Positive tensor products of maps andn-tensor-stable positive qubit maps

Linear Maps Preserving Numerical Radius of Tensor Products of Matrices

Let m,n ≥ 2 be positive integers. Denote by Mm the set of m×m complex matrices and by w(X) the numerical radius of a square matrix X. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map φ : Mmn →Mmn satisfies w(φ(A⊗B)) = w(A⊗B) for all A ∈Mm and B ∈Mn if and only if there is a unitary matrix U ∈Mmn and a complex unit ξ such that φ(A⊗B) = ξU(φ1(...

Positive tensor factorization

A novel fixed point algorithm for positive tensor factorization (PTF) is introduced. The update rules efficiently minimize the reconstruction error of a positive tensor over positive factors. Tensors of arbitrary order can be factorized, which extends earlier results in the literature. Experiments show that the factors of PTF are easier to interpret than those produced by methods based on the s...

Tensor network complexity of multilinear maps

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. This model turns out to be strong enough to capture the current best algorithms for a variety of problems, e.g., O(n ) time matrix multiplication, O(n logn) time discrete Fourier transform, O(n ) time 3t-clique counting and O∗(2n) time for computing the permanent of a matrix. For counting homomorphism...

Normal CP-Maps Admit Weak Tensor Dilations

K-decomposability of Positive Maps

For any C-algebra A let A denote the set of all positive elements in A. A state on a unital C-algebra A is a linear functional ω : A → C such that ω(a) ≥ 0 for every a ∈ A and ω(I) = 1 where I is the unit of A. By S(A) we will denote the set of all states on A. For any Hilbert space H we denote by B(H) the set of all bounded linear operators on H . A linear map φ : A → B between C-algebras is c...