The Equivalence between Hardy-type paradox and Logical Contextuality

TL;DR

This paper establishes a unified logical framework for Hardy-type paradoxes, proving their equivalence to logical contextuality across finite scenarios and analyzing specific quantum scenarios like KCBS.

Contribution

It introduces logical Hardy-type paradoxes, proving their equivalence to logical contextuality, and generalizes prior results to all finite scenarios.

Findings

Logical Hardy-type paradoxes are equivalent to logical contextuality.

Strong contextuality corresponds to paradoxes with success probability 1.

KCBS scenario admits only one Hardy-type paradox with success probability ~10.56%.

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.