We provide an operational reinterpretation of wave-particle complementarity in the low-gain Zou-Wang-Mandel (ZWM) induced-coherence interferometer. In the low gain limit, each photon pair is emitted by either one of two nonlinear crystals. Preparing nonorthogonal conditional idler states that encode which-crystal information. While previous studies inferred distinguishability indirectly from signal visibility with undetected idler photons. We show that the idler states naturally define a binary quantum hypothesis-testing problem. By performing optimal measurements on the idler, we analyze this task using both zero-error measurement unambiguous discrimination (Ivanovic-Dieks-Peres (IDP)) and minimum-error discrimination (Helstrom bound). We show that the signal visibility equals the optimal inconclusive probability of unambiguous discrimination. The Helstrom bound gives the optimal probability of identifying the emitting crystal. While signal visibility is an ensemble-averaged expectation value, the IDP and Helstrom strategies correspond to optimal single-photon decision measurements on the idler. The decision problem concerns inferring a past source event from a present measurement outcome This establishes wave-particle duality in induced coherence as a manifestation of optimal quantum decision strategies rather than a purely geometric constraint. We further extend the analysis to the presence of thermal photons introduced in the object arm, which render the conditional idler states mixed. In this case, both the visibility and the achievable distinguishability are reduced, reflecting the fundamental limitations imposed by mixed-state discrimination. The approach is model-independent and applies to general two-path interferometers with markers.
We provide an operational reinterpretation of wave-particle complementarity in the low-gain Zou-Wang-Mandel (ZWM) induced-coherence interferometer. In the low gain limit, each photon pair is emitted by either one of two nonlinear crystals. Preparing nonorthogonal conditional idler states that encode which-crystal information. While previous studies inferred distinguishability indirectly from signal visibility with undetected idler photons. We show that the idler states naturally define a binary quantum hypothesis-testing problem. By performing optimal measurements on the idler, we analyze this task using both zero-error measurement unambiguous discrimination (Ivanovic-Dieks-Peres (IDP)) and minimum-error discrimination (Helstrom bound). We show that the signal visibility equals the optimal inconclusive probability of unambiguous discrimination. The Helstrom bound gives the optimal probability of identifying the emitting crystal. While signal visibility is an ensemble-averaged expectation value, the IDP and Helstrom strategies correspond to optimal single-photon decision measurements on the idler. The decision problem concerns inferring a past source event from a present measurement outcome This establishes wave-particle duality in induced coherence as a manifestation of optimal quantum decision strategies rather than a purely geometric constraint. We further extend the analysis to the presence of thermal photons introduced in the object arm, which render the conditional idler states mixed. In this case, both the visibility and the achievable distinguishability are reduced, reflecting the fundamental limitations imposed by mixed-state discrimination. The approach is model-independent and applies to general two-path interferometers with markers.
We propose a novel quantum interferometer based on two stimulated Parametric Down-Conversion (PDC) crystals seeded by weak coherent idler fields to investigate the relationship between the nonclassicality of the signal photon state and the fidelity of the conjugate idler photon state. We show that the purity of the signal photon state is tightly bounded by the entanglement between the signal photon and its conjugate idler photon, i.e., the fidelity of the idler state in the bipartite system, resulting in a novel complementarity relation for wave-particle duality. Moreover, we find that the nonclassical measure of the signal photon state is identical to the fidelity of the conjugate idler photon states prepared in each crystal. In our novel interferometer, the input beam splitter of the conventional SU(2) interferometer is replaced with two seeded SU(1,1) PDC systems that emit only a single photon at once from two crystals, thus producing an entangled superposition state where the conjugate idler photon states act as path detectors with controllable fidelity. We demonstrate that the presented quantum interferometer consisting of two stimulated PDC crystals positioned in parallel is an ideal setup for investigating and quantitatively understanding the complementarity relation of wave-particle duality.
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.
Abstract Based on the concept of geometric coherence, we propose a wave-particle duality relation that establishes an equation between distinguishability and visibility for any pure state in the composite system of a general n-path interferometer and a which-path detector. This relation has the desired property that any increase of distinguishability will inevitably cause a decrease in visibility, thereby quantitatively capturing the essence of complementarity. Additionally, we reveal a connection between the fidelity function and discrimination of pure states. Specifically, we generalize the previous result in Xiong and Wu (2018 J. Phys. A 51 414005) by showing that for any quantum state ρ, the geometric coherence can be exactly expressed as the optimal guessing probability in the discrimination of the corresponding ensemble of pure states.
To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the source purity μs is tightly bounded by the entanglement measure E by the relation μs=1−E2 and the visibility V and detector fidelity F determine the coherence of the quantons, i.e., C=V∣F∣. The quantitative complementarity of the double-path interferometer we developed recently is explained in terms of the quanton-detector entanglement or quanton source purity that are expressed as functions of injected seed photon numbers. We further suggest that the experimental scheme utilizing two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and complementarity quantitatively.
Induced-coherence interferometry, first introduced in the Zou-Wang-Mandel (ZWM) setup, enables retrieval of object information from the interference pattern of light that never interacted with the object. This scheme relies on two identically correlated photon pairs and the absence of "which-way" information about the photons illuminating the object to induce coherence in their companions. In previous studies, the effect of thermal background on the ZWM interferometer was considered; here we explicitly include background noise and analyze the interference visibility in both low- and high-gain regimes, revealing how thermal photons introduce an incoherent offset that lowers the observed interference contrast. We show that the visibility can be restored either by optimal attenuation or by extending the geometry to a three-SPDC configuration. Furthermore, we demonstrate that introducing heralded detection removes the detrimental effect of thermal background noise, restoring high-contrast interference fringes.
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.