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The chain rule implies Tsirelson's bound: an approach from generalized mutual information

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.