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arXiv:2511.22871

Wave-Particle Complementarity as the Optimal Limit of Unambiguous Quantum-State Discrimination

Theerthagiri L

quant-phSubmitted Nov 28, 2025• Announced Dec 1, 2025• 0 citations• Revised Dec 2, 2025 (v2)
v1 · Nov 28, 2025v2 · Dec 2, 2025v3 · Jan 2, 2026
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Abstract

We establish a direct operational interpretation of wave--particle complementarity in induced-coherence interferometry by linking interference visibility to the optimal performance of unambiguous quantum-state discrimination. In the low-gain Zou--Wang--Mandel interferometer, the idler modes of the two SPDC crystals act as nonorthogonal which-crystal marker states, and we show that the single-photon visibility equals the minimal inconclusive probability of the optimal Ivanovic--Dieks--Peres (IDP) strategy, $V = P_I^{\mathrm{opt}}$, so that the complementarity relation $D^{2}+V^{2}=1$ expresses a measurement-optimality boundary rather than a geometric constraint. Our results reveal that wave--particle duality in induced coherence is not limited by interferometer geometry but by measurement optimality in zero-error discrimination on the idler, with experimentally testable consequences predicting distinguishability beyond that inferred from visibility alone. In the presence of thermal noise we derive the hierarchy $V \le F(ρ_A,ρ_B) \le P_I^{\mathrm{opt}}$, linking visibility, fidelity, and optimal discrimination. These results identify distinguishability as fundamentally measurement-limited and motivate discrimination-enhanced induced-coherence imaging and sensing.

Comments
14 pages, 2 figures

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We establish a direct operational interpretation of wave--particle complementarity in induced-coherence interferometry by linking interference visibility to the optimal performance of unambiguous quantum-state discrimination. In the low-gain Zou--Wang--Mandel interferometer, the idler modes of the two SPDC crystals act as nonorthogonal which-crystal marker states, and we show that the single-photon visibility equals the minimal inconclusive probability of the optimal Ivanovic--Dieks--Peres (IDP) strategy, V=PIoptV = P_I^{\mathrm{opt}}V=PIopt​, so that the complementarity relation D2+V2=1D^{2}+V^{2}=1D2+V2=1 expresses a measurement-optimality boundary rather than a geometric constraint. Our results reveal that wave--particle duality in induced coherence is not limited by interferometer geometry but by measurement optimality in zero-error discrimination on the idler, with experimentally testable consequences predicting distinguishability beyond that inferred from visibility alone. In the presence of thermal noise we derive the hierarchy V≤F(ρA,ρB)≤PIoptV \le F(ρ_A,ρ_B) \le P_I^{\mathrm{opt}}V≤F(ρA​,ρB​)≤PIopt​, linking visibility, fidelity, and optimal discrimination. These results identify distinguishability as fundamentally measurement-limited and motivate discrimination-enhanced induced-coherence imaging and sensing.
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 μsμ_sμs​ is tightly bounded by the entanglement measure EEE by the relation μs=1−E2μ_s=\sqrt{1-E^2 }μs​=1−E2​ and the visibility VVV and detector fidelity FFF determine the coherence of the quantons, i.e., C=V∣F∣C = V|F|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 NNN 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]ξ\in[0,1]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=0ξ=0ξ=0, both approaches reduce to the ME measurement, which typically yields the tightest relation. For ξ>0ξ>0ξ>0, each approach yields a distinct relation that generally becomes less tight as ξξξ 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.