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On the Sharpness and Bias of Quantum Effects

Paul Busch

Published Jul 1, 2009• 23 citations

Abstract

The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given. Dedication The impossibility of measuring jointly certain pairs of observables is an intriguing non-classical feature of quantum theory that Pekka Lahti identified as a candidate for a rigorous formulation of the principle of complementarity. While he was investigating this fundamental no-go statement in the early 1980s, he learned from Peter Mittelstaedt that one of his students was aiming to prove the positive possibility of approximate joint measurements of complementary quantities such as position and momentum. Pekka joined our group as an Alexander von Humboldt Fellow, and together we found that a reconciliation between complementarity and (approximate) joint measurability is possible on the basis of the generalized representation of observables as positive operator measures (POMs). Since then we have pursued together our aspirations of understanding quantum mechanics and understanding Nature. I have benefited much from Pekka’s intellectual rigor and have been privileged ever since to enjoy his warm humanity. It is a great pleasure to present this paper to Pekka as a token of thanks and friendship on the occasion of his 60th birthday, with all good wishes for many happy recurrences and productive years to come. 1

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The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given. Dedication The impossibility of measuring jointly certain pairs of observables is an intriguing non-classical feature of quantum theory that Pekka Lahti identified as a candidate for a rigorous formulation of the principle of complementarity. While he was investigating this fundamental no-go statement in the early 1980s, he learned from Peter Mittelstaedt that one of his students was aiming to prove the positive possibility of approximate joint measurements of complementary quantities such as position and momentum. Pekka joined our group as an Alexander von Humboldt Fellow, and together we found that a reconciliation between complementarity and (approximate) joint measurability is possible on the basis of the generalized representation of observables as positive operator measures (POMs). Since then we have pursued together our aspirations of understanding quantum mechanics and understanding Nature. I have benefited much from Pekka’s intellectual rigor and have been privileged ever since to enjoy his warm humanity. It is a great pleasure to present this paper to Pekka as a token of thanks and friendship on the occasion of his 60th birthday, with all good wishes for many happy recurrences and productive years to come. 1

refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be

In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg's famous but often challenged measurement uncertainty relation. These relations take the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Here we review some of these error measures, examine their properties and suitability, and compare their relative strengths as criteria for "good" approximations.

We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity—some of which new—in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position–momentum, position–energy, number–phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul’s work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.

For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.

Abstract The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of observables and of the statistical and individual aspects of quantum measurements.

In the conventional axiomatic formalism of quantum mechanics, a physical measurement of a quantum observable is mathematically represented by the spectral decomposition of the Hermitian operator associated with the observable. This constitutes the scenario of a L\"uders (projective) measurement, which includes a von Neumann (rank-one projective) measurement as a special and prominent instance. In this context, the measurement is called sharp in the sense that each measurement operator is an orthogonal projection (eigenprojection of the observable). In the modern operational formalism, a measurement is represented by a positive-operator valued measure (POVM), which consists of a family of non-negative definite operators (measurement operators, effects) summing to the identity. In this scenario, a measurement is called unsharp (fuzzy) if some measurement operators are not orthogonal projections. A natural question arises as to how to quantify unsharpness of a measurement. In this work, we address this issue in terms of uncertainty. For this purpose, we study a family of observables associated with a measurement and their uncertainty. By exploiting the difference between the (total) measurement uncertainty and the observable uncertainty, we are led to some information-theoretic quantifiers of unsharpness. We reveal their basic properties and illustrate them through some important measurements. In particular, we characterize L\"uders measurements and equiangular POVMs as extreme measurements in terms of unsharpness.