Starting from the recoiling double slit gedanken experiment, protagonist of the debates between Einstein and Bohr, we study the main steps that led to the discussion of the complementarity between path information and interference visibility at intermediate domains, where each quantity is partially known. Trying to highlight the strategies of quantification of path knowledge, we emphasize the existence of two regimes of complementarity. The first regime, which leads to the formulation of Greenberger-Yasin duality, is based on predictive strategies. The second one, which leads to the formulation of Englert-Greenberger-Yasin duality (EGY), is based on the retrodiction of the path through the result of a measurement on an auxiliary system. In this second regime, we stress the point that the description of the auxiliary system through nonorthogonal quantum states is the essential ingredient in the derivation of EGY duality. Thus, the quantitative study of complementarity opens space for a close relationship with the strategies of discrimination of quantum states, which for a long time was mentioned in the literature only in a very subtle way. In addition, we also study recent works, which make use of a coherence quantifier and take important steps in obtaining interferometric dualities in a multipath scenario. At the end we present a preliminary form of generalization of these works in a three-way interferometer.
Starting from the recoiling double slit gedanken experiment, protagonist of the debates between Einstein and Bohr, we study the main steps that led to the discussion of the complementarity between path information and interference visibility at intermediate domains, where each quantity is partially known. Trying to highlight the strategies of quantification of path knowledge, we emphasize the existence of two regimes of complementarity. The first regime, which leads to the formulation of Greenberger-Yasin duality, is based on predictive strategies. The second one, which leads to the formulation of Englert-Greenberger-Yasin duality (EGY), is based on the retrodiction of the path through the result of a measurement on an auxiliary system. In this second regime, we stress the point that the description of the auxiliary system through nonorthogonal quantum states is the essential ingredient in the derivation of EGY duality. Thus, the quantitative study of complementarity opens space for a close relationship with the strategies of discrimination of quantum states, which for a long time was mentioned in the literature only in a very subtle way. In addition, we also study recent works, which make use of a coherence quantifier and take important steps in obtaining interferometric dualities in a multipath scenario. At the end we present a preliminary form of generalization of these works in a three-way interferometer.
The issue of interference and which-way information is addressed in the context of three-slit interference experiments. A new path distinguishability |DQ| is introduced, based on unambiguous quantum state discrimination. An inequality connecting the interference visibility and path distinguishability, |V+3−DQ2DQ≤1|, is derived which puts a bound on how much fringe visibility and which-way information can be simultaneously obtained. It is argued that this bound is tight. For two-slit interference, we derive a new duality relation which reduces to Englert's duality relation and the Greenberger–Yasin duality relation, in different limits.
We critically analyse the problem of formulating duality between fringe visibility and which-way information, in multibeam interference experiments. We show that the traditional notion of visibility is incompatible with any intuitive idea of complementarity, but for the two-beam case. We derive a number of new inequalities, not present in the two-beam case, one of them coinciding with a recently proposed multibeam generalization of the inequality found by Greenberger and YaSin. We show, by an explicit procedure of optimization in a three-beam case, that suggested generalizations of Englert's inequality do not convey, differently from the two-beam case, the idea of complementarity, according to which an increase of visibility is at the cost of a loss in path information, and vice versa.
The quantitative formulation of Bohr's complementarity proposed by Greenberger and Yasin is applied to some physical situations for which analytical expressions are available. This includes a variety of conventional double-slit experiments, but also particle oscillations, as in the case of the neutral-kaon system, and Mott scattering of identical nuclei. For all these cases, a unified description can be achieved including a new parameter $\ensuremath{\nu},$ which quantifies the effective number of fringes one can observe in each specific interferometric setup.
We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known complementarity relation. The duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.
A quantum system (quanton) traverses an interferometer with N equally probable paths and interacts with another quantum system (detector) that stores path information in a set of symmetric states. In this framework, we present a systematic quantitative study of wave-particle duality based on entropic trade-off relations between quantum coherence, characterized by the relative entropy of coherence of the quanton state, and which-path knowledge, quantified by the mutual information obtained through detector-state discrimination. Unlike prior upper-bound-based formulations, we obtain an exact quantification of which-path knowledge for the discrimination strategies considered here, encompassing previous strategies and extending beyond them. A general optimal measurement is implemented through a state-separation map governed by a continuous parameter ξ∈[0,1], followed by minimum-error (ME) discrimination of either the successful outputs only (standard approach) or both successful and failure outputs (concatenated approach). Our framework identifies a strategy-dependent hierarchy that generally emerges in the tightness of the resulting duality relations. At ξ=0, both approaches reduce to the ME measurement, which typically yields the tightest relation. For ξ>0, each approach yields a distinct relation that generally becomes less tight as ξ increases, with the concatenated strategy always yielding a relation at least as tight as its standard counterpart. We further characterize saturation of these relations by identifying the detector-state sets that lead to it and show that nontrivial saturation occurs only for interferometers with a nonprime number of paths. From these sets, we elucidate how the underlying structure of the quanton-detector correlations gives rise to this behavior and explains the lack of universal saturation in multipath duality relations for pure states.
Complementarity, or more specifically, the wave-particle duality, and quantum coherence are two fundamental concepts in quantum mechanics. Recently, motivated by the progress of the quantification of quantum coherence, the complementary relation between coherence and path information is investigated by many authors, and various duality relations between them are established. Such relations not only provide insights into fundamental problems of quantum mechanics, such as the understanding of quantum coherence and wave-particle duality; but also are important in applications of quantum technologies. In this paper, based on the Bures distance and unambiguous quantum state discrimination, systematic analysis of the complementarity between the quantum coherence and path information in two path interferometers is carried out. Similarly as other related works, the wave aspect, or the visibility of the interferometer, is quantified by the l1-norm measure of quantum coherence, and the path information is considered via unambiguous quantum state discrimination. In this way, a novel duality relation in two path interferometers is obtained. Compared with known results, our work considers mixed states as well as pure states; considers the path predictability resulting from the intrinsic path asymmetry of the quantum state, as well as the path distinguishability resulting from the use of a which-path detector. Therefore, our work systematically generalizes known results in two path interferometers by removing all the unnecessary restrictions. Specifically, the most general form of quantum states in two path interferometers is considered and the duality relation between quantum coherence and path information is proved based on the positivity property of density matrices. The cases of path predictability and path distinguishability are considered separately. For path predictability, the proof is straightforward; whereas some advanced mathematical techniques, such as the Schur-Weyl inequality, properties of the fidelity and properties of positive matrices, are required in order to give a rigorous proof of the duality relation between coherence and path distinguishability. Concrete examples are provided to illustrate the abstract method and results. Our work concerns about two path interferometers exclusively and depends heavily that the dimensionality is two, therefore it would be an interesting task to generalize the results in this paper to n-path interferometers.