  In order to analyze an information theoretical derivation of Tsirelson's
bound based on information causality, we introduce a generalized mutual
information (GMI), defined as the optimal coding rate of a channel with
classical inputs and general probabilistic outputs. In the case where the
outputs are quantum, the GMI coincides with the quantum mutual information. In
general, the GMI does not necessarily satisfy the chain rule. We prove that
Tsirelson's bound can be derived by imposing the chain rule on the GMI. We
formulate a principle, which we call the no-supersignalling condition, which
states that the assistance of nonlocal correlations does not increase the
capability of classical communication. We prove that this condition is
equivalent to the no-signalling condition. As a result, we show that
Tsirelson's bound is implied by the nonpositivity of the quantitative
difference between information causality and no-supersignalling.
