A constructive solution to the $n$-representability problem

2026-10-16 · arXiv Quantum Physics

arXiv:2610.20737v1 Announce Type: new

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.

原文链接：https://arxiv.org/abs/2610.20737

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