arXiv subjects

Igor Klep

Publications and source records attributed to Igor Klep.

At least 19 recordsLinked to original sources

arXiv2026

Addendum to "Factoring non-negative operator valued trigonometric polynomials in two variables"

Factorization for positive semidefinite matrix-valued polynomials over a nonsingular compact affine real surface is established. Corollaries include Fej\'er-Riesz factorization for bivariate matrix polynomials that take positive semidefinite values and resolutions to questions posed by Mehta-Slofstra-Zhao and Savchuk-Schm\"udgen. 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].

math.FA↗

arXiv2026

The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds

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-$\frac{1}{2}$ kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.

quant-ph↗

arXiv2026

Quantitative semidefinite certificates for ground-state energies of Pauli Hamiltonians

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 $C(k)\xi^{n,4}_{d+1}/n$, where $\xi^{n,4}_{d+1}$ 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.

quant-ph↗

arXiv2026

Operator-Valued Positivstellens\"atze on Matrix Convex Sets and Free Products of Finite Abelian Groups

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)\otimes C $ of degree at most $2d+1$, where $H$ is separable and infinite-dimensional. Let $L(x)=I+\sum_{j=1}^{g} A_j x_j$ be a monic linear operator pencil, and let $D_L=\{X: L(X) \geq 0\}$ be the associated matrix convex set. We show that $p$ is positive on $D_L$ if and only if $p=r^*r+q^*\pi(L)q$, where $q$ and $r$ have degree at most $d$, and $\pi$ 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.

math.FA↗

arXiv2026

Robust self-testing with CHSH mod 3

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 $\varepsilon$ of the optimum is $O(\sqrt{\varepsilon})$-close, up to local isometries, to a direct sum of optimal irreducible strategies.

math.OC↗

arXiv2026

The Effective Lasserre's Perturbative Positivstellensatz

We study sum-of-squares (SOS) certificates for nonnegative polynomials $p$ on $\mathbb{R}^d$ 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 $\sum_{n=0}^N (x\cdot x)^n/(n!)^t$ for $0 < t < 1$. Our main result provides an explicit quantitative bound on the truncation order $N$ required to achieve an $\varepsilon$-accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that $N$ grows polynomially in $1/\varepsilon$, with rate $N = O((\|p\|/\varepsilon)^{1/(1-t)})$.

math.OC↗

arXiv2025

Inclusion constants for free spectrahedra with applications to quantum incompatibility

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.

quant-ph↗

arXiv2025

Exploiting Term Sparsity in Symmetry-Adapted Basis for Polynomial Optimization

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.

math.OC↗

arXiv2025

Fej\'er--Riesz factorization for positive noncommutative trigonometric polynomials

We prove a Fej\'er-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on $\mathscr{W}\times\mathfrak{Y}$, where $\mathscr{W}$ is either the free semigroup $\langle x \rangle_g$ or the free product group $\mathbb{Z}_2^{g}$, and $\mathfrak{Y}$ is a discrete group. More precisely, using the shortlex order, if $A$ has degree at most $w$ in the $\mathscr{W}$ variables and is uniformly strictly positive on all unitary representations of $\mathscr{W}\times\mathfrak{Y}$, then $A=B^{*}B$ with $B$ analytic and of $\mathscr{W}$-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 $\mathbb{C}[\mathbb{Z}_2^{*g}\times \mathbb{Z}_2^{*h}]$ from quantum information theory. In the special case $\mathscr{W}=\mathbb{Z}^h$ we recover, in the matrix-valued setting, the classical commutative multivariable Fej\'er-Riesz factorization. For trivial $\mathfrak{Y}$ we obtain a ``perfect'' group-algebra Positivstellensatz on $\mathbb{Z}_2^{*g}$ that does not require strict positivity; this result is sharp, as demonstrated by counterexamples in $\mathbb{Z}_2*\mathbb{Z}_3$ and $\mathbb{Z}_3^{*2}$. 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 $\langle x \rangle_g$ or the free product group $\mathbb{Z}_2^{*g}$ indexed by words in $\mathscr{W}$ of length $\le w$.

math.FA↗

arXiv2025

Positive operator-valued noncommutative polynomials are squares

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. 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 $Y$.

math.FA↗

arXiv2025

Quantitative quantum soundness for all multipartite compiled nonlocal games

Compiled nonlocal games transfer the power of Bell-type multi-prover tests into a single-device setting by replacing spatial separation with cryptography. Concretely, the KLVY compiler (STOC'23) maps any multi-prover game to an interactive single-prover protocol, using quantum homomorphic encryption. A crucial security property of such compilers is quantum soundness, which ensures that a dishonest quantum prover cannot exceed the original game's quantum value. For practical cryptographic implementations, this soundness must be quantitative, providing concrete bounds rather than merely asymptotic. While quantitative quantum soundness has been established for the KLVY compiler in the bipartite case, it has only been shown asymptotically for multipartite games. This is a significant gap, as multipartite nonlocality exhibits phenomena with no bipartite analogue, and the difficulty of enforcing space-like separation makes single-device compilation especially compelling. This work closes this gap by providing quantitative upper bounds for all multipartite compiled nonlocal games via a new sequential NPA-like hierarchy. In particular, finite-level convergence yields quantitative quantum soundness with respect to the commuting quantum value, and flat optimality yields the same with respect to the tensor-product quantum value. On the way, we introduce an NPA-like hierarchy for quantum instruments and prove its completeness, thereby characterizing correlations from operationally-non-signaling sequential strategies. This NPA-like hierarchy can be seen to complement previous multipartite generalizations of the S-G-HJW purification theorem, which takes a central role in quantum information, nonlocality, and contextuality. We further develop novel geometric arguments for the decomposition of sequential strategies into their signaling and non-signaling parts, which might be of independent interest.

quant-ph↗

arXiv2025

Quantitative Quantum Soundness for Bipartite Compiled Bell Games via the Sequential NPA Hierarchy

Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for bipartite compiled Bell games via a newly formalized convergent sequential Navascu\'es-Pironio-Ac\'in (NPA) hierarchy: any polynomial-time prover's score is controlled by a finite-level hierarchy value, and finite-level convergence gives a negligible gap to the commuting quantum value, or to the tensor-product quantum value under flat optimality. Second, we provide a full characterization of this sequential NPA hierarchy, establishing it as a robust numerical tool that is of independent interest. Finally, for games without such finite-level certificates, we explore the necessity of NPA approximation error for quantitatively bounding their compiled scores, linking these considerations to the complexity conjecture $\mathrm{MIP}^{\mathrm{co}}=\mathrm{coRE}$ and open challenges such as quantum homomorphic encryption correctness for "weakly commuting" quantum registers.

quant-ph↗

arXiv2025

Quantitative Tsirelson's Theorems via Approximate Schur's Lemma and Probabilistic Stampfli's Theorems

Whether an almost-commuting pair of operators must be close to a commuting pair is a central question in operator and matrix theory. We investigate this problem for pairs of $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$ of $M_d(\mathbb{C})$, showing that each operator in $\mathcal{B}$ is $O(d^2\epsilon)$-close in operator norm to an operator in the commutant $\mathcal{A}'$ under two complementary formulations of "$\epsilon$-almost commutation." One formulation is probabilistic, requiring that the operators of $\mathcal{B}$ have small commutators for most Haar-random unitaries acting on $\mathcal{A}$. This first formulation leads to two novel probabilistic generalizations of Stampfli's theorem, which relates an operator's distance from the scalars to the norm of its inner derivation. The second formulation is deterministic, requiring small commutators between the generators of $\mathcal{A}$ and $\mathcal{B}$; we analyze this using an approximate Schur's lemma formulated in terms of Weyl-Heisenberg (clock-and-shift) matrices. As an application of our results to quantum information theory, we obtain a quantitative Tsirelson's theorem: in dimension $d$, every $\epsilon$-almost quantum commuting observable model is well approximated by a quantum tensor-product model with error $O(d^2\epsilon)$.

quant-ph↗

arXiv2025

Quantum Max d-Cut via qudit swap operators

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 $(\mathbb C^d)^{\otimes n}$. 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.

quant-ph↗

arXiv2024

Duality, extreme points and hulls for noncommutative partial convexity

This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of $\Gamma$-convexity. Here $\Gamma$ is a tuple of free symmetric polynomials determining the geometry of a $\Gamma$-convex set. The paper introduces the notions of $\Gamma$-operator systems and $\Gamma$-ucp maps and establishes a Webster-Winkler type categorical duality between $\Gamma$-operator systems and $\Gamma$-convex sets. Next, a notion of an extreme point for $\Gamma$-convex sets is defined, paralleling the concept of a free extreme point for a matrix convex set. To ensure the existence of such points, the matricial sets considered are extended to include an operator level. It is shown that the $\Gamma$-extreme points of an operator $\Gamma$-convex set $K$ are in correspondence with the free extreme points of the operator convex hull of $\Gamma(K).$ From this result, a Krein-Milman theorem for $\Gamma$-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the $\Gamma$-convex hull of the matricial positivity domain {(also known as a free semialgebraic set)} $D_p$ of a free symmetric polynomial $p$ is given. The approximation consists of a decreasing family of $\Gamma$-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the $\Gamma$-convex hull of $D_p.$

math.OA↗

arXiv2024

Monoid algebras and graph products

In this note, we extend results about unique $n^{\textrm{th}}$ roots and cancellation of finite disconnected graphs with respect to the Cartesian, the strong and the direct product, to the rooted hierarchical products, and to a modified lexicographic product. We show that these results also hold for graphs with countably many finite connected components, as long as every connected component appears only finitely often (up to isomorphism). The proofs are via monoid algebras and generalized power series rings.

math.CO↗

arXiv2024

Upper bound hierarchies for noncommutative polynomial optimization

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.

math.OC↗

arXiv2024

Cross-positive linear maps, positive polynomials and sums of squares

A linear map $\Phi$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle \Phi(U),V\rangle\geq0$, and is completely cross-positive if all its ampliations $I_n\otimes \Phi$ 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 $\{(x,y)\mid x^Ty=0\}$. Moreover, $\Phi$ is shown to be completely cross-positive if and only if $p_\Phi$ is a sum of squares modulo the principal ideal $(x^Ty)$. 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 $\Phi$ mapping between $3\times 3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented.

math.FA↗