The Equivalence between Hardy-type paradox and Logical Contextuality

TL;DR

This work develops a unified, inequality-free framework linking Hardy-type paradoxes to logical contextuality within an event-based contextuality formalism. By grounding the analysis in partial Boolean algebras, atom graphs, and incidence matrices, it proves that logical Hardy-type paradoxes exist if and only if a system is logically contextual, and that strong contextuality precisely corresponds to paradoxes with SP = 1. The authors provide algorithmic tools to identify logically contextual states and demonstrate concrete results for the KCBS and (2,2,2) scenarios, including a KCBS paradox with SP ≈ 10.56% and a standard Hardy paradox in the CHSH-type setting. They also outline how to extend the framework to Cabello-type paradoxes and emphasize the framework’s applicability to arbitrary finite quantum scenarios, offering a systematic route to classify and optimize contextuality-based paradoxes.

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.

Figures (5)

• Figure 1: Atom graph $\mathcal{G}_a(\mathcal{Q}_{(2,2,2)})$. Notations $\overline{a_i}$, $\overline{b_j}$ and $a_ib_j$ represent $\neg a_i$, $\neg b_j$ and $a_i\land b_j$ respectively ($i,j\in\{0,1\}$). Two atoms are adjacent if and only if they are compatible. Each straight line or circumference represents a maximal clique.
• Figure 2: Atom graph $\mathcal{G}_a(\mathcal{Q}_{\mathrm{KCBS}})$. For projectors $\hat{P}$ and $\hat{Q}$, the notation $\neg\hat{P}\neg\hat{Q}$ represents $(\neg\hat{P})\land(\neg\hat{Q})$. Two atoms are adjacent if and only if they are compatible.
• Figure 3: The Boolean vectors $\{\mathbf{b}_i\}_{i=1}^5$ corresponding five types of logically contextual states on $\mathcal{Q}_{\mathrm{KCBS}}$.
• Figure 4: Part 1. The Boolean vectors $\{b_i\}_{i=1}^{10}$ corresponding the logically contextual states on $\mathcal{Q}_{(2,2,2)}$ (omitting the values 1).
• Figure 5: Part 2.

Theorems & Definitions (36)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Definition 4
• Definition 5
• Definition 6
• Theorem 2
• proof