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arXiv:2601.01445

The Equivalence between Hardy-type paradox and Logical Contextuality

Songyi Liu, Yongjun Wang, Baoshan Wang, Chang He, Yunyi Jia

quant-phSubmitted Jan 4, 2026• Announced Jan 6, 2026• 0 citations• Revised Jan 6, 2026 (v2)
v1 · Jan 4, 2026v2 · Jan 6, 2026v3 · Jul 14, 2026
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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.

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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.
We present a logical type of proof of contextuality for a two-qubit state. We formulate a paradox that cannot be used by a two-qubit system with local measurements to test contextuality while it is possible by using entanglement measurements. With our scheme we achieve ${p}_{\mathrm{Hardy}}\ensuremath{\approx}0.167$. Our approach uses graph theory and the exclusivity principle to give an interpretation of a logical type of proof of quantum correlations. We review the Hardy paradox and find a connection to the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky inequality. We apply the same method to build a paradox based on the Clauser-Horne-Shimony-Holt inequality.
Quantum contextuality is a fundamental nonclassical property of quantum systems, regarded as a key resource that demonstrates the computational and informational advantages of quantum over classical systems. Our present work aims to construct Hardy's paradoxes, a set of possibilistic conditions witnessing contextuality, for Yu-Oh set, which is the state-independent contextual quantum system with the least number of vectors. To achieve the aim, we systematically enumerate all logically contextual pure states on Yu-Oh set, and theoretically prove that no mixed states in this scenario are logically contextual. Based on the identified logically contextual quantum states, we construct 12 Hardy's paradoxes with identical success probability SP=11.1%. Furthermore, we present corresponding observables to experimentally witness these Hardy's paradoxes.
In the present master's thesis we study logical contextuality and inequality-free proofs of quantum contextuality and nonlocality.The former concept can be understood as the possibilistic version of contextuality, while the latter refers to proofs of quantum probabilistic non-classicalities that are not based on violation of any noncontextuality (or Bell) inequality.The present work aims to build a bridge between these two concepts through what we call possibilistic paradoxes, which are sets of possibilistic conditions whose occurrence implies contextuality.As our main result, we demonstrate the existence of possibilistic paradoxes whose occurrence is a necessary and sufficient condition for logical contextuality in a very important class of scenarios.We discuss some consequences arising from the completeness of such possibilistic paradoxes.Finally, we review some inequality-free proofs under the light of the developed formalism.
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.
Contextuality and nonlocality are nonclassical properties exhibited by quantum statistics whose implications profoundly impact both the foundations and applications of quantum theory. In this paper we provide some insights into logical contextuality and inequality-free proofs. The former can be understood as the possibility version of contextuality, while the latter refers to proofs of quantum contextuality and nonlocality that are not based on violations of some noncontextuality (or Bell) inequality. By ``possibilistic'' we mean a description in terms of possibilities for the outcomes, which are Boolean variables assuming value one when the corresponding probability is strictly larger than zero and zero otherwise. The present work aims to build a bridge between these two concepts from what we call possibilistic paradoxes, which are sets of possibilistic conditions whose occurrence implies contextuality and nonlocality. As the main result, we demonstrate the existence of possibilistic paradoxes whose occurrence is a necessary and sufficient condition for logical contextuality in a very important class of scenarios. Finally, we discuss some interesting consequences arising from the completeness of these possibilistic paradoxes.
Non-locality and contextuality are key features of quantum mechanics that distinguish it from classical physics. We aim to develop a deeper, more structural understanding of these phenomena, underpinned by robust and elegant mathematical theory with a view to providing clarity and new perspectives on conceptual and foundational issues. A general framework for logical non-locality is introduced and used to prove that 'Hardy's paradox' is complete for logical non-locality in all (2,2,l) and (2,k,2) Bell scenarios, a consequence of which is that Bell states are the only entangled two-qubit states that are not logically non-local, and that Hardy non-locality can be witnessed with certainty in a tripartite quantum system. A number of developments of the unified sheaf-theoretic approach to non-locality and contextuality are considered, including the first application of cohomology as a tool for studying the phenomena: we find cohomological witnesses corresponding to many of the classic no-go results, and completely characterise contextuality for large families of Kochen-Specker-like models. A connection with the problem of the existence of perfect matchings in k-uniform hypergraphs is explored, leading to new results on the complexity of deciding contextuality. A refinement of the sheaf-theoretic approach is found that captures partial approximations to locality/non-contextuality and can allow Bell models to be constructed from models of more general kinds which are equivalent in terms of non-locality/contextuality. Progress is made on bringing recent results on the nature of the wavefunction within the scope of the logical and sheaf-theoretic methods. Computational tools are developed for quantifying contextuality and finding generalised Bell inequalities for any measurement scenario which complement the research programme. This also leads to a proof that local ontological models with `negative probabilities' generate the no-signalling polytopes for all Bell scenarios.