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