arXiv subjects

Jurij Volčič

Publications and source records attributed to Jurij Volčič.

At least 37 records · Page 2Linked to original sources

arXiv2022

Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem

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 operator variables, and are widespread in mathematics and physics. This paper delivers the first Positivstellensatz for global trace positivity of nc polynomials. Analogously to Hilbert's 17th problem in real algebraic geometry, trace-positive nc polynomials are shown to be weakly sums of hermitian squares and commutators of regular nc rational functions. In two variables, this result is strengthened further using a new sum-of-squares certificate with concrete univariate denominators for nonnegative bivariate polynomials. The trace positivity certificates in this paper are obtained by convex duality through solving the so-called unbounded tracial moment problem, which arises from noncommutative integration theory and free probability. Given a linear functional on nc polynomials, the tracial moment problem asks whether it is a joint distribution of integral operators affiliated with a tracial von Neumann algebra. A counterpart to Haviland's theorem on solvability of the tracial moment problem is established. Moreover, a variant of Carleman's condition is shown to guarantee the existence of a solution to the tracial moment problem. Together with semidefinite optimization, this is then used to prove that every trace-positive nc polynomial admits an explicit approximation in the 1-norm on its coefficients by sums of hermitian squares and commutators of nc polynomials.

math.OA↗

arXiv2022

Dimension-free entanglement detection in multipartite Werner states

Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints.

quant-ph↗

arXiv2022

Norms on complex matrices induced by complete homogeneous symmetric polynomials

We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension-independent tracial inequalities. Their potential merits further investigation.

math.CO↗

arXiv2021

Hilbert's 17th problem in free skew fields

This paper solves the rational noncommutative analog of Hilbert's 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of hermitian matrices in its domain, then it is a sum of hermitian squares of noncommutative rational functions. This result is a generalization and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq0$ if and only if it belongs to the rational quadratic module generated by $L$. The essential intermediate step towards this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

math.RA↗

arXiv2021

Optimization over trace polynomials

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme [New J. Phys., 2008] for optimization of noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

math-ph↗

arXiv2021

Positive univariate trace polynomials

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

math.RA↗

arXiv2020

Multipartite rational functions

Consider a tensor product of free algebras over a field $k$, the so-called multipartite free algebra $A=k \langle X^{(1)}\rangle\otimes\cdots\otimes k\langle X^{(G)}\rangle$. It is well-known that $A$ is a domain, but not a fir nor even a Sylvester domain. Inspired by recent advances in free analysis, formal rational expressions over $A$ together with their matrix representations in $M_{n_1}(k)\otimes\cdots\otimes M_{n_G}(k)$ are employed to construct a skew field of fractions $U$ of $A$, whose elements are called multipartite rational functions. It is shown that $U$ is the universal skew field of fractions of $A$ in the sense of Cohn. As a consequence a multipartite analog of Amitsur's theorem on rational identities relating evaluations in matrices over $k$ to evaluations in skew fields is obtained. The characterization of $U$ in terms of matrix evaluations fits naturally into the wider context of free noncommutative function theory, where multipartite rational functions are interpreted as higher order noncommutative rational functions with an associated difference-differential calculus and linear realization theory. Along the way an explicit construction of the universal skew field of fractions of $D\otimes k\langle X\rangle$ for an arbitrary skew field $D$ is given using matrix evaluations and formal rational expressions.

math.RA↗

arXiv2020

Noncommutative rational functions invariant under the action of a finite solvable group

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.

math.RA↗

arXiv2020

Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $Z_n(f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $Z^{re}(h)=\{X: \det h(X,X^*)=0\}$. A polynomial $f$ which depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $Z(f) \subseteq Z^{re}(h)$ is equivalent to each factor of $f$ being "stably associated" to a factor of $h$ or of $h^*$. For perspective, classical Hilbert type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above "algebraic certificate" does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018) 589-626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros $V(f)=\{X: f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

math.RA↗

arXiv2019

Bianalytic free maps between spectrahedra and spectraballs

Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron. In this article, the bianalytic maps between a very general class of ball-like free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this leads to a classification of automorphism groups of ball-like free spectrahedra. The results depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra.

math.FA↗

arXiv2019

Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation

Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities. This paper investigates the local theory of noncommutative functions. The first main result shows that for a scalar point $Y$, the ring $O_Y$ of uniformly analytic noncommutative germs about $Y$ is an integral domain and admits a universal skew field of fractions, whose elements are called meromorphic germs. A corollary is a local-global rank principle that connects ranks of matrix evaluations of a matrix $A$ over $O_Y$ with the factorization of $A$ over $O_Y$. Different phenomena occur for a semisimple tuple of non-scalar matrices $Y$. Here it is shown that $O_Y$ contains copies of the matrix algebra generated by $Y$. In particular, there exist nonzero nilpotent uniformly analytic functions defined in a neighborhood of $Y$, and $O_Y$ does not embed into a skew field. Nevertheless, the ring $O_Y$ is described as the completion of a free algebra with respect to the vanishing ideal at $Y$. This is a consequence of the second main result, a free Hermite interpolation theorem: if $f$ is a noncommutative function, then for any finite set of semisimple points and a natural number $L$ there exists a noncommutative polynomial that agrees with $f$ at the chosen points up to differentials of order $L$. All the obtained results also have analogs for (non-uniformly) analytic germs and formal germs.

math.RA↗

arXiv2019

Free Bertini's theorem and applications

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a noncommutative polynomial such that $f-λ$ factors for infinitely many scalars $λ$, then there exist a noncommutative polynomial $h$ and a nonconstant univariate polynomial $p$ such that $f=p\circ h$. Two applications of free Bertini's theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of $f$ is the set of all matrix tuples $X$ where $f(X)$ attains some given eigenvalue. It is shown that eigenlevel sets of $f$ and $g$ coincide if and only if $fa=ag$ for some nonzero noncommutative polynomial $a$. The second application pertains quasiconvexity and describes polynomials $f$ such that the connected component of $\{X \text{ tuple of symmetric $n\times n$ matrices}: λI\succ f(X) \}$ about the origin is convex for all natural $n$ and $λ>0$. It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial.

math.RA↗

arXiv2019

Plurisubharmonic Noncommutative Rational Functions

A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no $x_j^*$) rational function. The proof is entirely constructive. Further, a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization.

math.FA↗

arXiv2019

Stable noncommutative polynomials and their determinantal representations

A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix pencils, i.e., pencils of the form $H+iP_0+P_1x_1+\cdots+P_dx_d$, where $H$ is hermitian and $P_j$ are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a strongly stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.

math.RA↗

arXiv2018

Positive trace polynomials and the universal Procesi-Schacher conjecture

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellensätze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmüdgen- and Putinar-type Positivstellensätze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellensätze is indispensable.

math.RA↗

arXiv2018

Matrix coefficient realization theory of noncommutative rational functions

Noncommutative rational functions, i.e., elements of the universal skew field of fractions of a free algebra, can be defined through evaluations of noncommutative rational expressions on tuples of matrices. This interpretation extends their traditional important role in the theory of division rings and gives rise to their applications in other areas, from free real algebraic geometry to systems and control theory. If a noncommutative rational function is analytic at the origin, it can be described by a linear object, called a {\em realization}. In this article we present a realization theory that is applicable to {\em arbitrary} noncommutative rationals function and is well-adapted for studying matrix evaluations. Of special interest are the minimal realizations, which compensate the absence of a canonical form for noncommutative rational functions. The non-minimality of a realization is assessed by obstruction modules associated with it, they enable us to devise an efficient method for obtaining minimal realizations. With them we describe the extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity. Using these results we determine concrete size bounds for rational identity testing, construct minimal symmetric realizations and prove an effective local-global principle of linear dependence for noncommutative rational functions.

math.RA↗

arXiv2018

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.

math.RA↗

arXiv2017

Free loci of matrix pencils and domains of noncommutative rational functions

Consider a monic linear pencil $L(x) = I - A_1x_1 - \cdots - A_gx_g$ whose coefficients $A_j$ are $d \times d$ matrices. It is naturally evaluated at $g$-tuples of matrices $X$ using the Kronecker tensor product, which gives rise to its free locus $Z(L) = \{ X: \det L(X) = 0 \}$. In this article it is shown that the algebras $A$ and $A'$ generated by the coefficients of two linear pencils $L$ and $L'$, respectively, with equal free loci are isomorphic up to radical. Furthermore, $Z(L) \subseteq Z(L')$ if and only if the natural map sending the coefficients of $L'$ to the coefficients of $L$ induces a homomorphism $A'/{\rm rad} A' \to A/{\rm rad} A$. Since linear pencils are a key ingredient in studying noncommutative rational functions via realization theory, the above results lead to a characterization of all noncommutative rational functions with a given domain. Finally, a quantum version of Kippenhahn's conjecture on linear pencils is formulated and proved: if hermitian matrices $A_1, \dots, A_g$ generate $M_d(\mathbb{C})$ as an algebra, then there exist hermitian matrices $X_1, \dots, X_g$ such that $\sum_i A_i \otimes X_i$ has a simple eigenvalue.

math.RA↗