The consensus regarding quantum measurements rests on two statements: (i) von Neumann’s standard quantum measurement theory leaves undetermined the basis in which observables are measured, and (ii) the environmental decoherence of the measuring device (the “meter”) unambiguously determines the measuring (“pointer”) basis. The latter statement means that the environment monitors (measures) selected observables of the meter and (indirectly) of the system. Equivalently, a measured quantum state must end up in one of the “pointer states” that persist in the presence of the environment. We find that, unless we restrict ourselves to projective measurements, decoherence does not necessarily determine the pointer basis of the meter. Namely, generalized measurements commonly allow the observer to choose from a multitude of alternative pointer bases that provide the same information on the observables, regardless of decoherence. By contrast, the measured observable does not depend on the pointer basis, whether in the presence or in the absence of decoherence. These results grant further support to our notion of Quantum Lamarckism, whereby the observer’s choices play an indispensable role in quantum mechanics.
A specific model of quantum measurement [D. F. Walls, M. J. Collet, and G. J. Milburn, Phys. Rev. D 32, 3208 (1985)] based on a generic system-meter-environment measurement scheme is investigated in terms of probability distributions on projective Hilbert space. Starting from the unitary evolution of the total system a stochastic process for the wave function of the reduced system-meter system is derived by elimination of the degrees of freedom of the environment. The stochastic process is interpreted in terms of the direct detection of the photoelectrons created by the meter quanta in the environment. Two general variances are introduced which represent, respectively, the average intrinsic quantum fluctuations and the statistical fluctuations of the quantum expectation of the measured observable. Involving fourth moments of the wave function, these variances allow to monitor the transformation from quantum to classical probabilities during the process of measurement. \textcopyright{} 1996 The American Physical Society.
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.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.
The form of the interaction Hamiltonian between the apparatus and its environment is sufficient to determine which observable of the measured quantum system can be considered "recorded" by the apparatus. The basis that contains this record---the pointer basis of the apparatus---consists of the eigenvectors of the operator which commutes with the apparatus-environment interaction Hamiltonian. Thus the environment can be said to perform a nondemolition measurement of an observable diagonal in the pointer basis.
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.