Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A One-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behavior of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe-particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle.
Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A One-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behavior of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe-particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle.
Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A one-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behaviors of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe–particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle. Hence, whereas the previous analysis was purely physical in nature, this one will be “knowledge-based” as well as physical.
Aims of the book The primary aim of this book is to develop a theory of measurement that incorporates the observer into the phenomenon under measurement. By this theory, the observer becomes both a collector of data and an activator of the phenomenon that gives rise to the data. These ideas have probably been best stated by J. A. Wheeler (1990; 1994): All things physical are information-theoretic in origin and this is a participatory universe … Observer participancy gives rise to information; and information gives rise to physics. The measurement theory that will be presented is largely, in fact, a quantification of these ideas. However, the reader might be surprised to find that the “information” that is used is not the usual Shannon or Boltzmann entropy measures, but one that is relatively unknown to physicists, that of R. A. Fisher. The measurement theory is simply a description of how Fisher information flows from a physical source effect to a data space. It therefore applies to all scenarios where quantitative data from repeatable experiments may be collected. This describes measurement scenarios of physics but, also, of science in general. The theory of measurement is found to define an analytical procedure for deriving all laws of science. The approach is called EPI, for “extreme physical information.” The secondary aim of the book is to show, by example, that most existing laws of science fit within the EPI framework. That is, they can be derived by its use. (Many can of course be derived by other approaches, but, apparently, no other single approach can derive all of them.) In this way the EPI approach unifies science under an umbrella of measurement and information.
Electron spin was derived by the use of EPI in Chapter 4, in the context of the Dirac equation. This was for the scenario of a single particle. Here, by comparison, we treat spin for a two-particle scenario, that of the well-known EPR–Bohm experiment. A general two-particle PDF can exhibit correlation. We show that the well-known probabilities of spin-pair combinations in the EPR–Bohm experiment follow simply from EPI; and that these do obey certain well-known correlation effects. See also a related derivation by S. Luo (2002). Information J was previously regarded as the “bound” information (Chapter 3), in the sense of the amount of information that is bound to the unobserved source phenomenon. Here we find that it can also represent a degree of information entanglement between two particles : At the EPI solution, the information J for the observed particle equals the information I for the un observed one. Or, the source of information for the observed particle is the unobserved one. A final point of interest will be the use, in Sec. 12.4, of the game corollary | I | = min . (Eq. (3.21)) to determine a vital free parameter of the problem. This approach is also used in Chapter 4 to establish the appropriate number N of amplitude functions to describe various quantum particles, and in Chapter 15 to determine the unknown exponent of a power law for cancer growth. All parameters in the preceding are unitless, as is required for the use of Eq. (3.21). See further discussion in Sec. 12.10.
A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to MMs on finite-dimensional Hilbert spaces. Suppose we want to measure an observable A whose outcomes Ax are represented by positive operators (effects) on a Hilbert Space H. We call H the base or object system. We interact H with a probe system on another Hilbert space K by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable F whose outcomes Fx are measured by an apparatus that is frequently (but need not be) classical in practice. The MM protocol gives a method for determining the probability of an outcome Ax for any state of H in terms of the outcome Fx. The interaction channel usually entangles this state with an initial probe state of K that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing MMs. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.
Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.
This chapter examines the reasons why measurement theory is a particularly controversial part of modern theoretical physics. The problem of measurement in quantum mechanics is considered as nonexistent or trivial by an impressive body of theoretical physicists and as presenting almost insurmountable difficulties by a somewhat lesser but steadily growing number of their colleagues. The law of evolution in time constitutes the essential part of the conventional measurement theory. In this theory, the existence of a Hamiltonian describing the instrument-system interaction is postulated, not derived. In other words, this assumption is considered as a legitimate idealization of the experimental facts. A wave packet reduction can be considered as a phenomenological procedure that, in many instances, can be applied in order to calculate the results of successive observations and the correlations between them without introducing into the formulas the complicated wave functions of the various instruments.