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A two-setting nonlocality argument in which several zero-probability events forbid a further coincidence in any deterministic local hidden-variable model, but quantum theory assigns that coincidence positive probability. For binary indicators, suppose $C_{+}^R{C}_{-}^R=0$, $D_{+}^I=1\Rightarrow C_{-}^R=1$ and $D_{-}^I=1\Rightarrow C_{+}^R=1$. Any assignment with $D_{+}^I{D}_{-}^I=1$ would violate the first relation. A quantum implementation can nevertheless give such a coincidence. Real imperfect experiments require an inequality that accounts for their nonzero error probabilities. Lucien Hardy gave a two-particle inequality-free argument in 1992.^[ref]

Lucien Hardy gave an argument against local hidden-variable theories using logical exclusions among two-particle measurement outcomes, rather than a statistical Bell inequality. The structure combines outcomes forbidden by quantum mechanics with another outcome that remains possible: assigning all the relevant local results in advance cannot satisfy these requirements together. His 1992 paper also examined an additional assumption about Lorentz-invariant elements of reality, keeping the role of that assumption explicit in its comparison with quantum mechanics.^[ref]

Homepage: perimeterinstitute.ca/people/lucien-hardy