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arXiv:2412.04713 · Journal of Physics A Mathematical and Theoretical

States violating both locality and noncontextuality inequalities in quantum theory

Yuichiro Kitajima

Published Mar 6, 2025• 1 citation
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Abstract

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.

DOI
10.1088/1751-8121/adbdb1

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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 quantum no-disturbance principle imposes a fundamental monogamy relation on quantum contextuality and quantum nonlocality. We experimentally demonstrated the simulation of this monogamy relation on a ququart-qubit system using three NMR qubits. We constructed experimental tests of the violation of the Klyachko-Can-Binicioglu-Shumovski (KCBS) inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality. We measured the average CHSH and KCBS values at the single point where the quantum boundary meets the no-disturbance boundary, at which point the tradeoff inequality becomes an equality. The tradeoff relation is always satisfied, implying that the violation of either the KCBS or the CHSH inequality automatically forbids the violation of the other.
In this work, we show that the well-known Bell inequality in the Clauser-Horne-Shimony-Holt (CHSH) formulation serves not only as a non-locality test but also as a tool to prove KCBS-type contextuality. For this purpose, we investigate the symmetric family of two-qubit states that can be mapped to qutrit states (three-level quantum states) and show that they exhibit a contextual description, as evidenced by the KCBS-type inequality. Later, we apply the CHSH test to them and find a new non-contextuality bound for this test. This shows that the CHSH inequality can be modified to serve as a KCBS-type contextuality test by changing the limit. Also, the number of measurements required is four, fewer than the KCBS test's five. We discuss the new bound concerning the monogamy problem.
Contextuality and nonlocality are distinct manifestations at the foundation of quantum mechanics, yet their coexistence within a single quantum state remains subtle. In a hybrid CHSH--KCBS scenario involving the entanglment of a qubit and a qutrit, the qutrit supports the KCBS contextuality test, and the CHSH nonlocality arises from correlations between the qubit and qutrit. Here, we derive the analytical closed-form expressions for both inequalities and also simulate this physics on a quantum circuit. We show that contextuality is governed solely by a population parameter p2p_2p2​, associated with the occupation of the qutrit subsystem in the ∣2⟩|2\rangle∣2⟩ level, which plays a distinguished role in the KCBS structure. In contrast, nonlocality depends irreducibly on coherence, involving both amplitudes and phases encoded in parameters (Xi,Yi)(X_i, Y_i)(Xi​,Yi​). This separation of physical resources reveals parameter regimes that optimize KCBS violation while suppress CHSH violation, and vice versa. As a result, the optimal regions do not overlap, and coexistence is restricted to a narrow intermediate regime in parameter space.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.
Contextuality and measurement incompatibility are two fundamental aspects of nonclassicality, and their manifestations in observed quantum correlations are often deeply interconnected. Recently, measurement incompatibility has been studied in connection with nonlocality, particularly in terms of their robustness under various quantum channels. This line of investigation helps establish a connection between the channels that break nonlocality and those that break incompatibility. In this study, we focus on an asymmetric bipartite Bell scenario involving three and four inputs on Alice and Bob sides, respectively, with each of these inputs having dichotomous outcomes. Under the assumption of locality, the observed statistics in this asymmetric scenario obeys the Elegant Bell inequality (EBI). Here, we use a different version of the EBI that relies on the assumption of the preparation noncontextuality. By taking the violation of this noncontextual version of EBI as a witness of preparation contextuality we establish a connection between the channels that break contextuality and the channels that break triple-wise measurement incompatibility. Our results suggest that any channel which breaks EBI contextuality will also break Clauser-Horne-Shimony-Holt (CHSH) nonlocality; however, the reverse does not hold. We also show that a depolarising channel that breaks N-wise incompatibility can also break a certain form of contextuality, witnessed by a generalised inequality involving N measurements on one wing of a bipartite Bell scenario.