arXiv subjects

Jurij Volčič

Publications and source records attributed to Jurij Volčič.

At least 19 recordsLinked to original sources

arXiv2026

Hereditary NC Polynomials & Positivity Domains

Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables $x_1,...,x_g,x_1^*,...,x_g^*$, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial $f$ is the sequence of hypersurfaces $\mathscr{Z}_n(f)=\{X\in M_n(\mathbb{C})^g:\det f(X) = 0\}$. When $f$ is Hermitian, the positivity domain of $f$ is the sequence of semialgebraic sets $\mathcal{D}_n(f)=\{X\in M_n(\mathbb{C})^g:f(X)\text{ is positive semidefinite}\}$. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through $Γ$-convexity.

math.RA↗

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

Global structure behind pointwise equivalences of noncommutative polynomials

This paper investigates the interplay between local and global equivalences on noncommutative polynomials, the elements of the free algebra. When the latter are viewed as functions in several matrix variables, a local equivalence of noncommutative polynomials refers to their values sharing a common feature point-wise on matrix tuples of all dimensions, such as rank-equivalence (values have the same ranks), isospectrality (values have the same spectrum), and pointwise similarity (values are similar). On the other hand, a global equivalence refers to a ring-theoretic relation within the free algebra, such as stable association or (elementary) intertwinedness. This paper identifies the most ubiquitous pairs of local and global equivalences. Namely, rank-equivalence coincides with stable association, isospectrality coincides with both intertwinedness and transitive closure of elementary intertwinedness, and pointwise similarity coincides with equality. Using these characterizations, further results on spectral radii and norms of values of noncommutative polynomials are derived.

math.RA↗

arXiv2026

Beyond real: Investigating the role of complex numbers in self-testing

We investigate complex self-testing, a generalization of standard self-testing that accounts for quantum strategies whose statistics is indistinguishable from their complex conjugate's. We show that many structural results from standard self-testing extend to the complex setting, including lifting of common assumptions. Our main result is an operator-algebraic characterization: complex self-testing is equivalent to uniqueness of the real parts of higher moments, leading to a basis-independent formulation in terms of real C* algebras. This leads to a classification of non-local strategies, and a tight boundary where standard self-testing does not apply and complex self-testing is necessary. We further construct a strategy involving quaternions, establishing the first standard self-test for genuinely complex strategy. Our work clarifies the structure of complex self-testing and highlights the subtle role of complex numbers in bipartite Bell non-locality.

quant-ph↗

arXiv2025

Invariants of finite groups acting on (free) skew fields

Let $M$ be a finitely generated skew field over a ground field $k$, and let $G$ be a finite group of $k$-linear automorphisms of $M$. This paper investigates finite generation of the skew subfield $M^G$ of $G$-invariants in $M$, and relations between the generators. The first main result shows that $M^G$ is finitely generated. Stronger conclusions hold when $M$ is a free skew field, i.e., the universal skew field of fractions of a free algebra. The second main result is the solution of the free Noether problem for non-modular linear group actions: if $G$ acts linearly on the free skew field $M$ on $m$ generators and the characteristic of $k$ does not divide $|G|$, then $M^G$ is the free skew field on $|G|(m-1)+1$ generators. In contrast, a nonlinear action of $Z_2$ on the free skew field $M$ on two generators is presented such that $M^{Z_2}$ is not a free skew field, resolving the free Lüroth problem. This action also exposes a non-scalar element of $M$ whose centralizer is not a rational field, refuting a conjecture of P. M. Cohn from 1978.

math.RA↗

arXiv2025

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↗

arXiv2025

A noncommutative generalization of Hunter's positivity theorem

Hunter proved that the complete homogeneous symmetric polynomials of even degree are positive definite. We prove a noncommutative generalization of this result, in which the scalar variables are replaced with hermitian operators. We provide a sharp lower bound and a sum of hermitian squares representation that are novel even in the scalar case.

math.FA↗

arXiv2025

Matrix evaluations of noncommutative rational functions and Waring problems

Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can be written as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variations of this result for the case where $r$ is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if $f$ and $g$ are nonconstant polynomials, then, for $n$ large enough, every nonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\in M_n(K)^m$.

math.RA↗

arXiv2024

Positivity of state, trace, and moment polynomials, and applications in quantum information

State, trace, and moment polynomials are polynomial expressions in several operator or random variables and positive functionals on their products (states, traces or expectations). While these concepts, and in particular their positivity and optimization, arose from problems in quantum information theory, yet they naturally fit under the umbrella of multivariate operator theory. This survey presents state, trace, and moment polynomials in a concise and unified way, and highlights their similarities and differences. The focal point is their positivity and optimization. Sums of squares certificates for unconstrained and constrained positivity (Positivstellensätze) are given, and parallels with their commutative and freely noncommutative analogs are discussed. They are used to design a convergent hierarchy of semidefinite programs for optimization of state, trace, and moment polynomials. Finally, circling back to the original motivation behind the derived theory, multiple applications in quantum information theory are outlined.

quant-ph↗

arXiv2024

Maximal device-independent randomness in every dimension

Random numbers are used in a wide range of sciences. In many applications, generating unpredictable private random numbers is indispensable. Device-independent quantum random number generation is a framework that makes use of the intrinsic randomness of quantum processes to generate numbers that are fundamentally unpredictable according to our current understanding of physics. While device-independent quantum random number generation is an exceptional theoretical feat, the difficulty of controlling quantum systems makes it challenging to carry out in practice. It is therefore desirable to harness the full power of the quantum degrees of freedom (the dimension) that one can control. It is known that no more than $2 \log(d)$ bits of private device-independent randomness can be extracted from a quantum system of local dimension $d$. In this paper we demonstrate that this bound can be achieved for all dimensions $d$ by providing a family of explicit protocols. In order to obtain our result, we develop new certification techniques that can be of wider interest in device-independent applications for scenarios in which complete certification ('self-testing') is impossible or impractical.

quant-ph↗

arXiv2024

Linear matrix pencils and noncommutative convexity

Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix inequality, the free spectrahedron, from the perspective of free real algebraic geometry. Namely, among matricial solution sets of noncommutative polynomial inequalities, free spectrahedra are precisely the convex ones. Furthermore, a procedure for detecting free spectrahedra and producing their representing linear matrix pencils is discussed. Finally, free spectrahedra admit a perfect Positivstellensatz, leading to a semidefinite programming formulation of eigenvalue optimization over convex matricial sets constrained by noncommutative polynomial inequalities.

math.FA↗

arXiv2024

Sums of squares certificates for polynomial moment inequalities

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 open nonlinear Bell inequalities from quantum physics are settled.

math.FA↗

arXiv2024

Constant-sized self-tests for maximally entangled states and single projective measurements

Self-testing is a powerful certification of quantum systems relying on measured, classical statistics. This paper considers self-testing in bipartite Bell scenarios with small number of inputs and outputs, but with quantum states and measurements of arbitrarily large dimension. The contributions are twofold. Firstly, it is shown that every maximally entangled state can be self-tested with four binary measurements per party. This result extends the earlier work of Mančinska-Prakash-Schafhauser (2021), which applies to maximally entangled states of odd dimensions only. Secondly, it is shown that every single binary projective measurement can be self-tested with five binary measurements per party. A similar statement holds for self-testing of projective measurements with more than two outputs. These results are enabled by the representation theory of quadruples of projections that add to a scalar multiple of the identity. Structure of irreducible representations, analysis of their spectral features and post-hoc self-testing are the primary methods for constructing the new self-tests with small number of inputs and outputs.

quant-ph↗

arXiv2024

Dimension-free matricial Nullstellensätze for noncommutative polynomials

Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry, and has inspired a plethora of noncommutative analogs. In last two decades, there has been an increased interest in understanding vanishing sets of polynomials in several matrix variables without restricting the matrix size, prompted by developments in noncommutative function theory, control systems, operator algebras, and quantum information theory. The emerging results vary according to the interpretation of what vanishing means. For example, given a collection of noncommutative polynomials, one can consider all matrix tuples at which the values of these polynomials are all zero, singular, have common kernel, or have zero trace. This survey reviews Nullstellensätze for the above types of vanishing sets, and identifies their structural counterparts in the free algebra.

math.RA↗

arXiv2023

State polynomials: positivity, optimization and nonlinear Bell inequalities

This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin's solution to Hilbert's 17th problem is proved showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Somewhat surprisingly, it is also established that a Krivine-Stengle Positivstellensatz fails to hold in the state polynomial setting. Further, archimedean Positivstellensätze in the spirit of Putinar and Helton-McCullough are presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués-Pironio-Acín scheme for optimization of noncommutative polynomials. The motivation behind this theory arises from the study of correlations in quantum networks. Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite and the bilocal tripartite scenario are analyzed. To reduce the size of the constructed SDPs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited. To obtain the above-mentioned results, techniques from noncommutative algebra, real algebraic geometry, operator theory, and convex optimization are employed.

math.FA↗

arXiv2023

All Real Projective Measurements Can be Self-tested

Self-testing is the strongest form of quantum functionality verification which allows a classical user to deduce the quantum state and measurements used to produce measurement statistics. While self-testing of quantum states is well-understood, self-testing of measurements, especially in high dimensions, has remained more elusive. We demonstrate the first general result in this direction by showing that every real projective measurement can be self-tested. The standard definition of self-testing only allows for the certification of real measurements. Therefore, our work effectively broadens the scope of self-testable projective measurements to their full potential. To reach this result, we employ the idea that existing self-tests can be extended to verify additional untrusted measurements. This is known as `post-hoc self-testing'. We formalize the method of post-hoc self-testing and establish a sufficient condition for its application. Using this condition we construct self-tests for all real projective measurements. Inspired by our construction, we develop a new technique of iterative self-testing, which involves using post-hoc self-testing in a sequential manner. Starting from any established self-test, we fully characterize the set of measurements that can be verified via iterative self-testing. This provides a clear methodology for constructing new self-tests from pre-existing ones.

quant-ph↗

arXiv2023

Ranks of linear matrix pencils separate simultaneous similarity orbits

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 $L(X_1,\dots,X_m)=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m$. It is shown that ranks of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, $m$-tuples $A$ and $B$ of $n\times n$ matrices are simultaneously similar if and only if the ranks of $L(A)$ and $L(B)$ are equal for all linear matrix pencils $L$ of size $mn$. Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced.

math.RA↗