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

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^{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 <= F(rho_A, rho_B) <= P_I^{opt}, linking visibility, fidelity, and optimal discrimination. These results identify distinguishability as fundamentally measurement-limited and motivate discrimination-enhanced induced-coherence imaging and sensing.