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Complementarity and Uncertainty in Mach-Zehnder Interferometry and beyond

Paul Busch, C. R. Shilladay

Published Jan 1, 2006• 58 citations
OpenAlex

Abstract

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.

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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.
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.
The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration shows that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.
We revisit the connection between the complementarity and uncertainty principles of quantum mechanics within the framework of Mach?Zehnder interferometry. We focus our attention on the trade-off relation between complementary path information and fringe visibility. This relation is equivalent to the uncertainty relation of Schrödinger and Robertson for a suitably chosen pair of observables. We show that it is equivalent as well to the uncertainty inequality provided by Landau and Pollak. We also study the relationship of this trade-off relation with a family of entropic uncertainty relations based on Rényi entropies. There is no equivalence in this case, but the different values of the entropic parameter do define regimes that provides us with a tool to discriminate between non-trivial states of minimum uncertainty. The existence of such regimes agrees with previous results of Luis (2011 Phys. Rev. A 84 034101), although their meaning was not sufficiently clear. We discuss the origin of these regimes with the intention of gaining a deeper understanding of entropic measures.
We revisit the connection between the complementarity and uncertainty principles of quantum mechanics within the framework of Mach–Zehnder interferometry. We focus our attention on the trade-off relation between complementary path information and fringe visibility. This relation is equivalent to the uncertainty relation of Schrödinger and Robertson for a suitably chosen pair of observables. We show that it is equivalent as well to the uncertainty inequality provided by Landau and Pollak. We also study the relationship of this trade-off relation with a family of entropic uncertainty relations based on Rényi entropies. There is no equivalence in this case, but the different values of the entropic parameter do define regimes that provides us with a tool to discriminate between non-trivial states of minimum uncertainty. The existence of such regimes agrees with previous results of Luis (2011 Phys. Rev. A 84 034101), although their meaning was not sufficiently clear. We discuss the origin of these regimes with the intention of gaining a deeper understanding of entropic measures.
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.