arXiv subjects

Tea Štrekelj

Publications and source records attributed to Tea Štrekelj.

9 recordsLinked to original sources

arXiv2026

Kadison duality for partially convex sets

This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is a set that is convex in a designated set of convex variables when the others are held fixed. We introduce the notion of a regular partially convex set and identify its dual as a finitely generated free module over a commutative C*-algebra endowed with a compatible Archimedean order unit structure. We call such spaces free order unit modules. We prove that for any compact regular partially convex set K, the space of continuous functions on K that are affine in the convex variables is the canonical example of such a module. Conversely, we show that the partially convex state space of a free order unit module is a compact regular partially convex set. Our main result establishes a duality between compact regular partially convex sets and free order unit modules. We also establish a Stone-Weierstrass-type theorem, demonstrating that partially affine polynomials are dense in the space of continuous partially affine functions on any compact regular partially convex set. Finally, we prove a Hahn-Banach-type separation theorem of compact partially convex sets from their outer points.

math.FA↗

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

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↗

arXiv2025

Quantum Max Cut for complete tripartite graphs

The Quantum Max-$d$-Cut ($d$-QMC) problem is a special instance of a $2$-local Hamiltonian problem, representing the quantum analog of the classical Max-$d$-Cut problem. The $d$-QMC problem seeks the largest eigenvalue of a Hamiltonian defined on a graph with $n$ vertices, where edges correspond to swap operators acting on $(\mathbb{C}^d)^{\otimes n}$. In recent years, progress has been made by investigating the algebraic structure of the $d$-QMC Hamiltonian. Building on this approach, this article solves the $d$-QMC problem for complete tripartite graphs for small local dimensions, $d \le 3$.

quant-ph↗

arXiv2025

Construction of exceptional copositive matrices

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a positive semidefinite matrix and a nonnegative one and the latter contains the cone of completely positive matrices. These are the matrices of the form $BB^T$ for some $n\times r$ matrix $B$ with nonnegative entries. The above inclusions are strict for $n\geq5.$ The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes $\geq 5$, i.e., copositive matrices that are not the sum of a positive semidefinite matrix and a nonnegative one. The second contribution of this paper addresses the asymptotic ratio of the volume radii of compact sections of the cones of copositive and completely positive matrices. In a previous work by the authors, it was shown that, by identifying symmetric matrices naturally with quartic even forms, and equipping them with the $L^2$ inner product and the Lebesgue measure, the ratio of the volume radii of sections with a suitably chosen hyperplane is bounded below by a constant independent of $n$ as $n$ tends to infinity. In this paper, we extend this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.

math.FA↗

arXiv2024

Extreme points of matrix convex sets and their spanning properties

This expository article gives a survey of matrix convex sets, a natural generalization of convex sets to the noncommutative (dimension-free) setting, with a focus on their extreme points. Mirroring the classical setting, extreme points play an important role in matrix convexity, and a natural question is, ``are matrix convex sets the (closed) matrix convex hull of their extreme points?" That is, does a Krein-Milman theorem hold in this setting? This question requires some care, as there are several notions of extreme points for matrix convex sets. Three of the most prevalent notions are matrix extreme points, matrix exposed points, and free extreme points. For each of these types of extreme points, we examine strengths and shortcomings in terms of a Krein-Milman theorem. Of particular note is the fact that these extreme points are all finite-dimensional in nature. As such, a large amount of our discussion is about free spectrahedra, which are matrix convex sets determined by a linear matrix inequality.

math.FA↗

arXiv2024

A random copositive matrix is completely positive with positive probability

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 the form $BB^T$ for some (possibly rectangular) matrix $B$ with nonnegative entries. The main result, proved using Blekherman's real algebraic geometry inspired techniques and tools of convex geometry, shows that asymptotically, as $n$ goes to infinity, the ratio of volume radii of the two cones is strictly positive. Consequently, the same holds true for the ratio of volume radii of any two cones sandwiched between them, e.g., the cones of positive semidefinite matrices, matrices with nonnegative entries, their intersection and their Minkowski sum.

math.FA↗

arXiv2021

Convexity of a certain operator trace functional

In this article the operator trace function $ Λ_{r,s}(A)[K, M] := {\operatorname{tr}}(K^*A^r M A^r K)^s$ is introduced and its convexity and concavity properties are investigated. This function has a direct connection to several well-studied operator trace functions that appear in quantum information theory, in particular when studying data processing inequalities of various relative entropies. In the paper the interplay between $Λ_ {r,s}$ and the well-known operator functions $Γ_{p,s}$ and $Ψ_{p,q,s}$ is used to study the stability of their convexity (concavity) properties. This interplay may be used to ensure that $Λ_{r,s}$ is convex (concave) in certain parameter ranges when $M=I$ or $K=I.$ However, our main result shows that convexity (concavity) is surprisingly lost when perturbing those matrices even a little. To complement the main theorem, the convexity (concavity) domain of $Λ$ itself is examined. The final result states that $Λ_{r,s}$ is never concave and it is convex if and only if $r=1$ and $s\geq 1/2.$

quant-ph↗