Training machine learning models to be robust against adversarial inputs poses seemingly insurmountable challenges. To better understand model robustness, we consider the underlying problem of learning robust representations. We develop a general definition of representation vulnerability that captures the maximum change of mutual information between the input and output distributions, under the worst-case input distribution perturbation. We prove a theorem that establishes a lower bound on the minimum adversarial risk that can be achieved for any downstream classifier based on this definition. We then propose an unsupervised learning method for obtaining intrinsically robust representations by maximizing the worst-case mutual information between input and output distributions. Experiments on downstream classification tasks and analyses of saliency maps support the robustness of the representations found using unsupervised learning with our training principle.