Macroscopic variables of the macroscopic system lose the quantum fluctuations in the macroscopic limit. A separate Hilbert space corresponds to each c-number value of these macroscopic variables. The measuring process in quantum mechanics consists of recording these c-number values and amounts to determination of the Hilbert space to which the system belongs. These observations provide a solution to the problem of measurement and lead to predictions concerning the c-number equations of motion of the macroscopic variables.
We present and discuss some models for the quantum measuring process. Our models indicate that the coherent state approach to classical limits in quantum theories becomes a powerful method for a unified description of various measuring processes. Each detector in the models has a parameter N , the number of its dynamical variables; the whole process of an interaction between an object system and the detector is calculable for any finite number N . It is shown that the detector states after the interaction are characterized by a macroscopic parameter, which is closely connected with the “reliability” of the detector. The reliability plays an essential role in eliminating the quantum interference between two states corresponding to two different positions of the pointer. Finally, the quantum mechanics of the detector reduces to classical mechanics on suitable “phase space” as N becomes large, as is required.
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.
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.