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.

Jan 1, 200658 citations

On the connection between complementarity and uncertainty principles in the Mach–Zehnder interferometric setting

G M Bosyk, M Portesi, F Holik +1 more

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.

arXiv:1206.2992Physica ScriptaMay 9, 201329 citations