Quantum

A constructive solution to the $n$-representability problem

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Based on a feed excerpt; full article content has not been acquired. Source evidence retrieved 16/10/2026, 04:13:24 UTC.

Briefings

A new arXiv preprint presents a constructive solution to the n-representability problem by using weighted sum-of-squares hierarchies. The authors construct complete families of two-body witnesses at each hierarchy level, combining positive-semidefinite Gram matrices with explicit linear nullspace constraints to cancel higher-body contributions. This approach replaces searching for individual representability conditions with a systematic algebraic procedure. The construction is numerically demonstrated by solving a semidefinite program to recover the exact ground-state energy of a Richardson Hamiltonian in the three-fermion sector.

Key points

• The authors construct complete families of two-body witnesses at each level of a weighted sum-of-squares hierarchy.
• The construction uses explicit linear nullspace constraints to cancel all higher-body contributions.

Context

This work provides a unified constructive framework for verifying the physicality of a candidate two-particle reduced density matrix by characterizing all two-body witnesses. It offers a systematic algebraic procedure for solving the fermionic quantum marginal problem.

Limitations

The evidence is an arXiv preprint (arXiv:2610.20737v1) with no publication date or peer review status. Numerical results are limited to the three-fermion sector of a Richardson Hamiltonian, and the paper does not provide a general proof for all n-representable states.

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