In the considered linear Gaussian sensor scheduling problem, only one sensor out of a set of sensors performs a measurement. To minimize the estimation error over multiple time steps in a computationally tractable fashion, the so-called information-based pruning algorithm is proposed. It utilizes the information matrices of the sensors and the monotonicity of the Riccati equation. This allows ordering sensors according to their information contribution and excluding many of them from scheduling. Additionally, a tight lower is calculated for branch-and-bound search, which further improves the pruning performance.