Characterising and bounding the set of quantum behaviours in contextuality scenarios

The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the peculiar extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In this work, we situate the set of contextual quantum behaviours in the landscape of general contextual behaviours by considering prepare-and-measure experiments. We formulate semi-definite programming relaxations for bounding the set and significant subsets of quantum contextual behaviours. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities, and identifying novel maximally contextual quantum strategies. Moreover, we discover contextuality scenarios, and associated noncontextuality inequalities that require for their violation individual quantum preparation and measurement procedures to be mixed states, and unsharp measurements. To further illustrate the versatility of these relaxations, we switch to a tripartite setting, wherein we present the possibility of secure quantum key distribution schemes powered by violation of noncontextuality inequalities.

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