Uncertainty relation and the measurement error – perturbation relation

Vadim I. Smirnov

Yu Watanabe

Yu Watanabe

Vladimir B. Braginsky, Farid Ya Khalili, Kip S. Thorne

The measurement process and the uncertainty relation In the unstarred sections of previous chapters, the back action of the measuring device on the measured object was analyzed by simply invoking the Heisenberg uncertainty relation for the perturbation and the measurement error. Is this simple method of analysis justified? The measurement process does not show up in the standard formulation of the uncertainty relations. Instead, the standard formulation, which follows from the fundamental postulates of quantum mechanics (section 2.1), couples the intrinsic rms uncertainties of observables as defined by the quantum state. Nevertheless, the examples of indirect measurements discussed above (the Heisenberg microscope and Doppler speed meter in chapter I, and the electron probe in chapter III) exhibit a tight coupling between the measurement process and the uncertainty relations. In particular, both the measurement error and the perturbation of the object can be traced to the uncertainty properties of the probe's initial state, and from the fact that this initial state is subject to the uncertainty relations, one can show that the measurement error and perturbation are also subject to it. Is it also possible, starting from the uncertainty relations as properties of the object's quantum state, to obtain the uncertainty relations for the measurement process? To answer this question, let us discuss the following thought experiment: We suppose that a generalized coordinate of some quantum object is measured, and that before the measurement the object was in a state with a very well defined momentum.

Yurii I. Vorontsov

The origins and physical consequences of the traditionally used relation between the position measurement error and the momentum perturbation, Δm2xΔp2p ≥ 2/4, are discussed. It is demonstrated that the corresponding increase in the momentum variance for the aposteriori state occurs only in some special cases. The product of Δm2A and Δp2B is shown to essentially differ from the one given by the uncertainty relation if the commutator is an operator. The error quantum limits for the joint homodyne measurement of quadrature amplitudes for an optical mode are found. It is shown that similar results can be obtained if the quadratures of a harmonic oscillator are estimated by means of continuous position measurement.

M. G. Raymer

Does the Heisenberg uncertainty principle refer to errors of measurement, or to the spread of values of the physical variables intrinsic to a particle’s state, or to some combination of these? The most commonly quoted form of the uncertainty principle relates the spread of an ensemble of separate measurements of some variable to the analogous spread of its conjugate variable. In contrast, Heisenberg’s original argument for the uncertainty principle involved the perturbation to a particle’s state by a measurement of one variable, which affects one’s ability to predict the outcome of a subsequent measurement of the conjugate variable. The relation between these two views of the uncertainty principle is discussed in this paper. A familiar example is considered: an ensemble of identically prepared particles passing through a slit, and after further propagation being detected. From this arrangement it is possible to infer joint (although necessarily imprecise) information on both transverse position and conjugate momentum for each member of the ensemble. It is shown that in this case of joint measurement the product of standard deviations for the measurement outcomes is at least twice as large as the lower bound implied by the usual uncertainty principle. The discussion is meant to help clarify the different roles played in the various statements of the uncertainty principle by the initial state and by measurement error.