We consider variations of the simplest one-dimensional linear control system in which communication between the sensor and the controller is constrained by a binary communication channel. The link between the controller and the plant is not constrained. We study the limits imposed by the channel on the controller's ability to estimate the state and achieve stability. For discrete-time systems, several such limits have been published in the literature. We first derive these limits as bounds on the state transition matrix of the linear system then we investigate extensions for the cases (1) unknown initial conditions, (2) unknown state transition matrix and (3) noisy binary communication channels. For continuous-time systems, we consider a binary queue as the communication link. For this case, we also derive the bounds on the gain matrix of the linear. system. To our knowledge this is the first time a binary queue has been studied as the communication link in the proposed setting.