Information causality beyond the random access code model

Information causality (IC) was one of the first principles that have been invoked to bound the set of quantum correlations. For some families of correlations, this principle recovers exactly the boundary of the quantum set; for others, there is still a gap. We close some of these gaps using a new quantifier for IC, based on the notion of "redundant information'". The new definition is still obeyed by quantum correlations. This progress was made possible by the recognition that the principle of IC can be captured without referring to the success criterion of random access codes.