Negative Shannon Information Hides Networks

Shannon information is defined for characterizing the uncertainty information of classical probabilistic distributions. As an uncertainty measure it is generally believed to be positive. This holds for any information quantity from two random variables because of the polymatroidal axioms. However, it is unknown why there is negative information for more than two random variables on finite dimensional spaces. It is first shown that the negative tripartite Shannon mutual information implies specific Bayesian network representations of its joint distribution. Then it is shown that the negative Shannon information is obtained from general tripartite Bayesian networks with quantum realizations. This provides a device-independent witness of negative Shannon information. Finally the result is extended for general networks. The present result shows new insights in the network compatibility from non-Shannon information inequalities.

Automated summary

Shannon information is commonly used to measure the uncertainty of classical probabilistic distributions and is generally believed to be positive. However, it is not clear why there can be negative information for more than two random variables in finite dimensional spaces. This research shows that the negative tripartite Shannon mutual information implies specific Bayesian network representations and can be obtained from general tripartite Bayesian networks with quantum realizations. This provides device-independent evidence of negative Shannon information and extends the result to general networks.