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

Abstract

In 1960, the mathematician Ernst Specker described a simple example of nonclassical correlations, the counterintuitive 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 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; 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 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; and lastly, 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 over-protective 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.

Parity Proofs of the Kochen-Specker Theorem Based on the 24 Rays of Peres

“…The 24 Peres rays arise as the simultaneous eigenstates of the six triads of commuting observables for a pair of qubits shown in Table 1. The rays form 24 bases of four mutually orthogonal rays that are shown within the circular boxes in Fig.1.…”

“…In an interesting development, Klyachko et al [23] have shown how the failure of the transitivity of implication for a closed loop of observables can be used to prove the K-S theorem. Two recent articles [24,25] expand on this theme and also explore the connection between the K-S theorem and a number of quantum paradoxes. In this connection it may be worth pointing out (see Appendix 2) that the Peres-Mermin square permits a simple state-independent demonstration to be given of the failure of the transitivity of implication for the observables occurring in it.…”

Violation of Bell inequalities from $$S_4$$ S 4 symmetry: the three orbits case

“…Their various forms have been derived [3][4][5][6][7][8][9] characterized by the number of parties, measurement settings and outcomes for each measurement (for a review, see [10,11]). …”

Bell inequalities for three systems and arbitrarily many measurement outcomes

“…We note, however, that our inequalities do not follow directly from the inequalities presented in Ref. [47] since the latter involve expectation values of the product of observables from the same party (and are thus better seen as examples of noncontextual inequalities [48]). There are nine inequivalent classes of facet-defining tripartite, two-input, three-output Bell inequalities which are symmetrical with respect to arbitrary permutations of parties and that involve only the sum of the outputs.…”