Related papers: Semidefinite descriptions of separable matrix cone…

We express the positive partial transpose (PPT) separability criterion for symmetric states of multi-qubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix…

Many important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set S of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in several low-dimensional…

A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard.…

We search for faces of the convex set consisting of all separable states, which are affinely isomorphic to simplices, to get separable states with unique decompositions. In the two-qutrit case, we found that six product vectors spanning a…

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 introduce a generalization of the set of completely positive matrices that we call "pairwise completely positive" (PCP) matrices. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive…

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R}) $, $n\geq 3$, that preserve the minus partial order in both directions.

We study the convex hull of $SO(n)$, thought of as the set of $n\times n$ orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of $SO(n)$ is doubly spectrahedral, i.e.…

We show that the closed convex hull of any one-dimensional semi-algebraic subset of R^n has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is…

In many contexts one encounters Hermitian operators $M$ on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such $M$ is positive…

The general expression with the physical significance and positive definite condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the $4\times 4$ density matrix…

We investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products. This can either be regarded as compression of positive semidefinite factorizations of…

We study $k$-positive linear maps on matrix algebras and address two problems, (i) characterizations of $k$-positivity and (ii) generation of non-decomposable $k$-positive maps. On the characterization side, we derive optimization-based…

We investigate the Peres-Horodecki positive partial transpose (PPT) criterion in the context of conserved quantities and derive a condition of in- separability for a composite bipartite system depending only on the dimen- sions of its…

The compact set of homogeneous quadratic polynomials in $n$ real variables with modulus bounded by 1 on the unit sphere $S^{n-1}$ is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded…

If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian product of the set of non-negative reals and $A$ is a positive semi-definite $m\times m$ matrix with eigenvalues $\lambda_j$, we set…

Let $\beta\equiv\beta^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive…

Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for non-commuting observables form a subset of these inequalities. In addition, criterion of positivity…

In this paper, the geometry properties of Hankel form are studied, including their positive semi-definite (PSD) cone and sum-of-squares (SOS) cone. We denote them by $HPSD(m,n)$ and $HSOS(m,n)$, respectively. We show that both $HPSD(m,n)$…

This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex…