The paper deals with the state estimation problem for nonlinear dynamical systems via communication channels with limited data rate. We introduce several minimum data-rate limits associated with various types of observability. A notion of the restoration entropy (RE) is also introduced and its relevance to the problem is outlined by a corresponding Data Rate Theorem. Theoretical lower and upper estimates for the RE are proposed in the spirit of the first and second Lyapunov methods, respectively. For three classic chaotic multi-dimensional systems, it is demonstrated that the lower and upper estimates for the RE coincide for one of them and are nearly the same for the others. (C) 2019 Elsevier Ltd. All rights reserved.