Exact and local compression of quantum bipartite states

We study exact local compression of a quantum bipartite state; that is, applying local quantum operations to reduce the dimensions of the Hilbert spaces while perfectly preserving the correlation. We provide a closed-form expression for the minimal achievable dimensions, formulated as a minimization of the Schmidt rank of a particular pure state constructed from the given state. Numerically tractable upper and lower bounds on this rank are also obtained. As an application, we consider the exact compression of quantum channels. This method enables the analysis of a post-processing step that reduces the output dimensions while retaining the information contained in the original channel's output.