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

The structural connections between the Kochen-Specker (KS) theorem, pre- and post-selection (PPS) paradoxes, and anomalous weak values are explored in detail. All PPS paradoxes, such as the 3-box paradox, the Quantum Cheshire Cat, and the Quantum Pigeonhole principle, construct a particular type of ontological model that assigns an eigenvalue to each observable (independent of context) of a system such that these assignments are consistent with the PPS. It is shown that such an ontological model must be explicitly contextual in the sense of the KS theorem, or otherwise implies either a restriction on free random choice or explicitly retrocausal behavior. We call such models PPS-contextual. The structure of each paradox is always such that there are particular contexts of mutually commuting observables that violate the product rule or sum rule, when the ontological model is extended to include observables that are not measured during the experiment. These paradoxes are counterfactual, in the sense that they are not directly observed, and also because the product and sum rules are always obeyed by projective measurements in actual experiments. It is shown that by adopting an alternate ontological model, where all hidden variables are weak values (which are not always eigenvalues, but obey the sum rule by definition), the same contexts that presented the original paradox must also contain observables with anomalous weak values. These anomalous weak values are not counterfactual because they can be probed through weak measurements on an ensemble of identically pre- and post-selected states, allowing this localized signature of KS contextuality to be experimentally observed. The weak values of all observables of a system can in principle be measured during an experiment, making this model a promising candidate for describing PPS-contextual ontological `elements of reality.'