Partial ignorance communication tasks in quantum theory

We introduce a generalization of communication of partial ignorance where both parties of a prepare-and-measure setup receive inputs from a third party before a success metric is maximized over the measurements and preparations. Various methods are used to obtain bounds on the success metrics, including SDPs, ultraweak monotones for communication matrices and frame theory for quantum states. Simplest scenarios in the new generalized prepare-and-measure setting, simply called partial ignorance communication tasks, are analysed exhaustively for bits and qudits. Finally, the new generalized setting allows the introduction of operational equivalences to the preparations and measurements, allowing us to analyse and observe a contextual advantage for quantum theory in one of the communication tasks.