The Equivalence between Hardy-type paradox and Logical Contextuality

Abstract: Hardy-type paradoxes offer elegant, inequality-free proof of quantum contextuality. In this work, we introduce a unified logical formulation for general Hardy-type paradoxes, which we term logical Hardy-type paradoxes. We prove that for any finite scenario, the existence of a logical Hardy-type paradox is equivalent to logical contextuality. Specially, strong contextuality is equivalent to logical Hardy-type paradoxes with success probability SP = 1. These results generalize prior work on (2,k,2), (2,2,d), and n-cycle scenarios, and resolve a misconception that such equivalence does not hold for general scenarios [1]. We analyse the logical Hardy-type paradoxes on the (2,2,2) and (2,3,3) Bell scenarios, as well as the Klyachko-Can-Binicioglu-Shumovsky (KCBS) scenario. We show that the KCBS scenario admits only one kind of Hardy-type paradox, achieving a success probability of SP \approx 10.56% for a specific parameter setting.