Information causality does not single out the set of quantum correlations
Information causality (IC) was introduced as a physical principle that imposes constraints on the non-local correlations between two parties (Alice and Bob) in a Bell scenario. Specifically, it states that the information that Bob can gain about an $n$-bit database held by Alice is bounded by the information she communicates to him. To date, the question whether IC for any $n \in \mathbb{N}$ singles out the set of quantum correlations remains open, meaning that it is unknown whether for any number of inputs $m$ and outputs $d$ of the two parties the set of correlations that satisfy IC exactly coincides with the quantum correlations. In this work, we answer this question in the negative. Firstly, we show that for any finite database length $n \geq 2$, there is a bipartite set of correlations that satisfies IC for all compatible protocols and that exhibits post-quantum correlations in the Bell scenario with $n+1$ inputs and $2$ outcomes per party. This result relies on constructing an entropic proof of IC for GPTs that may be of independent interest. We further show that in the Bell scenario with $n+1$ inputs and $2$ outcomes per party, our constructed sets of correlations lead to neither a superset nor a subset of the quantum correlations. Secondly, we construct a set of correlations in the Bell scenario with $3$ inputs and $2$ outputs per party that satisfies IC (for all database sizes $n \in \mathbb{N}$) and that includes post-quantum points. Consequently, all known formulations of information causality fail to fully characterize the bipartite set of quantum correlations, already in the $3$-input, $2$-outcome Bell scenario.