PHYSICAL REVIEW A 84, 044102 (2011) Hardy’s paradox and the entanglementlike structure of forward-scattered waves M´ aty´ as Koniorczyk and Levente Szab´ o Institute of Physics, University of P´ ecs, H-7624 P´ ecs, Ifj ´ us´ ag ´ utja 6, Hungary Peter Adam Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 49, Hungary and Institute of Physics, University of P´ ecs, H-7624 P´ ecs, Ifj ´ us´ ag ´ utja 6, Hungary (Received 20 June 2011; published 31 October 2011) We analyze Hardy’s paradox from the point of view of scattering theory. This approach has been useful for the understanding of interaction-free measurement, which is a similar setup. We calculate the forward-scattered waves generated by the beam splitters, which are replaceable in the gedanken experiment. These two-mode waves appear to have an entanglementlike structure. DOI: 10.1103/PhysRevA.84.044102 PACS number(s): 03.65.Ud I. INTRODUCTION Quantum correlations are perhaps the most characteristic features of quantum mechanics. Since the seminal paper of Einstein, Podolsky, and Rosen [1], the study of quantum correlations has been a focus not only of physics, but also of philosophy. The starting point of the quantitative study of such phenomena is the Bell inequality [2]. Clauser et al. [3] were the first to provide an experimental setup in which Bell inequalities could be verified. An extensive literature is devoted to the quantification and observation of related phenomena, including Bell’s theorems and the Greenberger- Horne-Zeilinger scenario. Hardy’s paradox [4] is a relevant phenomenon of this type. Bell-type correlations appear in an interferometric setup in which there are two Mach-Zehnder interferometers, with one containing a particle and the other containing an antiparticle, at least in the gedanken-experimental scenario. The exit beam splitters of the interferometers are optionally replaceable, with their presence playing the role of the local setting in this Bell- type setup. It has been shown that the measurement results are correlated in a way that cannot be explained classically or by local hidden variable theories. Recently, optical experiments realizing Hardy’s scenario have been carried out by Lundeen and Steinberg [5] and independently by Yokota et al. [6]. These experiments have confirmed the existence of the nonclassical correlations. These works also introduce a link between Hardy’s paradox and weak measurement. Interaction-free measurement (IFM) is another very in- teresting and experimentally verified feature of quantum mechanics. The presence of an absorber in one arm of an interferometer can be detected without the event of absorption. A very clarifying paper by Geszti [7] explains this phenomenon within the framework of scattering theory: he points out that it is the so-called forward-scattered wave, introduced by the presence of the absorber according to the optical theorem of scattering theory, which carries the information regarding the presence of the absorber. The motivation of the present study is the observation that the setup for Hardy’s paradox, at least in one of the four possible local choices, consists of two IFM setups playing the role of an absorber for each other. Since the concept of forward-scattered waves was useful in the study of IFM itself, we consider the notion and behavior of the forward-scattered waves in the case of Hardy’s paradox. However, there is no replaceable detector in this setup, but there are replaceable beam splitters there, which is not the case for IFM. Hence, the concept of the forward-scattered waves appears differently in this case. This Brief Report is organized as follows: in Secs. II and III, we recall the description of interaction-free measurement in terms of scattering theory and Hardy’s paradox, respectively. Section IV is devoted to the analysis of the forward-scattered waves in Hardy’s setup. In Sec. V, the results are summarized and conclusions are drawn. II. INTERACTION-FREE MEASUREMENT For the sake of consistency, let us recall the IFM setup and the explanation of Geszti in Ref. [7]. The scenario is depicted in Fig. 1. In this Mach-Zehnder interferometer, a destructive interference is observed at the detector D in the absence of the absorber, optionally placeable in either of the arms. If the absorber resides in one of the paths, then the detector, which is idle otherwise, will fire. This is strange since if the absorber had absorbed the particle, there would have been nothing to make the detector fire. Hence, there is a possibility of detecting the presence of the absorber without absorption or, in other words, without interaction. This phenomenon is called interaction-free measurement. This phenomenon can be viewed as the consequence of the optical theorem of quantum scattering theory: the unitary nature of quantum mechanics does not allow the absorption to be the only effect of an object. The incoming wave should be extinguished by emitting a forward-scattered wave. In this case, the forward-scattered wave can be obtained as a difference between the outgoing wave with an absorber and the outgoing wave with no absorber. This forward-scattered wave gets to the detector, making the interaction-free measurement possible. III. HARDY’S PARADOX The interferometric setup for Hardy’s paradox is depicted in Fig. 2; we use the notation introduced there in what follows. 044102-1 1050-2947/2011/84(4)/044102(4) ©2011 American Physical Society