TY - JOUR

T1 - Specker's parable of the overprotective seer

T2 - A road to contextuality, nonlocality and complementarity

AU - Liang, Yeong Cherng

AU - Spekkens, Robert W.

AU - Wiseman, Howard M.

N1 - Funding Information:
This project was inspired by Ernst Specker’s talk at the workshop “Information Primitives and Laws of Nature”, which took place at ETH, Zürich in May 2008. On the topic of the joint measurability of POVMs, we thank Robin Blume-Kohout for a motivating discussion and David Pegg for helpful comments. On the topic of nonlocal OS correlations, we acknowledge useful discussions with Ben Toner and Jean-Daniel Bancal. We also thank Allen Stairs for comments on a draft of this article and Lucien Hardy for bringing Ref. [40] to our attention and for suggesting the connection between Specker’s parable and the failure of transitivity of implication in proofs of nonlocality. Finally, we thank Adán Cabello for pointing out the connection between our Kochen–Specker proof based on the failure of transitivity of implication and Clifton’s proof. YCL and HMW acknowledge funding from the Australian Research Council. Part of this work was conducted during visits by YCL and HMW to Perimeter Institute, and RWS to Australia, through the PIAF (Perimeter Institute Australia Foundations) collaboration. Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. YCL was also supported by the Swiss NCCR “Quantum Photonics” and the European ERC-AG QORE.

PY - 2011/9

Y1 - 2011/9

N2 - 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.

AB - 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.

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U2 - 10.1016/j.physrep.2011.05.001

DO - 10.1016/j.physrep.2011.05.001

M3 - Review article

AN - SCOPUS:80051895678

VL - 506

SP - 1

EP - 39

JO - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 1-2

ER -