Related papers: Positive maps and extendibility hierarchies from c…
We study the cone of completely positive (cp) matrices for the first interesting case $n = 5$. This is a semialgebraic set, which means that the polynomial equalities and inequlities that define its boundary can be derived. We characterize…
In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, we must consider an associated self-adjoint map. Here, we show that if $A$ is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC)…
A linear map $\Phi$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle…
Quantum states that remain separable (i.e., not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely…
The paper is devoted to a study of the cone $\cop$ of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone $\cop$ and use them…
We construct a family of map which is shown to be positive when imposing certain condition on the parameters. Then we show that the constructed map can never be completely positive. After tuning the parameters, we found that the map still…
We show that each positive map from B(K) to B(H) with K and H finite dimensional Hilbert spaces is a scalar multiple of a map of the form $Tr - \psi$ with $\psi$ completely positive. This is used to give necessary and sufficient conditions…
We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with $n$ copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show…
Decorated cospans provide a categorical framework for composing open systems along shared interfaces. This paper is a computational proof of concept: we show that the framework produces a working coupled dynamical system when the…
In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the…
A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = V V ^T$. In this paper, we study the CP-matrix approximation problem of projecting a matrix onto the intersection of a set…
A convex cone $\mathcal{K}$ is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD)…
The Promise Constraint Satisfaction Problem (PCSP for short) is a generalization of the well-studied Constraint Satisfaction Problem (CSP). The PCSP has its roots in such classic problems as the Approximate Graph Coloring and the…
An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the…
We introduce cosurfaces with values in the group \(\PC_n(H)\) of \(H\)-valued reciprocal pairwise comparison matrices. The composition law is covariant on upper triangular coefficients and contravariant on lower triangular coefficients,…
An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a…
We investigate questions about the cone $\mathrm{SEP}_n$ of separable bipartite states, consisting of the Hermitian matrices acting on $\mathbb{C}^n\otimes\mathbb{C}^n$ that can be written as conic combinations of rank one matrices of the…
We give conditions for when the tensor product of two positive maps between matrix algebras is a positive map. This happens when one map belongs to a symmetric mapping cone and the other to the dual cone. Necessary and sufficient conditions…
Capturing the composition patterns of relations is a vital task in knowledge graph completion. It also serves as a fundamental step towards multi-hop reasoning over learned knowledge. Previously, several rotation-based translational methods…
There is a profound connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and solving various challenging graph-related problems. Conversely, graph theory provides a rich set of…