Quantum Coherence and Distinguishability as Complementary Resources: A Resource-Theoretic Perspective from Wave-Particle Duality

Wave-particle duality, a fundamental principle of quantum mechanics, encapsulates the complementary relationship between the wave and particle behaviors of quantum systems. In this paper, we treat quantum coherence and classical distinguishability as complementary resources and uncover a novel duality relation, which is explored through quantum state discrimination under incoherent operations, extending beyond typical interference scenarios. We prove that in an ensemble of mutually orthogonal pure states, the sum of `co-bits', quantifying the coherence preserved under incoherent free operations, and classical bits, representing the distinguishability extracted via quantum state discrimination, is bounded. This coherence-distinguishability duality relation exposes an inherent trade-off between the simultaneous preservation of a system's quantum coherence (wave-like property) and the extraction of its classical distinguishability (particle-like property). Our findings provide a fresh perspective on wave-particle duality through quantum resource theories, offering complementary insights into manipulating quantum and classical resources, with implications for quantum foundations and quantum technologies.