Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values

Abstract

The Kochen–Specker (KS) theorem shows that noncontextual hidden variable models of reality that allow random choice are inconsistent with quantum mechanics. Such noncontextual models predict certain outcomes for specific experiments that are not observed in practice, and this is how the theorem is proved. A realist hidden-variable model suggested by the Aharonov–Bergmann–Lebowitz reformulation of quantum mechanics is introduced to explain why those outcomes are never observed. Just as the KS theorem requires them due to noncontextuality, this model requires independent truth-value assignments for each observable, but now allows that the entire set of assignments depends on both the pre-selected and post-selected quantum states in a time-symmetric manner. Using sets that prove the KS theorem, along with pre- and post-selected states, we find that particular projectors within the set cannot be assigned logically consistent truth-values, and furthermore that the weak values of these projectors have a corresponding signature. The contextual behavior has effectively been confined to a particular context by the pre- and post-selection. This inconsistency is never observed because it is never in the pre-selected or post-selected quantum state, but its signature can be experimentally revealed through weak measurements. We also show that for specific cases where the logical inconsistency is in a ‘classical basis,’ this gives rise to the quantum pigeonhole effect. As a related issue, we show using weak values that any KS set can be used to ensure that the Mean King always wins the game.