Hardy-type paradoxes provide elegant, inequality-free proofs of quantum contextuality. We introduce a unified logical formalism for these paradoxes, termed logical Hardy-type paradoxes. For any finite quantum scenario of ideal measurements, we prove that the existence of a logical Hardy-type paradox is equivalent to logical contextuality. Specifically, 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. We analyze logical Hardy-type paradoxes in the Mansfield and Klyachko-Can-Binicioglu-Shumovsky (KCBS) scenarios. In the KCBS scenario, we show that there is exactly one type of logical Hardy-type paradox, achieving SP\approx 10.56% for a specific parameter setting.
Hardy's paradox is an all-versus-nothing logical contradiction for bipartite quantum systems: three jointly satisfiable conditions on measurement probabilities each force a fourth probability to zero by classical logic, yet quantum mechanics permits that fourth probability to be strictly positive. Unlike Bell inequalities, no statistical averaging is required — the contradiction is exact and logical. This paper presents a five-part numerical study connecting Hardy's paradox to the non-associative geometry of the Fano plane
PG(2,2) and the octonion associator
A(ei,ej,ek)=(eiej)ek−ei(ejek). Part 1 finds the Hardy state by constrained optimisation (Nelder-Mead, 100 initialisations), confirming all three zero-conditions to within
10−6 and the impossible event
P(Z+,Z+)=0.10>0 with residual
∼10−36. Part 2 shows that the Fano-SYK ground state restricted to the measurement node pair
0,1 is entangled, with von Neumann entropy
S=0.8762 ebits, connecting the Fano dynamical structure to the Hardy contextual structure. Part 3 sweeps a one-parameter entangled state family and demonstrates that the Hardy impossible-event probability and the CHSH witness are correlated (Pearson
r=0.982), both vanishing at the separable limit. Part 4 verifies the full octonion associator table:
∣A(ei,ej,ek)∣=0 for all 7 Fano lines (associative triples) and
∣A∣=2 for all 28 non-Fano triples, exact to
<10−14. Part 5 maps the Hardy measurement settings
Z,X to octonion directions
e0,e1, which lie on the unique Fano line
0,1,3, and argues that entanglement forces the bipartite system to sample non-Fano directions in
R7⊗R7 where no consistent classical truth-value assignment exists. The central claim — stated as the Fano-Contextuality Conjecture — is that a measurement triple admits a consistent joint classical value assignment if and only if the corresponding octonion directions lie on a Fano line (
∣A∣=0). Hardy contextuality is diagnosed as the consequence of entanglement distributing quantum weight into the non-associative region of measurement space. A formal proof connecting this partition to the Kochen-Specker theorem is identified as the primary open problem. This is the first paper in the ASA Quantum Foundations series (Portfolio F), accompanied by the Spacelike Associator Paradox (doi:10.5281/zenodo.20058013) and the Fano Monogamy Paradox (doi:10.5281/zenodo.20058092). Working paper in the Adelic Simplicial Architecture (ASA) series.
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.
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.
The strength of quantum contextuality is closely related to quantum computation power. Yu-Oh set is the minimal quantum system with state-independent contextuality(SIC). However, its strength of the contextuality has not been taken into account. In this paper, we present a general method to determine whether there is a quantum state with strong contextuality in the quantum system composed of rank-one projectors. Based on this method, we conclude that Yu-Oh set does not have quantum states with strong contextuality. This indicates that strong contextuality and SIC are mutually independent.
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.