Suppose P, Q are probability distributions on the same sample space. Their relative entropy is defined as S(P||Q) = \sum_i P(i) \log (P(i) / Q(i)). The relative entropy is an important information theoretic quantity and is related to mutual information of two systems. The substate theorem states that if S(P||Q) < c, then there is a probability distribution P' close to P such that P'(i) / 2^{O(c)} < Q(i) for all i. A similar substate theorem holds for a pair of quantum states, under suitable definitions of the quantum information theoretic quantities.
        The substate theorem gives us a powerful tool for several questions in classical and quantum communcation and information. Very roughly, the power of the substate theorem comes from the fact that if the entropy of P relative to Q is at most c, then Q can be used as a substitute for P with a 2^{O(c)} loss in efficiency. Using the substate theorem, one can prove rounds versus privacy tradeoffs as well as rounds versus communication tradeoffs for several problems, as well as message compression and direct sum resuts in communication complexity, besides other information theoretic results.
        The talk will give an introduction to the classical and quantum statements of this theorem, together with an illustration of one of its applications. No prior knowledge of quantum information is required.