Is the Matrix Completion of Reduced Density Matrices Unique?

Reduced density matrices are central to describing observables in many-body quantum systems. In electronic structure theory, the two-particle reduced density matrix (2-RDM) suffices to determine the energy and other key properties. Recent work has used matrix completion, leveraging the low-rank structure of RDMs and approximate theoretical models, to reconstruct the 2-RDM from partial data and thus reduce the computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina's theorem (Rosina, M. Queen's Papers on Pure and Applied Mathematics, 1968, No. 11, 369), we here show that the matrix completion is unique under certain conditions, identifying the subset of 2-RDM elements that enables its exact reconstruction from incomplete information. Building on this, we introduce a hybrid quantum-stochastic algorithm that achieves exact matrix completion, demonstrated through applications to the Fermi-Hubbard model.