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Journal of the Japan Association for Philosophy of Science

Contextuality in Algebraic Quantum Theory

Yuichiro Kitajima

Published Jan 1, 2018• 0 citations
Publisher

Abstract

Klyachko, Can, Binicioğlu, and Shumovsky (KCBS) have introduced an inequality which holds if there is a noncontextual hidden variable theory. It is called KCBS inequality. Its violation shows a contradiction between predictions of quantum theory and noncontextual hidden variable theories. Bell-Kochen-Specker (BKS) theorem also shows that there are no noncontextual hidden variable theories in quantum theory by showing that there is no value assignment on a set of all observables. Thus, its approach is different from KCBS inequality. In the present paper, we show a logical relation between KCBS inequality and a value assignment, and examine a consequence of this logical relation.

DOI
10.4288/kisoron.45.1-2_23

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Klyachko, Can, Binicioğlu, and Shumovsky (KCBS) have introduced an inequality which holds if there is a noncontextual hidden variable theory. It is called KCBS inequality. Its violation shows a contradiction between predictions of quantum theory and noncontextual hidden variable theories. Bell-Kochen-Specker (BKS) theorem also shows that there are no noncontextual hidden variable theories in quantum theory by showing that there is no value assignment on a set of all observables. Thus, its approach is different from KCBS inequality. In the present paper, we show a logical relation between KCBS inequality and a value assignment, and examine a consequence of this logical relation.
The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.
Klyachko–Can–Binicioglu–Shumovsky (KCBS) inequality is a Bell-like inequality,the violation of which can be used to confirm the existence of quantum contextuality.However,the imperfection of detection efficiency may cause the so-called loophole in actual KCBS’s experiments.We derive an alternative KCBS inequality to deal with the loophole in actual KCBS’s experiments.We prove that if the experimental data violate this KCBS inequality,the loophole-free violation of the original KCBS inequality will occur.We show that the minimum detection efficiency needed for a loophole-free violation of the KCBS inequality is about 0.9738.
Klyachko—Can—Binicioğlu—Shumovsky (KCBS) inequality is a Bell-like inequality, the violation of which can be used to confirm the existence of quantum contextuality. However, the imperfection of detection efficiency may cause the so-called loophole in actual KCBS's experiments. We derive an alternative KCBS inequality to deal with the loophole in actual KCBS's experiments. We prove that if the experimental data violate this KCBS inequality, the loophole-free violation of the original KCBS inequality will occur. We show that the minimum detection efficiency needed for a loophole-free violation of the KCBS inequality is about 0.9738.
Abstract The Clauser–Horne–Shimony-Holt (CHSH) inequality tests locality in quantum theory and is recognized as one of Bell’s inequalities. In contrast, the Klyachko–Can–Binicioğlu–Shumovsky (KCBS) inequality tests noncontextuality, which refers to the idea that measurement outcomes are determined by intrinsic properties independent of other measurements that could be performed simultaneously. While individual quantum states can violate these inequalities separately, it was believed that no single state could violate both simultaneously. This relationship suggests a trade-off between nonlocality and contextuality: if one is violated, the other cannot be. Such a relationship is known as a monogamy relation. Xue et al demonstrated, however, that simultaneous violations of both the CHSH and KCBS inequalities are possible with specific choices of observables. This finding challenges the universal validity of the monogamy relation. They further showed that numerous scenarios exist in which both the CHSH inequality and a noncontextuality inequality involving more than five observables are violated. Nevertheless, the number of such scenarios is finite. In this paper, we extend these findings in two significant ways. First, we show that infinitely many scenarios exist where both the CHSH inequality and a noncontextuality inequality involving an odd number of observables are violated. This demonstrates that the monogamy relation between the CHSH inequality and the noncontextuality inequality fails in infinitely many cases. Second, we identify quantum states that violate both the KCBS inequality and a locality inequality distinct from the CHSH inequality. Our results highlight that the so-called ‘monogamy relation’ between nonlocality and contextuality is not universally valid. Instead, it appears that both phenomena can coexist under broader conditions than initially expected, thus challenging the conventional viewpoint and paving the way for new theoretical and experimental research.
Bell nonlocality and Kochen-Specker contextuality are two remarkable nonclassical features of quantum theory, related to strong correlations between outcomes of measurements performed on quantum systems. Both phenomena can be witnessed by the violation of certain inequalities, the simplest and most important of which are the Clauser-Horne-Shimony-Holt (CHSH) and the Klyachko-Can-Binicioğlu-Shumovski (KCBS), for Bell nonlocality and Kochen-Specker contextuality, respectively. It has been shown that, using the most common interpretation of Bell scenarios, quantum systems cannot violate both inequalities concomitantly, thus suggesting a monogamous relation between the two phenomena. In this Letter, we show that the joint consideration of the CHSH and KCBS inequalities naturally calls for the so-called generalized Bell scenarios, which, contrary to the previous results, allows for joint violation of them. In fact, this result is not a special feature of such inequalities: We provide very strong evidence that there is no monogamy between nonlocality and contextuality in any scenario where both phenomena can be observed. We also implement a photonic experiment to test the synchronous violation of both CHSH and KCBS inequalities. Our results agree with the theoretical predictions, thereby providing experimental proof of the coexistence of Bell nonlocality and contextuality in the simplest scenario, and lead to novel possibilities where both concepts could be jointly employed for quantum information processing protocols.
The Kochen-Specker theorem shows the incompatibility of noncontextual hidden variable theories with quantum mechanics. Quantum contextuality is a more general concept than quantum non-locality which is quite well tested in experiments using Bell inequalities. Within neutron interferometry we performed an experimental test of the Kochen-Specker theorem with an inequality, which identifies quantum contextuality, by using spin-path entanglement of single neutrons. Here entanglement is achieved not between different particles, but between degrees of freedom of a single neutron, i.e., between spin and path degree of freedom. Appropriate combinations of the spin analysis and the position of the phase shifter allow an experimental verification of the violation of an inequality derived from the Kochen-Specker theorem. The observed violation 2.291±0.008≰1 clearly shows that quantum mechanical predictions cannot be reproduced by noncontextual hidden variable theories.