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Igor Klep

Publications and source records attributed to Igor Klep.

At least 19 recordsLinked to original sources
arXiv2026

Fejer-Riesz factorization for free products

We establish an operator-valued Fejer-Riesz factorization theorem for the algebraic free product A of C[F], the group algebra on the free group F on finitely many generators, with finitely many finite dimensional C-star algebras. With respect to a natural reduced-word length, called extent, we show that every linear combination F of elements of A with coefficients from the algebra of operators on a Hilbert space of extent at most 2d that is positive under all finite-dimensional unital *-representations of A admits a factorization F=B^*B with B of extent at most d. This sharp half-extent bound holds without strict positivity. The proof combines a new finite-dimensional realization theorem for positive truncated kernels with operator-system methods. We also establish factorization results for tensor products with group algebras of arbitrary discrete groups under strict positivity in the maximal C-star-tensor product; for amenable groups, strict positivity in the minimal tensor product suffices.

math.FA↗
arXiv2026

The bulk spectral gap is certifiable from above but uncomputable from below

Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.

quant-ph↗
arXiv2026

Fejér--Riesz factorization for positive noncommutative trigonometric polynomials

We prove a Fejér-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 Fejér-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↗
arXiv2026

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 all bipartite compiled Bell games via a newly formalized convergent sequential Navascués-Pironio-Acín (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 develop the sequential NPA hierarchy as a computational tool of independent interest, establishing its convergence and a stopping criterion based on flatness, and identifying its conic dual as a sparse SOS hierarchy.

quant-ph↗
arXiv2026

Convergence Rates for Lasserre's Upper-Bound Hierarchy on $\mathbb{R}^d$ with Gaussian Measure

We establish quantitative convergence rates for Lasserre's measure-based upper-bound hierarchy on $\mathbb{R}^d$ using the standard Gaussian measure as the reference measure. For any polynomial $f$ attaining its global infimum $f_{\min}$, we prove that the order-$r$ upper bounds satisfy $$ λ_r-f_{\min}=O\Big(\frac{1}{r}\Big), $$ yielding, to the best of our knowledge, the first explicit convergence rate for this hierarchy on a noncompact domain. We further show that the rate is sharp by proving an $Ω(1/r)$ rate for $f=x^2$, while demonstrating that, without attainment of the infimum, convergence can be arbitrarily slow: for every $m\ge0$, we construct a polynomial whose upper bounds converge at rate $Θ(r^{-1/(m+2)})$.

math.OC↗
arXiv2026

Quantum inflation and source-transfer hierarchies are complete for tree networks

Understanding the capabilities and limits of quantum systems is central to quantum information processing. This includes determining which correlations can arise in networks of quantum systems distributed by independent sources. Yet for general quantum networks, no systematic, dimension-free method is known to bound these correlations arbitrarily well. We resolve this problem for all quantum tree networks through a new source-transfer model, which we prove equivalent to a mixed quantum model. The mixed model places no restriction on dimension and reduces to the standard tensor-product model in finite dimensions; the source-transfer model instead describes these correlations through states on $C^*$-algebras and completely positive maps composed recursively along the tree. We prove the equivalence by reconstructing local measurements as Radon-Nikodym derivatives in the spatial tensor products prescribed by the network. Building on this characterization, we establish two convergent outer hierarchies of semidefinite programs (SDPs): a new source-transfer hierarchy and the inflation-NPA hierarchy. For the source-transfer hierarchy, we extend noncommutative real algebraic geometry to multi-state polynomials with completely positive maps, prove a reconstruction theorem establishing its convergence, and give a sufficient stopping criterion for extracting finite-dimensional realizations. For the inflation-NPA hierarchy, we reconstruct source-transfer realizations from weak-operator limits of averages over source copies produced by quantum inflation. Consequently, the game-value promise problem for quantum tree networks in the mixed model is $\mathsf{coRE}$-complete.

quant-ph↗
arXiv2026

Facial structure of matrix convex sets

This article investigates the notions of exposed points and (exposed) faces in the matrix convex setting. Matrix exposed points in finite dimensions were first defined by Kriel in 2019. Here this notion is extended to matrix convex sets in infinite-dimensional vector spaces. Then a connection between matrix exposed points and matrix extreme points is established: a matrix extreme point is ordinary exposed if and only if it is matrix exposed. This leads to a Krein-Milman type result for matrix exposed points that is due to Straszewicz-Klee in classical convexity: a compact matrix convex set is the closed matrix convex hull of its matrix exposed points. Several notions of a fixed-level as well as a multicomponent matrix face and matrix exposed face are introduced to extend the concepts of a matrix extreme point and a matrix exposed point, respectively. Their properties resemble those of (exposed) faces in the classical sense, e.g., it is shown that the $C^\ast$-extreme (matrix extreme) points of a matrix face (matrix multiface) of a matrix convex set $K$ are matrix extreme in $K$. As in the case of extreme points, any fixed-level matrix face is ordinary exposed if and only if it is a matrix exposed face. From this it follows that every fixed-level matrix face of a free spectrahedron is matrix exposed. On the other hand, matrix multifaces give rise to the noncommutative counterpart of the classical theory connecting (archimedean) faces of compact convex sets and (archimedean) order ideals of the corresponding function systems.

math.FA↗
arXiv2026

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↗
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 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].

math.FA↗
arXiv2026

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 $Γ$-convexity. Here $Γ$ is a tuple of free symmetric polynomials determining the geometry of a $Γ$-convex set. The paper introduces the notions of $Γ$-operator systems and $Γ$-ucp maps and establishes a Webster-Winkler type categorical duality between $Γ$-operator systems and $Γ$-convex sets. Next, a notion of an extreme point for $Γ$-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 $Γ$-extreme points of an operator $Γ$-convex set $K$ are in correspondence with the free extreme points of the operator convex hull of $Γ(K).$ From this result, a Krein-Milman theorem for $Γ$-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the $Γ$-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 $Γ$-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the $Γ$-convex hull of $D_p.$

math.OA↗
arXiv2026

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↗
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 $C(k)ξ^{n,4}_{d+1}/n$, 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.

quant-ph↗
arXiv2026

Operator-Valued Positivstellensätze 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 $p=r^*r+q^*π(L)q$, 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.

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↗
arXiv2026

First-order optimality conditions for non-commutative optimization problems

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.

quant-ph↗
arXiv2026

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

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↗