The 2016 quantum pigeonhole principle shows three particles can occupy only two boxes with no two ever found together
In a 2026 oral-history interview, Yakir Aharonov explains the mechanism behind Aharonov, Colombo, Popescu, Sabadini, Struppa, and Tollaksen (2016, PNAS 113:532): starting from a spin pre- and post-selection that gives an imaginary weak value for σz, two non-entangled particles with identical pre- and post-selected states yield a weak value of −1 for the product of their σz measurements. Reinterpreted as a which-box observable, this implies that for three such particles, every pair fails to occupy the same box — even though only two boxes exist for three particles to be sorted into. The result is a counterintuitive consequence of pre- and post-selection in the two-state vector formalism rather than of entanglement, since the particles in question are explicitly not entangled.
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“Now you have two particles, separated from each other, but identical in the same pre- and post-selected spin states. They are not entangled, so the weak value of the product equals the product of the weak values, so the weak value of the product of the z component of the spin equals −1.”
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