State estimation is a fundamental problem for monitoring and controlling systems. Engineering systems interconnect sensing and computing devices over a shared bandwidth-limited channels, and therefore, estimation algorithms should strive to use bandwidth optimally. We present a notion of entropy for state estimation of switched nonlinear dynamical systems, an upper bound for it and a state estimation algorithm for the case when the switching signal is unobservable. Our approach relies on the notion of topological entropy and uses techniques from the theory for control under limited information. We show that the average bit rate used is optimal in the sense that, the efficiency gap of the algorithm is within an additive constant of the gap between estimation entropy of the system and its known upper-bound. We apply the algorithm to two system models and discuss the performance implications of the number of tracked modes.