A $\ast$-linear map $\Phi$ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations $I_n\otimes \Phi$ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.
COBISS.SI-ID: 18670425
An operator $C$ on a Hilbert space $\mathcal H$ dilates to an operator $T$ on a Hilbert space $\mathcal K$ if there is an isometry $V:\mathcal H\to \mathcal K$ such that $C= V^* TV$. A main result of this paper is, for a positive integer $d$, the simultaneous dilation, up to a sharp factor $\vartheta (d)$, expressed as a ratio of $\Gamma$ functions for $d$ even, of all $d\times d$ symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.
COBISS.SI-ID: 18571865
This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822-843, 2006) and Waki et al. (SIAM J Optim 17(1):218-242, 2006). The Gelfand-Naimark-Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.
COBISS.SI-ID: 49537283
New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in $\mathbb{P}^2 \times \mathbb{P}^2$ are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros; our examples show that the upper bound is obtained. Such biquadratic forms define positive linear maps on real symmetric $3 \times 3$ matrices that are not completely positive. Our constructions are explicit, and moreover we are able to determine which of the examples are extremal. We extend the examples to positive maps on complex matrices and find families of extreme rays in the cone of positive maps.
COBISS.SI-ID: 16956675
For every associative algebra $A$ and every class $\mathcal{C}$ of representations of $A$ the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors $\pi(a_1)v,\ldots,\pi(a_n)v$ are linearly dependent for every $\pi \in \mathcal{C}$ and every $v$ from the representation space of $\pi$. We answer this question in the following cases: (1) $A=U(L)$ is the enveloping algebra of a finite-dimensional complex Lie algebra $L$ and $\mathbb{C}$ is the class of all finite-dimensional representations of $A$. (2) $A=U(\mathfrak{sl}_2(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$. (3) $A=U(\mathfrak{sl}_3(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$ with sufficiently high weights. In case (1) the answer is: tuples that are linearly dependent over $\mathbb{C}$ while in cases (2) and (3) the answer is: tuples that are linearly dependent over the center of $A$. Similar results have been proved before for free algebras and Weyl algebras.
COBISS.SI-ID: 18542339