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Physical Review A

Difficulty in detecting deviations from wave-function collapse

Seth Lloyd

Published Oct 1, 1988• 1 citation
Publisher

Abstract

We use the example of a Stern-Gerlach apparatus to show that the quantum-mechanical measurement problem ceases to be a physical problem for a measuring device with many degrees of freedom in the sense that deviations from wave-function collapse become virtually impossible to detect.
DOI
10.1103/physreva.38.3161

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We provide a reasonable explanation for measurement where no wave function in the sense of quantum mechanics is collapsed. The wave functions of the particles in the measuring device should be considered together with the measured particle as one system. The measurement is actually a physical process which makes the wave functions of different paths orthogonal. The collapsed wave function is just an equivalent one to explain certain physical phenomena. Our understanding of measurement also provides reasonable explanations for both EPR paradox and the experiments of delayed quantum eraser.
The measurement problem is to explain why a system which is in a linear combination of states appears, upon measurement, to be in just one of those states. The solution given here is to first show that if one assumes linear, unitary, no collapse, no particle quantum mechanics, the different versions of reality are completely isolated from each other. This then implies only the eigenstates appropriate to the measurement setup will be perceived. In a Stern-Gerlach experiment, for example, only spin up or spin down will be perceived, but never a combination of the two.
A dynamical model for the collapse of the wave function in a quantum measurement process is proposed by considering the interaction of a quantum system (spin-1/2) with a macroscopic quantum apparatus interacting with an environment in a dissipative manner. The dissipative interaction leads to decoherence in the superposition states of the apparatus, making its behaviour classical in the sense that the density matrix becomes diagonal with time. Since the apparatus is also interacting with the system, the probabilities of the diagonal density matrix are determined by the state vector of the system. We consider a Stern-Gerlach type model, where a spin- 1/2 particle is in an inhomogeneous magnetic field, the whole set up being in contact with a large environment. Here we find that the density matrix of the combined system and apparatus becomes diagonal and the momentum of the particle becomes correlated with a spin operator, selected by the choice of the system-apparatus interaction. This allows for a measurement of spin via a momentum measurement on the particle with associated probabilities in accordance with quantum principles.
In the context of the measurement problem, we propose to model the interaction between a quantum particle and an "apparatus" through a non-Hermitian Hamiltonian term. We simulate the time evolution of a normalized quantum state split into two spin components (via a Stern-Gerlach experiment) and that undergoes a wave-function collapse driven by a non-Hermitian Hatano-Nelson Hamiltonian. We further analyze how the strength and other parameters of the non-Hermitian perturbation influence the time-to-collapse of the wave function obtained under a Schrödinger-type evolution. We finally discuss a thought experiment where manipulation of the apparatus could challenge standard quantum mechanics predictions.