Specker's Parable of the Over-protective Seer: A Road to Contextuality, Nonlocality and Complementarity (PLUS AN ERRATUM)

Understand

In 1960, the mathematician Ernst Specker described a simple example of nonclassical correlations which he dramatized using a parable about a seer who sets an impossible prediction task to his daughter's suitors.

• We revisit this example here, using it as an entree to three central concepts in quantum foundations: contextuality, Bell-nonlocality, and complementarity.
• Specifically, we show that Specker's parable offers a narrative thread that weaves together a large number of results, including: the impossibility of measurement-noncontextual and outcome-deterministic ontological models of quantum theory (the Kochen-Specker theorem), in particular the proof of Klyachko; the impossibility of Bell-local models of quantum theory (Bell's theorem), especially the proofs by Mermin and Hardy; the impossibility of a preparation-noncontextual ontological model of quantum theory; and the existence of triples of positive operator valued measures (POVMs) that can be measured jointly pairwise but not triplewise.
• Along the way, several novel results are presented, including: a generalization of a theorem by Fine connecting the existence of a joint distribution over outcomes of counterfactual measurements to the existence of a noncontextual model; a generalization of Klyachko's proof of the Kochen-Specker theorem; a proof of the Kochen-Specker theorem in the style of Hardy's proof of Bell's theorem; a categorization of contextual and Bell-nonlocal correlations in terms of frustrated networks; a new inequality testing preparation noncontextuality; and lastly, some results on the joint measurability of POVMs and the question of whether these can be modeled noncontextually.