum measurement. I show how the results lead to negative consequences with respect to the interpretive questions in quantum mechanics. The discussion of these specific aspects of quantum measurements does, on the other hand, suggest a particular strategy for solving the problems. This is found in Arthur Fine's solution to the measurement problem, which is based on the idea of a selective interaction. The discussion of Fine's solution emphasises in general how simply implementing technical strategies is not sufficient to solve the measurement problem in quantum mechanics: further arguments must be given for why the strategy is appropriate, rather than just mathematically satisfactory. I claim that the arguments given by Fine are far from sufficient. The thesis concludes that, although the quantum theory of measurement is immune from Margenau's critique, and retains a theoretical autonomy, it is still plagued by numerous problems: the thesis identifies clearly what some of these problems are and considers some solutions, most of which, however, raise serious philosophical questions about the interpretation of quantum mechanics.
We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.