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Journal of Mathematical Physics

Canonical approximate quantum measurements

Masanao Ozawa

Published Dec 1, 1993• 30 citations
Publisher

Abstract

In order to generalize the Wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables, a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable, which generalizes von Neumann’s model of measuring interaction and Davies’s covariant instrument for approximate position measurement. Calculations of two types of root-mean-square errors show that this model clears certain requirements for good approximate measurements. A modified Wigner formula for successive canonical approximate measurements of arbitrary observables is established. It is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von Neumann–Lüders collapse postulate by a minor modification in the process of measurement.

DOI
10.1063/1.530272

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In order to generalize the Wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables, a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable, which generalizes von Neumann’s model of measuring interaction and Davies’s covariant instrument for approximate position measurement. Calculations of two types of root-mean-square errors show that this model clears certain requirements for good approximate measurements. A modified Wigner formula for successive canonical approximate measurements of arbitrary observables is established. It is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von Neumann–Lüders collapse postulate by a minor modification in the process of 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.
A systematic construction of exactly solvable models of interactions between the measuring apparatus and the particle whose position is to be measured is shown. The two types of root-mean-square errors are calculated for these models and many new models with better statistics than the standard von Neumann model are continuously found. The best among them is the model which has been shown to circumvent the standard quantum limit for the position monitoring. A feature of the models similar to Bondurant's improvement of the interferometric position measurement is also pointed out.
Many textbooks on Quantum Mechanics are not very precise as to the meaning of making a measurement: as a consequence, they frequently make assertions which are not based on a dynamical description of the measurement process. A model proposed by von Neumann allows a dynamical description of measurement in Quantum Mechanics, including the measuring instrument in the formalism.In this article we apply von Neumann’s model to illustrate the measurement of an observable by means of a measuring instrument and show how various results, which are sometimens postulated without a dynamical basis, actually emerge.We also investigate the more complex, intriguing and fundamental problem of two successive measurements in Quantum Mechanics, extending von Neumann’s model to two measuring instruments. We present a description which allows obtaining, in a unified way, various results that have been given in the literature.
Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.
We previously remarked that when an observable A has a continuous spectrum, then von Neumann's formula for the post-measurement state needs to be extended and the correct formula ineluctably involves the resolution of the detector used in the measurement. We generalize previous results to compute the uncertainties in successive measurements of more general pairs of observables. We also show that this extended von Neumann's formula for the post-measurement state is a completely positive map and, moreover, that there is a completely-positive interpolation between the pre- and post-measurement states. Weinberg has advocated that the time-evolution during the measurement process should be modeled as an open quantum system and governed by a Lindblad equation. We verify that this is indeed the case for an arbitrary observable, A, and a fairly general class of interpolations.
We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called "Wigner's formula" emerges in the strong-coupling limit, while "Kirkwood's quasi-probability distribution" is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.