A constructive solution to the n-representability problem

arXiv:2610.20737 · quant-ph, physics.chem-ph · Submitted 2026-10-14 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "A constructive solution to the n-representability problem".

Mira: The gist The paper presents a constructive solution to the fermionic quantum marginal problem by constructing a complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: To recap, "A constructive solution to the n-representability problem" presents a method for constructing all two-body witnesses within a weighted sum-of-squares hierarchy. The core claim is that they combine Gram matrices with explicit linear nullspace constraints to exactly cancel out all contributions from higher bodies.

Mira: The thesis is that this construction replaces the search for individual representability conditions with a systematic algebraic procedure, and they prove this system is complete in two stages.

Lev: Stage one involves providing linear nullspace equations that capture every at-most-twobody witness admitting a degree-d weighted sum-of-squares representation, including those joint cancellations between the sum of squares and particle number multiplier terms.

Kai: And stage two is the proof that at d = n, this weighted sum of squares cone contains every Hermitian, particle-conserving, at most two-body operator that is nonnegative on the n-particle subspace.

Mira: This means they show that the terminal cone of their hierarchy satisfies W eta,≤two = wn eta,≤two which implies that degree d = eta is sufficient to generate all physical two-body witnesses, including those at lower levels <ref:2610.20737#pg1>.

Lev: The construction also shows that the difference between the constructed sum of squares and a target witness is just a particle-number shift of the form f r + f r where its degree is less than or equal to two eta - two <ref:2610.20737#pg1>.

Kai: So, what this means for us in terms of real science is that we have a complete way to check if a candidate two-particle reduced density matrix is physical using this framework.

Mira: It formalizes the search space for these matrices, giving chemists and physicists a systematic tool rather than just relying on enumerating conditions one by one.

Lev: For hardware engineers like us, having such a complete characterization helps us understand exactly what kind of constraints need to be enforced when we try to build error correction codes that operate on these reduced density matrices.

Conclusion: Kai: So, looking at "A constructive solution to the n-representability problem" by Rubin and DePrince III, they’ve managed to build a complete framework for verifying two-particle reduced density matrices.

Mira: The main implication is that we have moved from an open search problem to a systematic construction method using weighted sum-of-squares hierarchies and linear constraints.

Lev: In simpler terms, it gives you a complete set of rules—a constructive way—to tell if a specific two-particle density matrix you're looking at is actually achievable in any valid n-fermion state.

Kai: It means we don't have to guess or check conditions individually; we use this algebraic construction to find every possible witness.

Mira: This tool should help solidify the connection between the mathematical description of quantum states and what we can physically measure or simulate in condensed matter systems.

Lev: From an engineering standpoint, it means when we design algorithms for these systems, we have a complete set of constraints to check against for physical validity, which simplifies the overhead calculations significantly.

Kai: The paper also showed that using the full sum-of-squares generating algebra is enough to avoid needing those extra particle number multipliers in their constructions.

Mira: It’s a unified way to handle these problems, whether you're looking at finding an energy bound or just checking representability.

Lev: Overall, this work provides a solid mathematical foundation for verifying the physicality of these matrices when dealing with fermionic systems.

Nicholas C. Rubin, A. Eugene DePrince III

Google Quantum AI · Florida State University

quant-ph, physics.chem-ph

Submitted: 2026-10-14

Updated: 2026-10-14

Code: https://github.com/edeprince3/libsdp

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 81/100

The gist: The gist The paper presents a constructive solution to the fermionic quantum marginal problem by constructing a complete family of two-body witnesses at each level of a weighted sum-of-squares

Key concepts

SOS Witnesses: These are mathematical tools used to verify if a quantum state or operator is physically valid. The construction uses these witnesses within a weighted sum-of-squares framework to systematically check for the existence of valid two-particle reduced density matrices.
Weighted SOS (wSOS): This is an extension of standard sum-of-squares constraints that allows for witnesses to be non-negative on a specific particle subspace without needing global positivity across the entire Fock space. This makes it easier to find relevant physical states.
n-representability: This problem asks which two-particle density matrices can actually arise from an ensemble of n fermions. The paper provides a complete set of algebraic constraints (the weighted SOS hierarchy) that precisely defines this valid set, solving the problem systematically.

Summary

The gist The paper presents a constructive solution to the fermionic quantum marginal problem by constructing a complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy.

SOS Witnesses and n-representability

The problem aims to describe the space of fermionic two-particle reduced density matrices (2-RDM) that can be obtained from an ensemble state of n fermions, which is usually characterized by enumerating witnesses that verify membership in the valid set<ref:2610.20737#pg5>. The 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<ref:2610.20737#pg7>. The construction establishes completeness in two stages: 1) we provide linear nullspace equations that capture every at-most-twobody witness admitting a degree-d weighted SOS representation, including joint cancellations between the SOS and particle-number multiplier terms, 2) we prove that at d = n this weighted SOS cone contains every Hermitian, particle-conserving, at-most-two-body operator that is nonnegative on the n-particle subspace<ref:2610.20737#pg6>.

Weighted SOS Witnesses

The weighted sum of squares (wSOS) features a set of constraints defined by wSOS:= (B = X i R†iRi + X ji x†ij qjxij where Ri ∈ Span n ak, a†l, 1 d, xij ∈ Span ak, a†l, 1 d′, d′ = d −⌈max(deg(qj))/2⌉<ref:2610.20737#pg10>. This weighted cone permits witnesses that are nonnegative on the target particle-number subspace without requiring global positivity on Fock space<ref:2610.20737#pg10>. The construction of the augmented nullspace condition captures every at-most-two-body operator admitting a degree-d weighted SOS representation<ref:2610.20737#pg9>.

Complete Physical Witnesses and n-representability

The terminal cone contains every physical two-body witness, as Theorem III.2 states that the complete weighted SOS hierarchy satisfies Wη,≤2 = wnΣη,≤2<ref:2610.20737#pg9>. This means d = η is sufficient to generate all physical two-body witnesses, including those available at lower levels<ref:2610.20737#pg9>. The proof shows that the difference between the constructed SOS and the target witness is a particle-number shift of the form fr + rf † where deg f ≤ 2η − 2<ref:2610.20737#pg9>.

Complete 3-representability

The paper provides a numerical verification by computing a lower bound certificate that is equivalent to the true ground state in a three-particle sector of a Richardson Hamiltonian<ref:2610.20737#pg5>. The complete weighted SOS formulation searches over all positive-semidefinite Gram matrices on V3 and all allowed particle-number multipliers of degree at most four, subject to the symmetry and coefficient-matching constraints<ref:2610.20737#pg6>. The numerical results show that the degree-at-most-two and degree-at-most-four multiplier formulations have identical supremal energy bounds when the complete cubic SOS generator space is used<ref:2610.20737#pg5>.

Degree-2 multipliers are sufficient

The numerical agreement between the degree-at-most-two and degree-at-most-four multiplier formulations is a consequence of the complete cubic SOS construction, rather than a special property of the Richardson Hamiltonian<ref:2610.20737#pg5>. The proof shows that for fixed H and r = N − η1, E∗d,2 = E∗d,4 when the SOS part is generated by the full space V3<ref:2610.20737#pg9>. This result confirms that degree-at-most-two multipliers suffice when the full wnΣd,≤2 set of constraints are used<ref:2610.20737#pg9>.

Molecular symmetries

Symmetry enters the construction in two distinct ways: requiring the witness to commute with conserved quantities allows the Gram matrix to be organized into blocks labeled by corresponding changes in quantum numbers<ref:2610.20737#pg5>. Restricting positivity to prescribed eigenvalues is accomplished by including the associated equality-constraint multipliers<ref:2610.20737#pg5>. The Gram blocks are labeled by (∆N, 2∆Sz), while the equality constraints are N − η1 and Sz − s1<ref:2610.20737#pg5>.

AI Statement

This work builds on the SOS representations of Ref. [22]. The key components of the constructive solution are the work of humans, but some key proofs come from LLMs<ref:2610.20737#pg10>. Specifically, the reduction of degree-4 multiplier polynomials to degree-2 multiplier polynomials and the generalized form in Appendix C along with the proof for the sufficiency of wnΣη,≤2 in Section III B were derived by OpenAI’s Astra-6 pro model<ref:2610.20737#pg10>. Early discussions with Gemini 3.1 pro indicated that weighted SOS could be used as a counterexample to the completeness of Ref. [17]<ref:2610.20737#pg10>. Gemini 4 and Astra aided in writing the manuscript<ref:2610.20737#pg10>. All authors assume full responsibility for the correctness of the results<ref:2610.20737#pg10>.

Improvements for AI systems

The improved system can perform exact ground-state energy calculations for Richardson Hamiltonians in three-fermion sectors by solving semidefinite programs derived from degree-3 weighted SOS formulations, as demonstrated in Section IV A. This surpasses conventional v2RDM relaxations like DQG and T1+T2' conditions by searching over all positive-semidefinite Gram matrices on V3 and all allowed particle-number multipliers of degree at most four.
The system can verify the n-representability of a candidate two-particle reduced density matrix through a complete characterization of all two-body witnesses, using the augmented nullspace construction in Theorem III.1. This ensures that every at-most-two-body operator admitting a degree-d weighted SOS representation is captured, even when global positivity is not guaranteed (The augmented nullspace construction therefore captures every at-most-two-body element of the prescribed degreed weighted SOS cone).
The improved system can rigorously prove the sufficiency of degree-at-most-two particle-number multipliers for preserving the supremal energy bound across all SOS levels, as shown in Proposition IV.1 and Equation (C17). This establishes that the complete cubic SOS construction yields bounds equivalent to those obtained with a lower multiplier cutoff, meaning degree-at-most-two multiplier formulations have identical supremal energy bounds when the complete cubic SOS generator space is used.
The system can generate explicit, exact ground state energies for the Richardson pairing Hamiltonian in the three-fermion sector by utilizing degree-2 multipliers within the complete weighted SOS framework, as shown in Section IV B. This demonstrates that degree-2 multiplier polynomials are sufficient when the full wnΣd,≤2 set of constraints are used, recovering exact energy within solver accuracy.
The system can construct symmetry-adapted theories by organizing the Gram matrix into blocks labeled by conserved quantities like particle number and spin projection, as detailed in Appendix D. This allows for Restriction of positivity to prescribed eigenvalues is accomplished by including the associated equality-constraint multipliers to enforce physical symmetries automatically.

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.