We analyze the performance of an adaptive chaotic synchronization system under information constraints assuming that some system parameters are unknown and only the system output is measured. Such a problem was studied previously in the absence of information constraints based on an adaptive observer scheme, allowing for its use in message transmission systems. We provide analytical bounds for the closed-loop system performance (asymptotic synchronization error) and conduct a numerical case study for a typical chaotic system, namely the Chua circuit, in the presence of information constraints. It is shown that the time-varying quantizer with one-step memory provides a reasonable approximation of the minimum transmission rate for adaptive state estimation.