We study an exact local compression of a quantum bipartite state; that is, applying local quantum operations to the state to reduce the dimensions of Hilbert spaces while perfectly maintaining the correlation. We provide a closed formula for calculating the minimal achievable dimensions, provided as a minimization of the Schmidt rank of a particular pure state constructed from the given state. Numerically more tractable upper and lower bounds of the rank are also obtained. Subsequently, we consider the exact compression of quantum channels as an application. Using this method, a post-processing step that can reduce the output dimensions while retaining information on the output of the original channel can be analyzed.