J. S. Bell’s well-known proofs of inequalities (and related work) are shown to be invalidated by two counter-arguments (-examples) that are based on Einstein-local propositions: Bell-type inequalities of Einstein-Podolsky-Rosen experiments must include, as virtually all physical theories do, elements of physical reality and their mathematical representations that relate to continua as opposed to exclusively finite numbers. Furthermore, Bell-type inequalities must be valid for all possible experimental geometries that lead to the quantum result. Based on these propositions, violations of Bell-type inequalities are demonstrated without violating Einstein locality, without conspiracy type theories and even for the case that all known “loopholes” are closed.
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.
The
I3322 inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for
I3322, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture.
An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.