Synchronous Observation of Bell Nonlocality and State-Dependent Contextuality

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.

Figure 1

Representation of the compatibility relations between all measurements in the experiment. Each measurement is represented by a vertex, and vertices connected by an edge represent compatible measurements. Both measurements of Alice (white vertices) are compatible with all measurements of Bob (black vertices); compatibility of measurements of Bob is represented by a pentagon. Sets of measurements that are two-by-two compatible are jointly compatible (e.g., $\{A_0,B_0,B_1\}$).

Figure 2

Illustration of the experimental setup. Polarization-entangled photon pairs are generated via type-I spontaneous parametric down-conversion where two joint $\beta$-BBO crystals are pumped by a continuous wave diode laser. Qubit is encoded in the horizontal and vertical polarizations of one photon of each pair, while qutrit is encoded in both polarizations and spatial modes of the other photons of the entangled pairs, which are split in different paths dependent on their polarizations via a beam displacer (BD). For Alice, observables $A_i$ are measured via standard polarization measurements using a half-wave place (HWP) and a BD. For Bob, cascade Mach-Zehnder interferometers for sequentially measuring observables $B_j$ and $B_{(j+1)\text{mod}5}$ are used to test the KCBS inequality.

Figure 3

Experimental results. The measurements in Eqs. (11) and (12) and the one-parameter family of states in Eq. (13) lead to the solid (red) line. Experimental data of ${\alpha }_{\mathrm{CHSH}}$ and ${\beta }_{\mathrm{KCBS}}$ for specific values of the parameter $\varphi$ are represented by the black dots and compared to their theoretical predictions (red circles). The points can be separated in three sets: Points 1–4 exhibit only contextuality; points 5–7 exhibit both nonlocality and contextuality; and points 8–11 exhibit only nonlocality. Error bars are due to the statistical uncertainty in photon-number counting. The traced (blue) curve is an outer bound to the set of quantum behaviors calculated by means of the Navascués-Pironio-Acín (NPA) hierarchy [42].