We study sum-of-squares (SOS) certificates for nonnegative polynomials p on ℝ^d and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS representations of p augmented by weighted polynomial tails of the form ∑_n=0^N (x· x)^n/(n!)^t for 0 < t < 1. Our main result provides an explicit quantitative bound on the truncation order N required to achieve an ε-accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that N grows polynomially in 1/ε, with rate N = O((p/ε)^1/(1-t)).

Factorization for positive semidefinite matrix-valued polynomials over a nonsingular compact affine real surface is established. Corollaries include Fejér-Riesz factorization for bivariate matrix polynomials that take positive semidefinite values and resolutions to questions posed by Mehta-Slofstra-Zhao and Savchuk-Schmüdgen. An explicit example shows a conclusion of [Dri25, Theorem, p. 519] that arises organically from its proof need not hold. The difficulty is traced to [Dri25, Theorem 3.7].

A (*)-linear map Phi between matrix spaces is cross-positive if it is positive on orthogonal pairs (U, V ) of positive semidefinite matrices in the sense that (U, V ) := tr(UV) = 0 implies (Phi(U), V ) >= 0, and is completely cross-positive if all its ampliations In 0 Phi are cross-positive. (Completely) cross-positive maps arise in the theory of operator semigroups, where they are sometimes called exponentially-positive maps, and are also important in the theory of affine processes on symmetric cones in mathematical finance. To each Phi as above a bihomogeneous form is associated by p(Phi)(x,y) = y(T) Phi(xx(T))y. Then Phi is cross-positive if and only if p(Phi) is nonnegative on the variety of pairs of orthogonal vectors {(x, y) | x(T)y = 0}. Moreover, 'T' is shown to be completely cross-positive if and only if p Phi is a sum of squares modulo the principal ideal (x(T)y). These observations bring the study of cross-positive maps into the powerful setting of real algebraic geometry. Here this interplay is exploited to prove quantitative bounds on the fraction of cross-positive maps that are completely cross-positive. Detailed results about cross-positive maps 'T' mapping between 3 x 3 matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

We consider the problem of optimizing the state average of a polynomial of non-commuting variables, over all states and operators satisfying a number of polynomial constraints, and over all Hilbert spaces where such states and operators are defined. Such non-commutative polynomial optimization (NPO) problems are routinely solved through hierarchies of semidefinite programming (SDP) relaxations. By formulating the general NPO problem in Lagrangian terms, we heuristically derive first-order optimality conditions via small variations in the problem variables. Although the derivation is not rigorous, it gives rise to two types of optimality conditions - state and operator - which are rigorously analyzed in the paper. Both types of conditions can be enforced through additional positive semidefinite constraints in the SDP hierarchies. State optimality conditions are shown to be satisfied by all NPO problems. For NPO problems with optimal solutions (such as, e.g., Archimedean ones) they allow enforcing a new type of constraints: namely, restricting the optimization over states to the set of common ground states of an arbitrary number of operators. Operator optimality conditions are the non-commutative analogs of the Karush-Kuhn-Tucker (KKT) conditions, which are known to hold in many classical optimization problems. In this regard, we prove that a weak form of operator optimality holds for all NPO problems; stronger versions require the problem constraints to satisfy some qualification criterion, just like in the classical case (e.g. Mangasarian-Fromovitz constraint qualification). We test the power of the new optimality conditions by computing local properties of ground states of many-body spin systems and the maximum quantum violation of Bell inequalities.

We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann. Math., 2002) and one of the authors (Linear Alg. Appl., 2001). Specifically, we show that every positive operator-valued noncommutative polynomial p admits a single-square factorization p=r^*r . An analogous statement holds for operator-valued noncommutative trigonometric polynomials (i.e., operator-valued elements of a free group algebra). Our approach follows the now standard sum-of-squares (sos) paradigm but requires new results and constructions tailored to operator coefficients. Assuming a positive p is not sos, Hahn–Banach separation yields a linear functional that is positive on the sos cone and negative on p; a Gelfand–Naimark–Segal (GNS) construction then produces a representing tuple Y leading to contradiction since p was assumed positive on Y. The main technical input is a canonical tuple A of self-adjoint operators and, in the unitary case, a canonical tuple U of unitaries, both constructed from the left-regular representation on Fock space. We prove that, up to a universal constant, the norms ‖ p(A)‖ and ‖ p(U)‖ bound the operator norm of any positive semidefinite Gram matrix G representing the sos polynomial p. This uniform control is the key input in showing that the cone of (sums of) squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then produces a separating functional that is continuous for the weak operator topology (WOT). This two-step passage between the ultraweak and WOT topologies constitutes our separation argument and yields the required WOT closedness of the sos cone. With this in hand, the GNS construction associates to such a separating linear functional a finite-rank positive semidefinite noncommutative Hankel matrix and, on its range, produces the desired tuple Y.

Quantum Max Cut (QMC) problem for systems of qubits is an example of a 2-local Hamiltonian problem, and a prominent paradigm in computational complexity theory. This paper investigates the algebraic structure of a higher-dimensional analog of the QMC problem for systems of qudits. The Quantum Max d-Cut (d-QMC) problem asks for the largest eigenvalue of a Hamiltonian on a graph with n vertices whose edges correspond to swap operators acting on (ℂ^d)^⊗ n. The algebra generated by the swap operators is identified as a quotient of a free algebra modulo symmetric group relations and a single additional relation of degree d. This presentation leads to a tailored hierarchy of semidefinite programs, leveraging noncommutative polynomial optimization (NPO) methods, that converges to the solution of the d-QMC problem. For a large class of complete bipartite graphs, exact solutions for the d-QMC problem are derived using the representation theory of symmetric groups and Littlewood-Richardson coefficients. Lastly, the paper addresses a refined d-QMC problem focused on finding the largest eigenvalue within each isotypic component (irreducible block) of the graph Hamiltonian. It is shown that the spectrum of the star graph Hamiltonian distinguishes between isotypic components of the 3-QMC problem. For general d, low-degree relations for separating isotypic components are presented, enabling adaptation of the global NPO hierarchy to efficiently compute the largest eigenvalue in each isotypic component.

This work addresses the problem of computing the spectral minimum (ground state energy) of a noncommutative polynomial subject to noncommutative polynomial constraints. Building on the Helton-McCullough Positivstellensatz, the Navascu & eacute;s-Pironio-Ac & iacute;n (NPA) hierarchy provides a sequence of lower bounds that converge to the spectral minimum under mild assumptions on the constraint set. Each of these bounds can be computed via semidefinite programming. In this paper, we develop complementary, complete hierarchies of upper bounds for the spectral minimum. These are noncommutative counterparts to Lasserre's upper bound hierarchies for classical polynomial optimization. Each upper bound is obtained by solving a generalized eigenvalue problem. The proposed hierarchies are applicable to optimization problems in both bounded and unbounded contexts, as demonstrated through a range of examples. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The k-local Hamiltonian problem is a central model for quantum many-body systems and Hamiltonian complexity. Semidefinite programming and noncommutative sum-of-squares hierarchies provide systematic certificates for ground-state energies, but existing finite-convergence results give no quantitative guarantee on the accuracy of the low hierarchy levels accessible in computation. We prove explicit finite-level convergence rates for these hierarchies in the Pauli setting. For k-local Hamiltonians whose Pauli expansion contains only even-weight terms, we show that both the NPA-type lower-bound hierarchy and the upper-bound hierarchy on the spectral minimum have error at most C(k)ξ^n,4_d+1/n, where ξ^n,4_d+1 is the smallest root of a Krawtchouk polynomial and C(k) is independent of the number of qubits n and the hierarchy level d. General k-local Hamiltonians reduce to this even-weight case by adding one ancilla qubit while preserving the spectrum. The proof constructs almost-reproducing kernels for the Pauli algebra and relates their spectra to Krawtchouk polynomials, giving a noncommutative analogue of recent kernel-based convergence analyses for commutative polynomial optimization. These results provide the first quantitative finite-level accuracy guarantees for noncommutative semidefinite relaxations of Pauli Hamiltonians.

We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let p be an operator-valued polynomial in B(H)⊗ C of degree at most 2d+1, where H is separable and infinite-dimensional. Let L(x)=I+∑_j=1^g A_j x_j be a monic linear operator pencil, and let D_L={X: L(X) ≥ 0} be the associated matrix convex set. We show that p is positive on D_L if and only if p=r^*r+q^*π(L)q, where q and r have degree at most d, and π is a unital completely positive map on the operator system generated by the coefficients of L. The proof combines a Hahn–Banach separation argument with a tailored GNS construction. The main challenge is that the separation occurs in the product ultraweak topology, so boundedness of the resulting GNS operators is not automatic. We first handle bounded matrix convex sets, using closedness of the cone of weighted squares in the product ultraweak topology as the key technical input, and then pass to the general unbounded case by an approximation argument. Finally, we apply this convex Positivstellensatz to prove an operator-valued noncommutative Fejer–Riesz theorem on free products of finite abelian groups. The key additional ingredients are the universal *-algebra povm(n) associated with POVMs, a perfect Positivstellensatz for povm(n), and Boca's theorem on free products of completely positive maps. As a consequence, every positive operator-valued trigonometric polynomial on a free product of finite abelian groups admits a sum-of-squares factorization with explicit complexity bounds.

The CHSH mod 3 Bell inequality is a natural testbed for higher-dimensional quantum nonlocality, yet its maximal quantum violation and self-testing properties have remained unresolved. We determine its exact maximal quantum value and show that, up to unitary equivalence and the natural symmetries of the inequality, it admits a unique optimal irreducible strategy; equivalently, there are four symmetry-related optimal irreducible strategies. Each of these strategies uses a maximally entangled two-qutrit state. We further prove that any strategy whose value is within ε of the optimum is O(√(ε))-close, up to local isometries, to a direct sum of optimal irreducible strategies.

Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. Existing rigorous methods are largely limited to special settings, while variational numerical approaches typically provide estimates rather than certified bounds. Here we introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems. These upper bounds are obtained by solving a series of semidefinite programs and they become arbitrarily tight at the cost of more computational resources. This shows that the bulk spectral gap is semi-decidable, in contrast to undecidability results for alternative notions of spectral gap based on sequences of finite systems with prescribed boundary conditions. As a proof of principle, we apply our algorithm to the spin-1/2 kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.

We prove a Fejér-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on 𝒲×𝔜, where 𝒲 is either the free semigroup ⟨ x ⟩_g or the free product group ℤ_2^g, and 𝔜 is a discrete group. More precisely, using the shortlex order, if A has degree at most w in the 𝒲 variables and is uniformly strictly positive on all unitary representations of 𝒲×𝔜, then A=B^*B with B analytic and of 𝒲-degree at most w; this degree bound is optimal, and strict positivity is essential. As an application, we obtain degree-bounded sums-of-squares certificates for Bell-type inequalities in ℂ[ℤ_2^*g×ℤ_2^*h] from quantum information theory. In the special case 𝒲=ℤ^h we recover, in the matrix-valued setting, the classical commutative multivariable Fejér-Riesz factorization. For trivial 𝔜 we obtain a “perfect” group-algebra Positivstellensatz on ℤ_2^*g that does not require strict positivity; this result is sharp, as demonstrated by counterexamples in ℤ_2*ℤ_3 and ℤ_3^*2. To establish our main results two novel ingredients of independent interest are developed: (a) a positive-semidefinite Parrott theorem with entries given by functions on a group; and (b) solutions to positive semidefinite matrix completion problems for ⟨ x ⟩_g or the free product group ℤ_2^*g indexed by words in 𝒲 of length ≤ w.

Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function and constraint polynomials are invariant under the action of a finite group. The present paper simultaneously exploits group symmetry and term sparsity in order to reduce the computational cost of the hierarchy. We first exploit symmetry by writing the semidefinite matrices in a symmetry-adapted basis according to an isotypic decomposition. The matrices in such a basis are block diagonal. Secondly, we exploit term sparsity on each block to further reduce the optimization matrix variables. This is a non-trivial extension of the term sparsity-based hierarchy related to sign symmetry that was introduced by two of the authors. Our method is compared with existing techniques via benchmarks on quartics with dihedral, cyclic and symmetric group symmetry.

This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. A positive solution to Hilbert's 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two nonlinear Bell inequalities from quantum physics are settled.

In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings M_n(K) in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This is used together with invariant theory to prove quantifier elimination when K is an intersection of real closed fields. On the other hand, it is shown that finding a natural definable expansion with quantifier elimination of the theory of M_n(ℂ) is closely related to the infamous simultaneous conjugacy problem in matrix theory. Finally, for various natural structures describing dimension-free matrices it is shown that no such elimination results can hold by establishing undecidability results.

This paper studies Positivstellensätze and moment problems for sets K that are given by universal quantifiers. Let Q be the closed set of universal quantifiers. Fix a finite nonnegative Borel measure whose support is Q and assume it satisfies the multivariate Carleman condition. First, we prove a Positivstellensatz with universal quantifiers: if a polynomial f is positive on K, then f belongs to the associated quadratic module, under the archimedeanness assumption. Second, we prove some necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in K. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set K. Third, we present applications of the above Positivstellensatz and moment problems in semi-infinite optimization, where feasible sets are given by infinitely many constraints with universal quantifiers. This results in a new hierarchy of Moment-SOS relaxations. Its convergence is shown under some usual assumptions. The quantifier set Q is allowed to be non-semialgebraic, which makes it possible to solve some optimization problems with non-semialgebraic constraints. Funding: X. Hu and J. Nie are partially supported by the NSF [Grant DMS-2110780]. I. Klep is supported by the Slovenian Research Agency program P1-0222 [also Grants J1-50002, J1-60011, J1-50001, J1-2453, N1-0217, and J1-3004] and was partially supported by the Marsden Fund Council of the Royal Society of New Zealand. I. Klep’s work was partly performed within the project COMPUTE, funded within the QuantERA II program that has received funding from the EU’s H2020 research and innovation program under the GA No 101017733.

Building on the matrix cube problem, inclusions of free spectrahedra have been used successfully to obtain relaxations of hard spectrahedral inclusion problems. The quality of such a relaxation is quantified by the inclusion constant associated with each free spectrahedron. While optimal values of inclusion constants were known in certain highly symmetric cases, no general method for computing them was available. In this work, we show that inclusion constants for Cartesian products of free simplices can be computed using methods from non-commutative polynomial optimization, together with a detailed analysis of the extreme points of the associated free spectrahedra. This analysis also yields new closed-form analytic expressions for these constants. As an application to quantum information theory, we prove new bounds on the amount of white noise that incompatible measurements can tolerate before they become compatible. In particular, we study the case of one dichotomic and one k-outcome measurement, as well as the case of four dichotomic qubit measurements.

This work focuses on minimizing the eigenvalue of a noncommutative polynomial subject to a finite number of noncommutative polynomial inequality constraints. Based on the Helton-McCullough Positivstellensatz, the noncommutative analog of Lasserre's moment-sum of squares hierarchy provides a sequence of lower bounds converging to the minimal eigenvalue, under mild assumptions on the constraint set. Each lower bound can be obtained by solving a semidefinite program. We derive complementary converging hierarchies of upper bounds. They are noncommutative analogues of the upper bound hierarchies due to Lasserre for minimizing polynomials over compact sets. Each upper bound can be obtained by solving a generalized eigenvalue problem.

Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large majority of a set of standard Bell inequalities, denoted A2--A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton-McCullough Positivstellensatz, the Gelfand--Naimark--Segal (GNS) construction with a novel use of the Artin-Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new ``Newton chip"-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented. This technique is based on conditional expectations. Finally, noncommutative Gro"\bner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution.

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form L-A(X) = I-d circle times I-n+A(1)circle times X-1+A(2)circle times X-2+center dot center dot center dot+A(g)circle times X-g >= 0, where the Ai and Xi are symmetric matrices and the Xi have any size n x n. Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue.We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme.