The two-particle reduced density matrix (2-RDM) encodes all two-body correlations in a quantum system — enough information to compute any observable that depends on at most pairwise interactions, which includes the total energy of most molecular Hamiltonians. Measuring the full 2-RDM on a quantum computer requires an exponential number of measurements. But if partial information suffices to reconstruct the full matrix uniquely, far fewer measurements are needed.
Massaccesi et al. (arXiv:2603.13087) establish conditions under which the matrix completion of the 2-RDM is unique — when knowing a subset of matrix elements determines all the others exactly. The conditions depend on which elements are measured: certain patterns of known entries lock in the remaining entries through the N-representability constraints (the requirement that the 2-RDM must be derivable from some valid N-particle wavefunction).
The N-representability constraints are what make this different from generic matrix completion. In the general problem, completing a partially observed matrix is usually not unique — many matrices fit the observed entries. But the 2-RDM isn't an arbitrary matrix. It must satisfy positivity conditions (the 2-RDM and certain contractions must be positive semidefinite) and consistency conditions (partial traces must reproduce the 1-RDM). These constraints reduce the space of valid completions, and for the right pattern of observations, reduce it to a single point.
The practical validation uses a hybrid quantum-stochastic approach on the Fermi-Hubbard model. Quantum hardware measures the subset of matrix elements identified by the uniqueness conditions. A classical optimizer completes the remaining elements subject to N-representability. The completion matches the exact 2-RDM, confirming that the uniqueness conditions identified theoretically are sufficient in practice.