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Annalen der Physik

Some Remarks on Unsharp Quantum Measurements, Quantum Non‐Demolition, and All That

Paul Busch, PEKKA J. LAHTI

Published Jan 1, 1990• 28 citations
Publisher

Abstract

Abstract The theory of unsharp quantum measurements is reviewed as a generalization of the von Neumann‐Lüders measurement theory and applied to measurements of continuous quantities. A generalized notion of repeatability is proposed which is applicable even to intrinsically unsharp continuous observables. As an illustration a precise formulation of a quantum non‐demolition (QND) measurement performed on a harmonic oscillator is given.

DOI
10.1002/andp.19905020502

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Abstract The theory of unsharp quantum measurements is reviewed as a generalization of the von Neumann‐Lüders measurement theory and applied to measurements of continuous quantities. A generalized notion of repeatability is proposed which is applicable even to intrinsically unsharp continuous observables. As an illustration a precise formulation of a quantum non‐demolition (QND) measurement performed on a harmonic oscillator is given.
This is a limited overview of quantum non-demolition (QND) measurements, with brief discussions of illustrative examples meant to clarify the essential features. In a QND measurement, the predictability of a subsequent value of a precisely measured observable is maintained and any random back-action from uncertainty introduced into a noncommuting observable is avoided. The fundamental ideas, relevant theory and the conditions and scope for applicability are discussed with some examples. Precision measurements have indeed gained from developing QND measurements. Some implementations in quantum optics, gravitational wave detectors and spin-magnetometry are discussed.
Summary form only given. The present techniques allow one to perform continuous or discrete quantum measurements at or even beyond the standard "quantum noise" level, even on single quantum objects. In this context, Kip Thorne and Vladimir Braginsky introduced in the seventies the concept of "Quantum Non Demolition (QND) measurement", which was initially meant to be applied to mechanical harmonic oscillators designed for detecting gravitational waves, the so-called "Weber's bars". But the QND ideas encountered a real success in quantum optics, where quantum fluctuations are readily accessible, and can also be circumvented. We review the state of the art of QND measurements, in this broad sense of sub-shot-noise optical information processing. Most of the initial objectives of QND measurements have now been successfully reached through proof-of-principle experiments. The domain is now opening on future perspectives related to quantum information processing (e.g. quantum cryptography), either with "discrete" or "continuous" quantum variables. In this context the close relationship between QND measurements and controlled-NOT (CNOT) gates are discussed and illustrated.
The question of phase measurement for a quantum oscillator using the quantum non-demolition principle (NDP), which does not need the use of the wave packet reduction postulate, is studied. It is shown that this approach is appropriate to introduce in a proper way the relevant observables for unlocalized (unsharp) measurement of quantum phase. Our analysis suggests that in contrast to other quantum-mechanical problems (e.g. The position and the momentum of a particle), the operator to represent the phase of a quantum field cannot be separated from the measurement process. Thus, the quantum phase operator must be defined for a corresponding measurement scheme. Moreover, it allows us to describe the time continuous process of quantum phase observations and also to study the stochastic dynamics of the oscillator under such measurement.
The purpose of this paper is to provide a basis of theory of measurements of continuous observables. We generalize von Neumann’s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables, and show how every such measuring process determines the state change caused by the measurement. We establish a one-to-one correspondence between completely positive instruments in the sense of Davies and Lewis and the state changes determined by the measuring processes. We also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics, so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis. A proof of the Wigner–Araki–Yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework. We also discuss the implication of these results for the recent results due to Srinivas and due to Mercer on measurements of continuous observables.