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arXiv:quant-ph/0204121 · Physical Review A

Consistent approach for quantum measurement

Peng Zhang, Xingfang Liu, C. P. Sun

Published Oct 7, 2002• 18 citations
arXivHTMLPDF

Abstract

In a close form without referring the time-dependent Hamiltonian to the total system, a consistent approach for quantum measurement is proposed based on Zurek's triple model of quantum decoherence [W. Zurek, Phys. Rev. D 24, 1516 (1981)]. An exactly solvable model based on the intracavity system is dealt with in detail to demonstrate the central idea in our approach: by peeling off one collective variable of the measuring apparatus from its many degrees of freedom, as the pointer of the apparatus, the collective variable decouples with the internal environment formed by the effective internal variables, but still interacts with the measured system to form a triple entanglement among the measured system, the pointer, and the internal environment. As another mechanism to cause decoherence, the uncertainty of relative phase and its many-particle amplification can be summed up to an ideal entanglement or a Shmidt decomposition with respect to the preferred basis.

DOI
10.1103/physreva.66.042104

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In a close form without referring the time-dependent Hamiltonian to the total system, a consistent approach for quantum measurement is proposed based on Zurek's triple model of quantum decoherence [W. Zurek, Phys. Rev. D 24, 1516 (1981)]. An exactly solvable model based on the intracavity system is dealt with in detail to demonstrate the central idea in our approach: by peeling off one collective variable of the measuring apparatus from its many degrees of freedom, as the pointer of the apparatus, the collective variable decouples with the internal environment formed by the effective internal variables, but still interacts with the measured system to form a triple entanglement among the measured system, the pointer, and the internal environment. As another mechanism to cause decoherence, the uncertainty of relative phase and its many-particle amplification can be summed up to an ideal entanglement or a Shmidt decomposition with respect to the preferred basis.
The decoherence approach to the measurement problem invokes dissipative interactions between a measuring apparatus and its environment to explain, within the context of ‘pure’ quantum mechanics (QM), why such devices appear to possess definite pointer readings. By ‘pure’ QM, I mean Schrödinger evolution with no wavefunction collapse. Several classes of interpretations of pure QM can rely on decoherence. One class is the ‘modal’ interpretations of van Fraassen (1979), Dieks (1989), Healey (1989), and others. Another class is the relative-state and many-world interpretations. I will argue that decoherence cannot help these interpretations address the general metaphysical challenges raised against them. But decoherence can help pick out a ‘special’ basis that determines which observables receive definite values. I'll explore to what extent decoherence rescues the modal (biorthogonal) basis-selection rule, and Zurek's (environmental interaction) basis-selection rule, from the basis degeneracy problem and the imperfect measurement problem.
A model of quantum measurement is proposed, which aims to describe statistical mechanical aspects of this phenomenon, starting from a purely Hamiltonian formulation. The macroscopic measurement apparatus is modeled as an ideal Bose gas, the order parameter of which, that is, the amplitude of the condensate, is the pointer variable. It is shown that properties of irreversibility and ergodicity breaking, which are inherent in the model apparatus, ensure the appearance of definite results of the measurement, and provide a dynamical realization of wave-function reduction or collapse. The measurement process takes place in two steps: First, the reduction of the state of the tested system occurs over a time of order $\ensuremath{\Elzxh}{/(TN}^{1/4}),$ where T is the temperature of the apparatus, and N is the number of its degrees of freedom. This decoherence process is governed by the apparatus-system interaction. During the second step classical correlations are established between the apparatus and the tested system over the much longer time scale of equilibration of the apparatus. The influence of the parameters of the model on nonideality of the measurement is discussed. Schr\"odinger kittens, EPR setups, and information transfer are analyzed.
Quantum Measurements regarded in Systems Selfdescription framework for measuring system (MS) consist of measured state S environment E and observer OOO processing input S signal. OOO regarded as quantum object which interaction with S,E obeys to Schrodinger equation (SE) and from it and Breuer selfdescription formalism S information for OOO reconstructed. In particular S state collapse obtained if OOO selfdescription state has the dual structure LT=H⨂LVL_T=\cal H \bigotimes L_VLT​=H⨂LV​ where H\cal HH is Hilbert space of MS states ΨMSΨ_{MS}ΨMS​. LV\cal L_VLV​ is the set with elements VO=∣Oj><Oj∣V^O=|O_j> < O_j|VO=∣Oj​><Oj​∣ describing random 'pointer' outcomes OjO_jOj​ observed by OOO in the individual events. The 'preferred' basis ∣Oj>|O_j>∣Oj​> defined by OOO state decoherence via OOO - E interactions. Zurek's Existential Interpretation discussed in selfmeasurement framework.
We present a general analytic method for evaluating the generally time-dependent pointer states of a subsystem, which are defined by their capability not to entangle with the states of another subsystem. We explore the conditions under which the pointer states of the system become independent of time; so that a preferred basis of measurement can be realized. We relate the mathematical conditions for having time-independent pointer states to some classes of possible symmetries in the Hamiltonian of the total composite system. Indeed, our theory would serve as a generalization of the existing theory for determination of the preferred basis of measurement. By exploiting this new theory we can obtain those regimes of the parameter space for a given total Hamiltonian defining our system-environment model for which a preferred basis of measurement can be realized. Moreover, we can predict the corresponding preferred basis of measurement for each regime. We can also obtain the time-dependent pointer states of the system and the environment in most of the other regimes where the pointer states of the system are time-dependent and a preferred basis of measurement cannot be realized at all. This ability to obtain time-dependent pointer states is specifically important in decoherence studies; as these pointer states, although they evolve with time and cannot represent the preferred basis of measurement, they correspond to those initial conditions for the state of the system and the environment for which we can have longer decoherence times. In the next step, we consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system and the interaction with the environment do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we calculate the time-dependent pointer states of the system and the environment for the case that the environment initially is prepared in the coherent state. We also obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We will show that for the case that the system initially is prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will be a sinusoidal function with a slow decaying envelop which is characterized by a decay time proportional to n-bar.
In a quantum measurement setting, it is known that environment-induced decoherence theory describes the emergence of effectively classical features of the quantum system–measuring apparatus composite system when the apparatus is allowed to interact with the environment. In [E. A. Galapon, Europhys. Lett. 113, 60007 (2016)], a measurement model is found to have the feature of inducing exact decoherence at a finite time via one internal degree of freedom of the apparatus provided that the apparatus is decomposed into a pointer and an inaccessible probe, with the pointer and the probe being in momentum-limited initial states. However, an issue can be raised against the model: while the factorization method of the time-evolution operator used there is formally correct, it is not completely rigorous due to some unstated conditions on the validity of the factorization in the Hilbert space of the model. Furthermore, no examples were presented there in implementing the measurement scheme in specific quantum systems. The goal of this paper is to re-examine the model and confirm its features independently by solving the von Neumann equation for the joint state of the composite system as a function of time. This approach reproduces the joint state obtained in the original work, leading to the same conditions for exact decoherence and orthogonal pointer states when the required initial conditions on the probe and pointer are imposed. We illustrate the exact decoherence process in the measurement of observables of a spin-1/2 particle and a quantum harmonic oscillator by using the model.