Instruments are necessarily of macroscopic size because our sense organs—which can be considered as instruments occupying a certain rank in the von Neumann hierarchy—are of macroscopic size. It is only natural, therefore, that physicists should have attempted to make use of this property for constructing quantum measurement theories. Moreover, one of the remarkable properties possessed by most macroscopic systems, including all instruments, but lacking in microscopic systems is irreversibility. This simple observation suggests very strongly that the phenomenon of irreversibility and the phenomenon of "wave packet reduction," which is associated in some way or other with the measurement process, should somehow be correlated. The theories in the chapter show that it is in practice always possible to attribute definite macroscopic properties to the measuring instruments. The danger that the error we thus make will ever be detected has been proved to be vanishingly small.
Instruments are necessarily of macroscopic size because our sense organs—which can be considered as instruments occupying a certain rank in the von Neumann hierarchy—are of macroscopic size. It is only natural, therefore, that physicists should have attempted to make use of this property for constructing quantum measurement theories. Moreover, one of the remarkable properties possessed by most macroscopic systems, including all instruments, but lacking in microscopic systems is irreversibility. This simple observation suggests very strongly that the phenomenon of irreversibility and the phenomenon of "wave packet reduction," which is associated in some way or other with the measurement process, should somehow be correlated. The theories in the chapter show that it is in practice always possible to attribute definite macroscopic properties to the measuring instruments. The danger that the error we thus make will ever be detected has been proved to be vanishingly small.
In a certain sense, object and apparatus were essentially on the same footing in von Neumann's measurement theory. They were systems with k and l degrees of freedom respectively (page 146). There were no constraints on the numbers k and l , which could, for example, be of the same order of magnitude. Quite possibly, it was this lack of differentiation between object and apparatus that led Wigner to assert that ‘the state of the apparatus has no classical description’ (page 137), a state of affairs that produced the infinite von Neumann chain which only ended in the observer's consciousness. In this and the following chapter we shall break with von Neumann and assume that k is small, i.e., of the order of unity, and that l is large, i.e., within a few orders of magnitude of Avogadro's number. The room for manoeuvre that this provides will allow us to break with Wigner and explore systems with states that do have classical descriptions. It will not surprise the informed reader that the room for manoeuvre created by our assumption will be filled, very substantially, by von Neumann's own work. This chapter is divided into three sections. Section 9.1 is devoted to a theorem of von Neumann on observables that commute with each other. This prepares the way to our treatment of macroscopic observables, which is based on the commuting approximations to P and Q devised by von Neumann; these results, together with their antecedents, are presented in Section 9.2. In Section 9.3, an attempt is made to resolve some of Wigner's doubts.
This chapter focuses on the notions of classical instruments and of classical measurements. It reviews the theories, based on the concept of macroscopic properties, that is, on the essential complexity of the measuring apparatus. One of the most distinctive features of quantum mechanics is the fact that it often assigns precise structures—or discrete "choices" among possible different structures—to the physical systems it describes. Hisenberg's uncertainty relations bear on the dispersions of ensembles as a whole, whereas the limitations that serve to define "classical measurements" essentially concern our possibilities of actually performing measurements in different elements of the ensemble. Thus from the uncertainty relations alone one can certainly not derive the conclusion that nonclassical measurements are impossible. The chapter considers the restricted set of observables. It is convenient to use the word "macrostate" to designate the whole set of microstates on which all the classical observables forming a definite complete set have definite values.
This chapter presents the measurement problem from the point of view of the thermal interpretation of quantum physics. Everything physicists measure is measured in a thermal environment, for which statistical thermodynamics is relevant. The thermal interpretation agrees with how one interprets measurements in thermodynamics, the macroscopic part of quantum physics, derived via statistical mechanics. By its very construction, the thermal interpretation naturally matches the classical properties of our quantum world: The thermal interpretation assigns states-and a realistic interpretation for them-to individual quantum systems, in a way that large quantum systems are naturally described by classical observables. Since measurement devices are large quantum systems, this is the key to solving the measurement problem. We discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. Since quantum physics makes many deterministic predictions, for example regarding observed spectra, but also assertions about probabilities, we distinguish deterministic and statistical measurements. To clarify the meaning of the concept of measurement, we postulate (in Section 11.1) a measurement principle that defines what it means in the thermal interpretation to measure a quantity with a specified uncertainty. This turns the measurement problem from a philosophical riddle into a scientific problem in the domain of quantum statistical mechanics, namely how the quantum dynamics correlates macroscopic readings from an instrument with properties of the state of a measured microscopic system. The essence of scientific practice is the reproducibility of measurements, discussed in Section 11.2. Unlike in traditional interpretations, single, nonreproducible observations do not count as measurements, since this would violate the reproducibility of measurements-the essence of scientific practice. As a consequence, the measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation {A} rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. This difference is most conspicuous in the interpretation of single discrete microscopic events. Except in very special circumstances, these are not reproducible. Thus, they have no scientific value in themselves and do not constitute measurement results. Scientific value is, however, in ensembles of such observations, which result in approximate measurements of q-probabilities and q-expectations. Sections 11.3 and 11.4 distinguish deterministic and statistical measurements, depending on whether a single observation is reproducible, and discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. How to measure the q-probabilities of events, described in terms of POVMs, is discussed in Section 11.5. Section 11.6 poses the measurement problem as a problem of the statistical mechanics of measurement devices, and discuss recent work by Allahverdyan, Balian, and Nieuwenhuizen on its resolution. Section 11.7 discusses the problem how, in the thermal interpretation, discrete events can arise, although qexpectations are by their very nature continuous. This is resolved by pointing to work by Breuer and Petruccione on piecewise deterministic processes (PDP). Section 11.8 sheds additional light on the origin of discrete events by an analysis of the role dissipative bistability plays in obtaining discreteness in classical physics.
Macroscopic systems being composed of atoms and fields, there is no reason why they should not obey the quantum laws. It therefore seems quite natural to consider them as quantum systems, albeit extremely complex ones. Measurement constitutes a riddle, and a great many theories were put forward as attempts to solve this riddle. The number is so considerable indeed that to try and review them all would be an almost impossible undertaking. A few essentially quantum notions, such as quantum tunneling, have been shown quite indispensable for explaining a number of experimental facts concerning some systems of macroscopic size and complexity. Small differences necessarily exist between a system as it really is and its description by means of simple statements Moreover, elaborate experimental setups can in general be conceived of that, if used as measurement devices, would enhance these small differences.
Irreversibility is often considered to characterize measurements in quantum mechanics. Fundamental problems with this characterization are addressed. First, whether a measurement is made in quantum mechanics is an arbitrary decision on the part of the experimenter concerning how the experimental circumstances are structured. Second, how is irreversibility that occurs in making a measurement explained in terms of a neurophysiological mechanism where a macroscopic measuring instrument is not required in principle to make the measurement, as in macroscopic quantum tunneling? Third, how does irreversibility characterize a negative observation where there is no physical interaction in the measurement process?
The question as to whether quantum mechanics can be applied to a macroscopic system, which has recently been raised through discussions of the problem of how to interpret the so-called “reduction of a wave packet”, is discussed. It is shown by a detailed analysis of the destruction of interference which brings about the reduction, that a supplementary assumption on the preciseness of measurements must at least be introduced in order to interpret the reduction process as an objective process, and it does not, however, necessarily imply inapplicability of quantum mechanics to a macroscopic system.