We investigate the set of qutrit states in terms of symmetric states of two qubits that violate the minimal contextual inequality, namely the Klyachko-Can-Binicoglu-Shumovsky (KCBS) inequality. The physical system that provides a natural framework for this problem is a biphoton, which consists of two photons in the same spatiotemporal mode and whose effective polarization behaves as a three-level quantum system. The relationship between the KCBS contextual inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality is investigated. We find that every biphotonic state that is contextual with respect to KCBS is nonlocal as per the CHSH test when the two photons are apart, but the converse is not true.
We investigate the set of qutrit states in terms of symmetric states of two qubits that violate the minimal contextual inequality, namely the Klyachko-Can-Binicoglu-Shumovsky (KCBS) inequality. The physical system that provides a natural framework for this problem is a biphoton, which consists of two photons in the same spatiotemporal mode and whose effective polarization behaves as a three-level quantum system. The relationship between the KCBS contextual inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality is investigated. We find that every biphotonic state that is contextual with respect to KCBS is nonlocal as per the CHSH test when the two photons are apart, but the converse is not true.
Quantum contextuality is a nonintuitive property of quantum mechanics that distinguishes it from any classical theory. A complementary quantum property is quantum nonlocality, which is an essential resource for many quantum information tasks. Here we experimentally study the contextual and nonlocal properties of polarization biphotons. First, we investigate the ability of the biphotons to exhibit contextuality by testing the violation of the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky (KCBS) inequality. In order to do so, we used the original protocol suggested in the KCBS paper and adjusted it to the real scenario, where some of the biphotons are distinguishable. Second, we transmitted the biphotons through different unital channels with a controlled amount of noise. We measured the decohered output states and demonstrated that the ability to exhibit quantum contextuality using the KCBS inequality is more fragile to noise than the ability to exhibit nonlocality.
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.
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.
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.
We present a study of quantum contextuality of three-dimensional mixed states for the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky (KCBS) and the Kurzy\ifmmode \acute{n}\else \'{n}\fi{}ski-Kaszlikowski (KK) noncontextuality inequalities. For any class of states whose eigenvalues are arranged in decreasing order, a universal set of measurements always exists for the KK inequality, whereas none exists for the KCBS inequality. This difference can be reflected from the spectral distribution of the overall measurement matrix. Our results would facilitate the error analysis for experimental setups, and our spectral method in the paper combined with graph theory could be useful in future studies on quantum contextuality.
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 p2, associated with the occupation of the qutrit subsystem in the ∣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). 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.