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Physical Review Letters

Uncertainty in Quantum Measurements

David Deutsch

Published Feb 28, 1983• 908 citations
Publisher

Abstract

The object of this Letter is to show that except in the case of canonically conjugate observables, the generalized Heisenberg inequality does not properly express the quantum uncertainty principle. It is, in general, too weak. An inequality is obtained which does express the principle.
DOI
10.1103/physrevlett.50.631

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Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.
The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.
In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.