Bounding and simulating contextual correlations in quantum theory

We introduce a hierarchy of semidefinite relaxations of the set of preparation-contextual quantum correlations. This constitutes a simple and versatile tool for bounding the magnitude of quantum contextuality in general experiments. To illustrate its utility, we use it to determine the maximal quantum violation of several noncontextuality inequalities whose maximum violations were previously unknown. We then go further and use it to prove that certain contextual correlations cannot be explained with pure states, thereby showing that mixed states are an indispensable resource for contextuality. In the second part of the paper, we turn our attention to the simulation of contextual correlations in general operational theories. We introduce the information cost of simulating contextuality, which quantifies the additional, otherwise forbidden, information required to simulate contextual correlations in either classical or quantum models. In both cases, we show that the simulation cost can be efficiently bounded using a variant of our hierarchy of semidefinite relaxations, and we calculate it exactly in the simplest contextuality scenario of parity-oblivious multiplexing.