Is the matrix completion of reduced density matrices unique?
Abstract
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 computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina's theorem [M. Rosina, Queen's Papers on Pure and Applied Mathematics No. 11, 369 (1968)], 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.
Cite
@article{arxiv.2603.13087,
title = {Is the matrix completion of reduced density matrices unique?},
author = {Gustavo E. Massaccesi and Ofelia B. Oña and Luis Lain and Alicia Torre and Juan E. Peralta and Diego R. Alcoba and Gustavo E. Scuseria},
journal= {arXiv preprint arXiv:2603.13087},
year = {2026}
}