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Physical Review Letters

Inhibition of the Measurement of the Wave Function of a Single Quantum System in Repeated Weak Quantum Nondemolition Measurements

Orly Alter, Y. Yamamoto

Published May 22, 1995• 29 citations
Publisher

Abstract

It is shown that a series of repeated weak quantum nondemolition measurements performed on a single quantum system gives no information about the wave function of the system. The physical explanation, based on the quantum Brownian motion and the continuous collapse of the wave function which originate in the projection postulate, is discussed in two specific examples.

DOI
10.1103/physrevlett.74.4106

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It is shown that a series of repeated weak quantum nondemolition measurements performed on a single quantum system gives no information about the wave function of the system. The physical explanation, based on the quantum Brownian motion and the continuous collapse of the wave function which originate in the projection postulate, is discussed in two specific examples.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.
This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.
We start from the reasonable assumption that a quantum measurement can be properly described only when the measurement apparatus itself is also treated quantum mechanically. Using idealized optical measurements as an example and introducing a small loss into the measurement apparatus, we show that the combined density matrix of the system and apparatus decoheres in a manner that the outcome of the measurement can be described classically. The loss can be arbitrarily small when the classical limit of the measurement apparatus is taken, i.e., the gain is made to approach infinity. Further, we show that the von Neumann postulate of the collapse of the wave function of the measured system into an eigenstate of the measured observable can be deduced from the analysis of two quantum nondemolition experiments in cascade and need not be introduced as a separate postulate. \textcopyright{} 1996 The American Physical Society.
Many so called paradoxes of quantum mechanics are clarified when the measurement equipment is treated as a quantized system. Every measurement involves nonlinear processes. Self consistent formulations of nonlinear quantum optics are relatively simple. Hence optical measurements, such as the quantum nondemolition (QND) measurement of photon number, are particularly well suited for such a treatment. It shows that the so called 'collapse of the wave function' is not needed for the interpretation of the measurement process. Coherence of the density matrix of the signal is progressively reduced with increasing accuracy of the photon number determination. If the QND measurement is incorporated into the double slit experiment, the contrast ratio of the fringes is found to decrease with increasing information on the photon number in one of the two paths.
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.