Positive maps are essential in the description of quantum systems. However, characterization structure set all positive is a challenge mathematics and mathematical physics. We construct linear map from M4 to M5 state conditions under which they completely (copositivity positive).

Thus the definition of the permanent is similar to that of the determinant except for the sign associated with each term in the summation. This minor difference in the definition makes the two functions quite unlike each other. Perhaps the permanent cannot compete with its cousin, the determinant, in terms of the depth of theory and the breadth of applications, but it is safe to say that the pe...

Let K ⊂ E, K′ ⊂ E′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E ⊗ E′ is K ⊗ K′-separable if it can be represented as finite sum y = P l xl ⊗ x ′ l, where xl ∈ K and x ′ l ∈ K ′ for all l. Let S(n), H(n), Q(n) be the spaces of n × n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S+(n), H+...

Factor widths of nonnegative integral positive semidefinite square matrices are investigated. The nonnegative factor width, the exact factor width and the binary factor width of such matrices are introduced. Some lower and upper bounds for these widths are obtained. Nonnegative symmetric (completely positive) matrices with some given nonnegative (binary) factor widths

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

In the convergence theory of multisplittings for symmetric positive definite (s.p.d.) matrices it is usually assumed that the weighting matrices are scalar matrices, i.e., multiples of the identity. In this paper, this restrictive condition is eliminated. In its place it is assumed that more than one (inner) iteration is performed in each processor (or block). The theory developed here is appli...

The structural properties of the completely positive semidefinite cone CS+, consisting of all the n×n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size, are investigated. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set Q of bipartite quantum corr...

Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An r-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. Th...

We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and that our geometric mean generalizes many inequalities satisfied by the geometric mean of two positive semidefinite matrices. We prove some new properties of the geometric mean of ...