Related papers: Infinite dimensional generalizations of Choi's the…
The main aim of this paper is to generalize the classical concept of positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The…
The physically allowed quantum evolutions on a single qubit can be described in terms of their geometry. From a simple parameterisation of unital single-qubit channels, the canonical form of all such channels can be given. The related…
We study classical structures in various categories of completely positive morphisms: on sets and relations, on cobordisms, on a free dagger compact category, and on Hilbert spaces. As an application, we prove that quantum maps with…
The theory of positive maps plays a central role in operator algebras and functional analysis, and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert…
We introduce three metrics on the set of quantum probability measures over a compact Hausdorff space and characterize them in terms of the completely bounded norm of the corresponding unital completely positive maps. We extend the existing…
We present a generalization of the family of linear positive maps in $M_3$ proposed thirty years ago by Cho et al. (Linear Algebra Appl. ${\bf 171}$, 213 (1992)) as a generalization of the seminal Choi non-decomposable map. The necessary…
In this work we present a general mathematical framework to deal with Quantum Networks, i.e. networks resulting from the interconnection of elementary quantum circuits. The cornerstone of our approach is a generalization of the Choi…
We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated…
We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in…
A linear map $\Phi :\mathbb{M}_n \to \mathbb{M}_k$ is called completely copositive if the resulting matrix $[\Phi (A_{j,i})]_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}]_{i,j=1}^m$. In…
Two-parameter generalizations of depolarizing channels are introduced and studied. These so-called Cartan-covariant channels have a covariance Lie group that forms a Cartan decomposition of SU$(D)$. The regions of completely positive and…
A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some…
Quantum correlation can be created by a local operation from some initially classical states. We prove that the necessary and sufficient condition for a local trace-preserving channel to create quantum correlation is that it is not a…
Positive maps that are not decomposable are a key resource in entanglement theory because they can detect bound entangled states, yet systematic methods for constructing them remain limited. We introduce an optimization framework based on…
Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact…
In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove…
It was shown in arXiv:0906.2527, that in finite-dimensional Hilbert spaces each operator system corresponds to some channel, for which this operator system will be an operator graph. This work is devoted to finding necessary and sufficient…
Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices…
Constructing all extreme instances of the set of completely positive trace-preserving (CPTP) maps, i.e., quantum channels, is a challenging valuable open problem in quantum information theory. Here we introduce a systematic approach that…
A system interacting with its environment will give rise to a quantum evolution. After tracing over the environment the net evolution of the system can be described by a linear Hermitian map. It has recently been shown that a necessary and…