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

TL;DR

This paper explores Specker's parable as a narrative framework to unify key quantum foundational concepts like contextuality, nonlocality, and complementarity, presenting new theoretical results and inequalities that deepen understanding of quantum nonclassical correlations.

Contribution

It offers a comprehensive analysis connecting Specker's parable to major quantum theorems and introduces novel generalizations, inequalities, and classifications related to contextuality and joint measurability.

Findings

Generalization of Fine's theorem linking joint distributions to noncontextual models

Extension of Klyachko's Kochen-Specker proof

New inequality for testing preparation noncontextuality

Abstract

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. Finally, we emphasize that Specker's parable provides a novel type of foil to quantum theory, challenging us to explain why the particular sort of contextuality and complementarity embodied therein does not arise in a quantum world.