Lattice structures for quantum channels

We suggest that a certain one-to-one parametrization of completely positive maps on the matrix algebra might be useful in the study of quantum channels. This is illustrated in the case of binary quantum channels. While the algorithm is quite intricate, it admits a simple, lattice structure representation.Comment: 6 pages, 1 figure, typos correcte

Orthogonal polynomials in several non-commuting variables. II

In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.Comment: 10 pages, 1figur

Research problem: The completion number of a graph

Motivated by the remarkable interplay between (chordal) graphs and matrix algebra, we associate to each graph a so-called completion number that might encode some aspects of that interplay. We show that this number is not trivial, and we ask for a graph theoretic characterization of those graphs with a given completion number.Comment: 6 page

On L. Schwartz's boundedness condition for kernels

In previous works we analysed conditions for linearization of hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real space generated by positive definite kernels. The aim of this note is to find more concrete expressions of the Schwartz type conditions: in the Hamburger moment problem for Hankel type kernels on the free semigroup, in dilation theory (Stinespring type dilations and Haagerup decomposability), as well as in multi-variable holomorphy. Among other things, we prove that any hermitian holomorphic kernel has a holomorphic linearization, and hence that holomorphic kernels automatically satisfy L. Schwartz's boundedness condition.Comment: 21 page

Tensor Algebras and Displacement Structure. III. Asymptotic properties

We continue to investigate some classes of Szeg\"o type polynomials in several variables. We focus on asymptotic properties of these polynomials and we extend several classical results of G. Szeg\"o to this setting.Comment: 19 page

Parametrization of Quantum States and Channels

In this manuscript, a parametrization of positive matrices together with some of its many applications in quantum information theory is given.Comment: 19 page

Tensor algebras, displacement structure, and the Schur algorithm

In this paper we explore the connection between tensor algebras and displacement structure. We describe a scattering experiment in this framework, we obtain a realization of the elements of the tensor algebra as transfer maps of a certain class of nonstationary linear systems, and we describe a Schur algorithm for the Schur elements of the tensor algebra.Comment: 17 page

Tensor algebras and displacement structure. II. Noncommutative Szego polynomials

In this paper we continue to explore the connection between tensor algebras and displacement structure. We focus on recursive orthonormalization and we develop an analogue of the Szego type theory of orthogonal polynomials in the unit circle for several noncommuting variables. Thus, we obtain the recurrence equations and Christoffel-Darboux type formulas, as well as a Favard type result. Also we continue to study a Szego kernel for the N-dimnesional unit ball of an infinite dimensional Hilbert space.Comment: 17 page

Relations on noncommutative variables and associated orthogonal polynomials

This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on some dilation theoretic techniques that are also described in some details.Comment: 29 pages, 4 figure

A note on noncommutative interpolation

In this paper we formulate and solve Nevanlinna-Pick and Carath\'eodory type problems for tensor algebras with data given on the N-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory.Comment: 10 page