A constructive solution to the $n$-representability problem

Abstract:We present a solution to the fermionic quantum marginal problem by constructing the complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy. Our construction combines positive-semidefinite Gram matrices with explicit linear nullspace constraints that exactly cancel all higher-body contributions, replacing the search for individual representability conditions with a systematic algebraic procedure. Beyond the completeness proof, we demonstrate the construction numerically by solving the associated semidefinite program and recovering the exact ground-state energy of a Richardson Hamiltonian in the three-fermion sector. We further prove that higher-degree particle-number multipliers are unnecessary when the full sum-of-squares generating algebra is used. These results establish a unified constructive framework for verifying the physicality of a candidate two-particle reduced density matrix through a complete characterization of all two-body witnesses.

Quantum Physics (quant-ph); Chemical Physics (physics.chem-ph)

arXiv:2610.20737 [quant-ph]

(or arXiv:2610.20737v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.20737 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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