The concept of objectification Weak and strong objectification Let S be a quantum system that is prepared in a pure state φ and A a discrete nondegenerate observable with eigenvalues a i and eigenstates φ a i . If the preparation φ is not an eigenstate of A , then quantum mechanics does not provide any information about the value of the observable A . The pair 〈φ, A 〉 merely defines a probability distribution p (φ, a i ), the experimental meaning of which is given by the statistical interpretation of quantum mechanics: the real positive number p (φ a i ) is the probability of obtaining the result a i after a measurement process of the observable A . This means that if one were to perform a large series of N measurements of this kind, then the relative frequency f N (φ, a i ) of the result a i would approach for almost all test series the probability p (φ, a i ), for N → ∞ (cf. chapter 3). One could, however, in addition to these well-established results tentatively assume that a certain value a i or even an eigenstate φ a i of A pertains objectively to the system S but that this value or state is subjectively unknown to the observer, who knows only the probability p (φ, a i ) and hence the distribution of possible measurement outcomes. The probability would then express the subjective ignorance (or knowledge) about an objectively decided value or eigenstate of A . The hypothetical attribution of a certain value or eigenstate of A to the system S will be called objectification .
The concept of objectification Weak and strong objectification Let S be a quantum system that is prepared in a pure state φ and A a discrete nondegenerate observable with eigenvalues a i and eigenstates φ a i . If the preparation φ is not an eigenstate of A , then quantum mechanics does not provide any information about the value of the observable A . The pair 〈φ, A 〉 merely defines a probability distribution p (φ, a i ), the experimental meaning of which is given by the statistical interpretation of quantum mechanics: the real positive number p (φ a i ) is the probability of obtaining the result a i after a measurement process of the observable A . This means that if one were to perform a large series of N measurements of this kind, then the relative frequency f N (φ, a i ) of the result a i would approach for almost all test series the probability p (φ, a i ), for N → ∞ (cf. chapter 3). One could, however, in addition to these well-established results tentatively assume that a certain value a i or even an eigenstate φ a i of A pertains objectively to the system S but that this value or state is subjectively unknown to the observer, who knows only the probability p (φ, a i ) and hence the distribution of possible measurement outcomes. The probability would then express the subjective ignorance (or knowledge) about an objectively decided value or eigenstate of A . The hypothetical attribution of a certain value or eigenstate of A to the system S will be called objectification .
The concept of a quantum state represents one of the most fundamental pillars of the paradigm of quantum theory.'3 Contrary to its mathematical elegance and convenience in calculations, the physical interpretation of a quantum state is not so transparent. The problem is that the quantum state (described either by a state vector, or density operator or a phase-space probability density distribution) does not have a well defined objective status, i.e. a state vector is not an objective property of a particle. According to Peres (see,1 p. 374): "There is no physical evidence whatsoever that every physical system has at every instant a well defined state. .. In strict interpretation of quantum theory these mathematical symbols [i.e., state vectorsi represent statistical information enabling us to compute the probabilities of occurrence of specific events." Once this point of view is adopted then it becomes clear that any "measurement" or a reconstruction of a density operator (or its mathematical equivalent) can be understood exclusively as an expression of our knowledge about the quantum mechanical state based on a certain set of measured data. To be more specific, any quantum-mechanical reconstruction scheme is nothing more than an a posteriori estimation of the density operator of a quantum-mechanical (microscopic) system based on data obtained with the help of a macroscopic measurement apparatus.3 The quality of the reconstruction depends on the "quality" of the measured data and the efficiency of the reconstruction procedure with the help of which the data analysis is performed. In particular, we can specify three different situations. Firstly, when all system observables are precisely measured. In this case the complete reconstruction of an initially unknown state can be performed (we will call this the reconstruction on the complete observation level) . Secondly, when just part of the system observables is precisely measured then one cannot perform a complete reconstruction of the measured state. Nevertheless, the reconstructed density operator still uniquely determines mean values of the measured observables (we will denote this scheme as reconstruction on incomplete observation levels) . Finally, when measurement does not provide us with sufficient information to specify the exact mean values (or probability distributions) but only the frequencies of appearances of eigenstates of the measured observables, then one can perform an estimation (e.g. reconstruction based on quantum Bayesian inference) which is the "best" with respect to the given measured data and the a priori knowledge about the state of the measured system.
The problem of objectification of measurement is investigated in orthodox quantum mechanics, that is assuming the validity of the Schrodinger equation (and the other postulates) for the composite object-plus-measuring-instrument system. It is argued that by giving physical meaning to basic entities, the positivist and the realist positions must be explicitly distinguished. It is shown that the objectification problem is satisfactorily solved within quantum mechanics for the positivist, but not for the realist. For the latter, a partially satisfactory solution is obtained by assuming that the quantum mechanical description of individual systems is incomplete and, by assuming further, that the pointer observable has definite values prior to any measurement in any quantum state (a so-called beable).
The subject–object distinction is indeed at the very root of the unease that many people still feel in connection with quantum mechanics. Some such distinction is dictated by the postulates of the theory, but exactly where or when to make it is not prescribed. Thus in the classic treatise of Dirac we learn the fundamental propositions: … any result of a measurement of a real dynamical variable is one of its eigenvalues …, … if the measurement of the observable ξ for the system in the state corresponding to | x 〈 is made a large number of times, the average of all the results obtained will be 〈 x |ξ| x 〉 …, … a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured …. So the theory is fundamentally about the results of ‘measurements’, and therefore presupposes in addition to the ‘system’ (or object) a ‘measurer’ (or subject). Now must this subject include a person? Or was there already some such subject–object distinction before the appearance of life in the universe? Were some of the natural processes then occurring, or occurring now in distant places, to be identified as ‘measurements’ and subjected to jumps rather than to the Schrödinger equation? Is ‘measurement’ something that occurs all at once? Are the jumps instantaneous? And so on.
The main theme of this book is the idea that quantum mechanics is valid not only for microscopic objects but also for the macroscopic apparatus used for quantum mechanical measurements. The author demonstrates the intimate relations that exist between quantum mechanics and its interpretation which are induced by the quantum mechanical measurement process. Consequently, the book is concerned both with the philosophical, metatheoretical problems of interpretations and with the more formal problems of quantum object theory. The consequences of this approach turn out to be partly very promising and partly rather disappointing. On the one hand, it is possible to give a rigorous justification of some important parts of interpretation, such as probability, by means of object theory. On the other hand, the problem of the objectification of measurement results leads to inconsistencies which cannot be resolved in an obvious way. This open problem has far-reaching consequences for the possibility of recognising an objective reality in physics.
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.
Given the controversial status of the various interpretations of non-relativistic quantum mechanics (QM for short), it may seem crazy to use the philosophy of quantum physics to try to learn some lessons about the prevailing problems in the literature on dispositions and causal powers. However, such an attempt seems worthwhile for at least three reasons. The first is more internal to the philosophy of QM. Apart from a seminal paper by Robert Clifton and Constantine Pagonis (1995), who investigated the issue of dispositional properties in Bohmian mechanics, very little has been done to shed This neglect seems particularly surprising, especially if one considers that some of the founding fathers of quantum physics, notably Werner Heisenberg, have made direct reference to Aristotelian potentiae to refer to and interpret the elusive nature of the atomic and subatomic world ‘in se’, that is, before measurement: ‘Such a probability function [i.e. the statistical algorithm of quantum theory] combines objective and subjective elements. It contains statements on possibilities, or better tendencies (“potentiae” in Aristotelian philosophy), and such statements are completely objective, they don’t depend on any observer the passage from the “possible” to the Some years before, Henry Margenau3 had already characterized the properties of quantum systems as merely latent, in contrast to the properties intervening in the description of classical systems, which are always definite even when they are merely dispositional. Similarly, Karl Popper4 and Nicholas Maxwell5 have defended some sort of a propensity interpretation of probability, and a view of quantum reality essentially characterized by irreducibly probabilistic propensities. Influenced by this tradition, Michael Redhead’s influential textbook on the philosophy of QM distinguishes among three different interpretations of the theory, the second of which presupposes real propensities and potentialities and attributes to measurements the function of ‘converting latent values in possessed values’.6 Clearly, for this second interpretation to make less than an instrumentalistic sense, one needs to construe potentialities and dispositions as real properties of systems. Consequently, it seems important to try to get a firmer grasp of the sense in which QM might need irreducible dispositions to shed light on the nature of the properties of quantum systems before measurements, in particular when the state of such systems is not an eigenstate of the relevant physical observable.