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Physical Review A

Experimental observation of quantum contextuality beyond Bell nonlocality

Zhenghao Liu, Hui-Xian Meng, Zhen‐Peng Xu, Jie Zhou, Sheng Ye, Qiang Li, Kai Sun, Hong-Yi Su, Adán Cabello, Jing‐Ling Chen, Jin‐Shi Xu, Chuan‐Feng Li, Guang‐Can Guo

Published Oct 22, 2019• 4 citations
Publisher

Abstract

Bell nonlocality and quantum contextuality are two important resources which, however, behave very differently in many tasks where the quantum correlation has an edge over its classical counterpart. In this work, we directly compare these two behaviors, which via the Cabello-Severini-Winter graph-theoretical approach, can be associated with a same exclusivity graph. In particular, we consider the exclusivity graph that leads to ${I}_{3322}$-type inequalities in the Bell and the contextual scenarios, respectively. We find that maximal violation of Bell inequality is around 6.251, and maximal violation of the noncontextuality inequality is around 6.588 for a five-dimensional system and 6.571 for a four-dimensional system, respectively. The results predict a gap of $\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}0.32$ between quantum contextuality and Bell nonlocality. We then present an experimental observation of the gap by employing both the maximally entangled photon pairs from the spontaneous parametric down-conversion process and the single photons encoded into qudits from an intrinsic defect in gallium nitride. Our results will further deepen the understanding of different types of quantum correlations.

DOI
10.1103/physreva.100.042118

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Bell nonlocality and quantum contextuality are two important resources which, however, behave very differently in many tasks where the quantum correlation has an edge over its classical counterpart. In this work, we directly compare these two behaviors, which via the Cabello-Severini-Winter graph-theoretical approach, can be associated with a same exclusivity graph. In particular, we consider the exclusivity graph that leads to I3322{I}_{3322}I3322​-type inequalities in the Bell and the contextual scenarios, respectively. We find that maximal violation of Bell inequality is around 6.251, and maximal violation of the noncontextuality inequality is around 6.588 for a five-dimensional system and 6.571 for a four-dimensional system, respectively. The results predict a gap of \ensuremathΔ\ensuremath≈0.32\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}0.32\ensuremathΔ\ensuremath≈0.32 between quantum contextuality and Bell nonlocality. We then present an experimental observation of the gap by employing both the maximally entangled photon pairs from the spontaneous parametric down-conversion process and the single photons encoded into qudits from an intrinsic defect in gallium nitride. Our results will further deepen the understanding of different types of quantum correlations.
Non-contextuality (NC) and Bell inequalities can be expressed as bounds ΩΩΩ for positive linear combinations SSS of probabilities of events, S≤ΩS \leq ΩS≤Ω. Exclusive events in SSS can be represented as adjacent vertices of a graph called the exclusivity graph of SSS. In the case that events correspond to the outcomes of quantum projective measurements, quantum probabilities are intimately related to the Grötschel-Lovász-Schrijver theta body of the exclusivity graph. Then, one can easily compute an upper bound to the maximum quantum violation of any NC or Bell inequality by optimizing SSS over the theta body and calculating the Lovász number of the corresponding exclusivity graph. In some cases, this upper bound is tight and gives the exact maximum quantum violation. However, in general, this is not the case. The reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in Bell-inequality (and similar) scenarios. An interesting question is whether there is a graph-theoretical concept which accounts for this problem. Here we show that, for any given NNN-partite Bell inequality, an edge-coloured multigraph composed of NNN single-colour graphs can be used to encode the relationships of exclusivity between each party's parts of the events. Then, the maximum quantum violation of the Bell inequality is exactly given by a refinement of the Lovász number that applies to these edge-coloured multigraphs. We show how to calculate upper bounds for this number using a hierarchy of semi-definite programs and calculate upper bounds for I3I_3I3​, I3322I_{3322}I3322​ and the three bipartite Bell inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles.
There are two powerful arguments against the possibility of extending quantum mechanics, the violation of Bell inequalities and the Kochen-Specker theorem, but the connection between the two remains confused. Following the distinctive strategy proposed by Cabello [Phys. Rev. Lett. 127, 070401 (2021)], Bell inequalities can be violated by state-independent contextuality sets. However, the experimental realization of such ideas is challenging as it requires high-dimensional entanglement. Orbital angular momentum provides an unlimited state space and the number of effective dimensions can be readily tailored as required. We performed an experimental test of non-locality based on Bell inequalities from contextuality, using orbital angular momentum entanglement in a bipartite photonic system. Our experiment not only shows a new way to produce non-locality but also connects contextuality and non-locality, two fundamental quantum resources that are critical for quantum computation and secure communication tasks.
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.
Cabello-Severini-Winter and Abramsky-Hardy (building on the framework of Abramsky-Brandenburger) both provide classes of Bell and contextuality inequalities for very general experimental scenarios using vastly different mathematical techniques. We review both approaches, carefully detail the links between them, and give simple, graph-theoretic methods for finding inequality-free proofs of nonlocality and contextuality and for finding states exhibiting strong nonlocality and/or contextuality. Finally, we apply these methods to concrete examples in stabilizer quantum mechanics relevant to understanding contextuality as a resource in quantum computation.
Great effort has been made in the investigation of contextual correlations between compatible observables due to their both fundamental and practical importance. The graph-theoretic approach to correlate events has been proved to be an effective method in the characterization of quantum contextuality, which implies that quantum violations of noncontextual inequalities derived in the noncontextual hidden-variable models should be achievable. Finding experimentally more friendly and theoretically more powerful noncontextual inequalities associated with specific graphs is of particular interest. Here we consider Platonic graphs to vindicate the quantum maximum predicted by graph theory and test the quantum violation against the mixedness of the state. Among these solids we refer particularly to the icosahedron to build the experiment, as it gives rise to the largest quantum-classical difference. The contextual correlations are demonstrated on quantum four-dimensional states encoded in the spatial modes of single photons generated from a defect in a bulk silicon carbide. Our results shed new light on the conflict between quantum and classical physics and may promote deep understanding of the connection between quantum theory, graph theory, and operator theory.
Which is the simplest logical structure for which there is quantum nonlocality? We show that there are only three bipartite Bell inequalities with quantum violation associated with the simplest graph of relationships of exclusivity with a quantum-classical gap. These are the most elementary logical Bell inequalities. We show that the quantum violation of some well-known Bell inequalities is related to them. We test the three Bell inequalities with pairs of polarization-entangled photons and report violations in good agreement with the quantum predictions. Unlike other experiments testing noncontextuality inequalities with pentagonal exclusivity, the ones reported here are free of the compatibility loophole.