Measurement scenarios containing events with relations of exclusivity represented by pentagons, heptagons, nonagons, etc., or their complements are the only ones in which quantum probabilities cannot be described classically. Interestingly, quantum theory predicts that the maximum values for any of these graphs cannot be achieved in Bell inequality scenarios. With the exception of the pentagon, this prediction remained experimentally unexplored. Here we test the quantum maxima for the heptagon and the complement of the heptagon using three- and five-dimensional quantum states, respectively. In both cases, we adopt two different encodings: linear transverse momentum and orbital angular momentum of single photons. Our results exclude maximally noncontextual hidden-variable theories and are in good agreement with the maxima predicted by quantum theory.
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.
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-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.
Leggett inequality states that nonlocal hidden-variable models might still be incompatible with the predictions of quantum physics. However, its theoretical and experimental demonstration is only in the scenario of 2-dimensional maximally entangled systems. An open question remains as to whether the Leggett inequality can be violated by nonmaximally entangled states. Here, we answer this question both in theory and experiment. Specifically, from the point of view of geometry, we theoretically map the problem of maximizing the correlation measure in the Leggett inequality to maximizing the sum of an ellipse’s diameter and semi-diameter axes, accordingly, demonstrating that the violation of the Leggett inequality requires a more robust entanglement than that of Bell’s theory. Experimentally, by leveraging the controllable photonic orbital angular momentum entanglement, we demonstrate the violation of Leggett-type inequalities by more than 8.7 and 4.5 standard deviations under concurrence C = 0.95 and 0.9, respectively. Our observations indicate that, the requirement for quantum correlation should be increased to exclude a particular class of non-local hidden variable theories that abide by Leggett’s model, providing insights into the boundaries of quantum correlation and the limitations imposed by non-local hidden variables.
Non-contextuality (NC) and Bell inequalities can be expressed as bounds
Ω for positive linear combinations
S of probabilities of events,
S≤Ω. Exclusive events in
S can be represented as adjacent vertices of a graph called the exclusivity graph of
S. 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
S 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
N-partite Bell inequality, an edge-coloured multigraph composed of
N 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
I3,
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.
Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.