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

Abstract

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.' As a related issue, we show using the mathematical properties of weak values, that any KS set can be used to ensure that the Mean King always wins his game against the stranded physicist. *

Quantum reality with negative-mass particles

“…There are a number of well-known PPS paradoxes, including the 3-box, the quantum Cheshire cat, and the quantum Pigeonhole Effect, all of which are demonstrations of quantum contextuality ( 2 , 8 , 17 , 18 , 28 – 30 ). We show here that all PPS paradoxes belong to a single general family of extended -box paradoxes.…”

“…We call this the ABL interpretation, and it gives rise to various logical PPS-paradoxes, which are directly connected to Kochen–Specker-type ( 2 ) quantum contextuality. Because of these logical paradoxes, there is no consistent physical interpretation of such PPS situations, which has led to various ad hoc explanations of their meaning, as in the cases of the 3-box paradox ( 3 , 4 ), the quantum Cheshire cat ( 5 , 6 ), and the quantum pigeonhole effect ( 7 – 9 ), among others.…”

Observing the quantum Cheshire cat effect with noninvasive weak measurement

“…It is noteworthy that the negative weak value of hΠ l Π V i w is anomalous in the sense that it lies outside the eigenvalue spectrum of the projectorΠ l Π V . The anomalous weak value has been regarded as a quantum signature as they are connected to the quantum contextuality [29][30][31][32][33][34][35] and the violation of classical macroscopic realism [36][37][38] .…”

Contextuality, coherences, and quantum Cheshire cats

“…While [24,25] mention in passing that the quantum Cheshire cat scenario is equivalent to a contextual scenario known as the 2-qubit Peres-Mermin square, this has not yet actually been shown, nor has the scenario been fully analysed using contextuality theory. We seek to correct this oversight.…”