Igor Klep
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View article: Quantitative quantum soundness for all multipartite compiled nonlocal games
Quantitative quantum soundness for all multipartite compiled nonlocal games Open
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…
View article: Quantitative Quantum Soundness for Bipartite Compiled Bell Games via the Sequential NPA Hierarchy
Quantitative Quantum Soundness for Bipartite Compiled Bell Games via the Sequential NPA Hierarchy Open
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 a…
View article: Quantitative Tsirelson's Theorems via Approximate Schur's Lemma and Probabilistic Stampfli's Theorems
Quantitative Tsirelson's Theorems via Approximate Schur's Lemma and Probabilistic Stampfli's Theorems Open
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…
View article: Quantum Max d-Cut via qudit swap operators
Quantum Max d-Cut via qudit swap operators Open
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…
View article: A model theoretic perspective on matrix rings
A model theoretic perspective on matrix rings Open
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 …
View article: Duality, extreme points and hulls for noncommutative partial convexity
Duality, extreme points and hulls for noncommutative partial convexity Open
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. T…
View article: Monoid algebras and graph products
Monoid algebras and graph products Open
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 le…
View article: Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators Open
The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationshi…
View article: Upper bound hierarchies for noncommutative polynomial optimization
Upper bound hierarchies for noncommutative polynomial optimization Open
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 L…
View article: Upper bound hierarchies for noncommutative polynomial optimization
Upper bound hierarchies for noncommutative polynomial optimization Open
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 L…
View article: Positivstellensätze and Moment problems with Universal Quantifiers
Positivstellensätze and Moment problems with Universal Quantifiers Open
This paper studies Positivstellensätze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$. Then $K$ is…
View article: First-order optimality conditions for non-commutative optimization problems
First-order optimality conditions for non-commutative optimization problems Open
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 ar…
View article: Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators Open
The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationshi…
View article: Sums of squares certificates for polynomial moment inequalities
Sums of squares certificates for polynomial moment inequalities Open
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…
View article: A random copositive matrix is completely positive with positive probability
A random copositive matrix is completely positive with positive probability Open
An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of th…
View article: Matrix Extreme Points and Free extreme points of Free spectrahedra
Matrix Extreme Points and Free extreme points of Free spectrahedra Open
A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$. This …
View article: Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem Open
A noncommutative (nc) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra has nonnegative trace. Such polynomials emerge as trace inequalities in several matrix or ope…
View article: Noncommutative Nullstellensätze and Perfect Games
Noncommutative Nullstellensätze and Perfect Games Open
The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades the basic analogous theorems for matrix and operator theory (noncommutative variables) ha…
View article: Noncommutative partially convex rational functions
Noncommutative partially convex rational functions Open
Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially co…
View article: Ranks of linear matrix pencils separate simultaneous similarity orbits
Ranks of linear matrix pencils separate simultaneous similarity orbits Open
This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples as $L(X_1…
View article: Noncommutative Polynomial Optimization
Noncommutative Polynomial Optimization Open
In this chapter we present the sums of Hermitian squares approach to noncommutative polynomial optimization problems. This is an extension of the sums of squares approach for polynomial optimization arising from real algebraic geometry. We…
View article: Facial structure of matrix convex sets
Facial structure of matrix convex sets Open
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…