Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity

• Published in: Physics reports Vol. 506; no. 1; pp. 1 - 39
• Main Authors: Liang, Yeong-Cherng, Spekkens, Robert W., Wiseman, Howard M.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.09.2011; Elsevier
• Subjects: Bell inequalities; Complementarity; Contextuality; Exact sciences and technology; Joint measurability; Nonlocality; Physics; Quantum foundations; Contextuality; Bell inequalities; Joint measurability; Nonlocality; Complementarity; Quantum foundations; Operator; Bell theorem; Joint distribution; Bell inequality; Correlations; Deterministic model; Polygon; Quantum theory
• ISSN: 0370-1573; 1873-6270
• DOI: 10.1016/j.physrep.2011.05.001

In 1960, the mathematician Ernst Specker described a simple example of nonclassical correlations, the counter-intuitive features of 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 entrée 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 following: the impossibility of measurement-noncontextual and outcome-deterministic ontological models of quantum theory (the 1967 Kochen–Specker theorem), in particular, the recent state-specific pentagram proof of Klyachko; the impossibility of Bell-local models of quantum theory (Bell’s theorem), especially the proofs by Mermin and Hardy and extensions thereof; the impossibility of a preparation-noncontextual ontological model of quantum theory; 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: a generalization of a theorem by Fine connecting the existence of a joint distribution over outcomes of counterfactual measurements to the existence of a measurement-noncontextual and outcome-deterministic ontological model; a generalization of Klyachko’s proof of the Kochen–Specker theorem from pentagrams to a family of star polygons; a proof of the Kochen–Specker theorem in the style of Hardy’s proof of Bell’s theorem (i.e., one that makes use of the failure of the transitivity of implication for counterfactual statements); a categorization of contextual and Bell-nonlocal correlations in terms of frustrated networks; a derivation of a new inequality testing preparation noncontextuality; some novel results on the joint measurability of POVMs and the question of whether these can be modeled noncontextually. Finally, we emphasize that Specker’s parable of the overprotective seer 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. ► Prior to his work on quantum contextuality, Ernst Specker introduced nonclassical correlations. ► He dramatized these in a parable about an overprotective seer. ► We use this to unify many results on contextuality, Bell-nonlocality, and complementarity. ► Specker’s parable provides a novel type of foil to quantum theory.