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The predictions of quantum theory resist generalised noncontextualexplanations. In addition to the foundational relevance of this fact, theparticular extent to which quantum theory violates noncontextuality limitsavailable quantum advantage in communication and information processing. In thefirst part of this work, we formally define contextuality scenarios viaprepare-and-measure experiments, along with the polytope of general contextualbehaviours containing the set of quantum contextual behaviours. This frameworkallows us to recover several properties of set of quantum behaviours in thesescenarios, including contextuality scenarios and associated noncontextualityinequalities that require for their violation the individual quantumpreparation and measurement procedures to be mixed states and unsharpmeasurements. With the framework in place, we formulate novel semidefiniteprogramming relaxations for bounding these sets of quantum contextualbehaviours. Most significantly, to circumvent the inadequacy of pure states andprojective measurements in contextuality scenarios, we present a novel unitaryoperator based semidefinite relaxation technique. We demonstrate the efficacyof these relaxations by obtaining tight upper bounds on the quantum violationof several noncontextuality inequalities and identifying novel maximallycontextual quantum strategies. To further illustrate the versatility of theserelaxations, we demonstrate monogamy of preparation contextuality in atripartite setting, and present a secure semi-device independent quantum keydistribution scheme powered by quantum advantage in parity oblivious randomaccess codes.

Characterising and bounding the set of quantum behaviours in contextuality scenarios