The quantum pigeonhole effect as a new form of Bell's theorem without inequality
Yanlong Sun, Haiqing Wei, Jack W. Smith, Hongbin Wang
TL;DR
The paper addresses an apparent paradox in the quantum pigeonhole effect (QPE), which seems to violate the classical pigeonhole principle. It shows that the effect derives from quantum contextuality and reframes weak measurements within a bipartite density-operator framework, deriving Bell-type inequalities from pigeonhole logic and demonstrating that QPE constitutes Bell's theorem without inequalities. Key contributions include two context-based Bell inequalities, a bipartite reformulation of the weak-measurement scheme, and a Pauli-decomposition perspective showing that certain post-selected states are mixtures of Bell states yet separable, all pointing to contextuality as the source of the paradox. The work clarifies the foundational role of contextuality in QPE, linking it to Bell-type no-go theorems and offering interpretational insights aligned with QBism and Kochen–Specker-type constraints.
Abstract
The quantum pigeonhole effect (QPE) appears to contradict the classical pigeonhole principle by allowing three quantum particles distributed between two boxes to exhibit no pairwise coincidence. We show that this effect does not signal a breakdown of classical counting, but instead arises from quantum contextuality. By deriving Bell-type inequalities directly from the pigeonhole principle and reformulating the weak-measurement protocol within a bipartite density-operator framework, we demonstrate that the QPE is a form of Bell's theorem without inequalities. The apparent paradox reflects the impossibility of non-contextual eigenvalue assignments rather than a violation of classical combinatorial logic.