Determination of tripartite pure states by two overlapping marginals

In many-body quantum physics, the global state encodes all correlations, but the information most naturally available is often local. This makes it important to understand when a small collection of reduced density matrices already fixes the whole state. Such uniqueness questions are especially important for tomography and for the study of entanglement, because agreement with local data from a pure state does not by itself exclude a different mixed global state. We prove a generic uniqueness theorem for finite-dimensional tripartite systems. Previous results were restricted either to uniqueness against pure competitors or to uniqueness against all competitors under stronger dimension assumptions or balanced subsystem dimensions. Our result removes both restrictions.