  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.
