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Classical and quantum complementarity, impossible distributions and how much quantumness is truly quantum

Raquel Galazo García, Irene Bartolomé, Laura Ares, Alfredo Luis

Published Nov 30, 2018• 1 citation
OpenAlex

Abstract

Complementary observables can be measured simultaneously with less than perfect accuracy, but the uncertainty added in the observation to each observable can be removed by a proper data inversion. We show that complementarity manifests in that the inferred joint distribution after the inversion is pathological in the sense of not being able to represent a true joint distribution. This is closely related to the fact that the observed joint distribution is not separable. We apply this program to the paradigmatic example of complementarity: the path-interference duality in a Young interferometer. A key feature is that the conclusions hold equally well in the quantum and classical theories. Complementarity is also examined in terms of Wigner-like functions and duality relations.

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Complementary observables can be measured simultaneously with less than perfect accuracy, but the uncertainty added in the observation to each observable can be removed by a proper data inversion. We show that complementarity manifests in that the inferred joint distribution after the inversion is pathological in the sense of not being able to represent a true joint distribution. This is closely related to the fact that the observed joint distribution is not separable. We apply this program to the paradigmatic example of complementarity: the path-interference duality in a Young interferometer. A key feature is that the conclusions hold equally well in the quantum and classical theories. Complementarity is also examined in terms of Wigner-like functions and duality relations.
One of the most remarkable features of quantum physics is that attributes of quantum objects, such as the wavelike and particlelike behaviors of single photons, can be complementary in the sense that they are equally real but cannot be observed simultaneously. Quantum measurements, serving as windows providing views into the abstract edifice of quantum theory, are basic tools for manifesting the intrinsic behaviors of quantum objects. However, a quantitative formulation of complementarity that highlights its manifestations in general measurement scenarios remains elusive. Here we develop a general framework for demonstrating quantum complementarity in the form of information exclusion relations (IERs), which incorporates the wave-particle duality relations as particular examples. Moreover, we explore the applications of our theory in entanglement detection and elucidate that our IERs lead to an extended form of entropic uncertainty relations, providing insights into the connection between quantum complementarity and the preparation uncertainty.
This thesis is concerned with the notions of complementarity and uncertainty encountered in quantum mechanics. Its starting point is an assessment of how these concepts have been represented and illustrated by various writers dating back to their inception. Following the survey a coherent account of the connections and contrasts between complementarity and uncertainty is developed in the context of Mach-Zehnder interferometry. The effect on the interference pattern contrast of path detection via entanglement with a probe system, is explored and a joint unsharp measurement scheme of the complementary pairs, path and interference, described. The Mach-Zehnder set-up proves sufficiently versatile to show that quantum erasure and quantitative quantum erasure constitute instances of joint unsharp measurement of complementary observables. The analysis uses the representation of observables as positive operator valued measures.Path detection and interference observation require different experimental set-ups but can be reconciled in the simultaneous unsharp measurement and preparation. This reconciliation is expressed as an uncertainty relation however the mutually exclusive feature of complementarity is not discarded. It is possible to recover strict complementarity as a limit case of the appropriate uncertainty relation.One motivation for this study is the effort some authors have made in trying to express the founding features of quantum mechanics in the form of a hierarchy of significance. Here it is shown that complementarity and uncertainty have separate identities but are not completely independent of each other. Consequently, establishing a hierarchy of these features within the present formalism of quantum mechanics is not possible.
Complementarity, e.g., the wave–particle duality of nature, lies at the heart of quantum mechanics. It distinguishes the world of quantum phenomena from the reality of classical physics. In the 1920s, quantum theory as we know it today was still new, and examples used to illustrate complementarity emphasized the position (particle-like) and momentum (wave-like) attributes of a quantum mechanical object, be it a photon or a massive particle. This is the historical reason why complementarity is often superficially identified with the so-called wave-particle duality of matter . Complementarity, however, is a more general concept. We say that two observables are complementary if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable. We may illustrate this by two extreme examples. The first example consists of the position and momentum (along one direction) of a particle: if, say, the position is predetermined then the result of a momentum measurement cannot be predicted, all momentum values are equally probable (in a large range). The second extreme involves two orthogonal spin components of a spin- 1/2 particle: if, say, the vertical spin component has a definite value ( up or down ) then upon measuring a horizontal component both values ( left or right , for instance) are found, each with a probability of 50%. Thus, in the microcosmos complete knowledge in the sense of classical physics is not available. The classic example of the merger of wave and particle behavior is provided by Young's double-slit experiment.
A coherent account of the connections and contrasts between the principles of com-plementarity and uncertainty is developed starting from a survey of the various formalizations of these principles. The conceptual analysis is illustrated by means of a set of experimental schemes based on Mach-Zehnder interferometry. In particu-lar, path detection via entanglement with a probe system and (quantitative) quan-tum erasure are exhibited to constitute instances of joint unsharp measurements of complementary pairs of physical quantities, path and interference observables. The analysis uses the representation of observables as positive-operator-valued measures (POVMs). The reconciliation of complementary experimental options in the sense of simultaneous unsharp preparations and measurements is expressed in terms of uncertainty relations of different kinds. The feature of complementarity, manifest in the present examples in the mutual exclusivity of path detection and interference observation, is recovered as a limit case from the appropriate uncertainty relation. It is noted that the complementarity and uncertainty principles are neither completely logically independent nor logical consequences of one another. Since entanglement is an instance of the uncertainty of quantum properties (of compound systems), it is moot to play out uncertainty and entanglement against each other as possible mechanisms enforcing complementarity.