This paper studies the minimal rate requirements for state estimation in linear time-invariant (LTI) systems where the controller and the plant are connected via a noiseless bandlimited channel. Using information theoretic arguments, we obtain first for scalar systems lower bounds on the data rates required for state estimation under three different stability criteria; monotonic boundedness of distortion, terminal-time distortion minimization and stability in support size. The minimum data rate achievable by any source-coder is computed under each of these criteria, and the best rate achievable with quantization (operational source-coding) is shown to be in agreement with the information-theoretic bounds in some specific cases (such as if the system coefficient is an integer or if the criterion is an asymptotic one). The optimal variable-rate and fixed-rate quantizers are studied and constructed for each of these criteria. We then extend these results to multi-dimensional systems, replacing the distortion measure in the first criterion with differential entropy. One byproduct of this analysis is the message that entropy is not an appropriate measure of uncertainty in multi-dimensional systems for control purposes.