Generalized reduction criterion for separability of quantum states

A new necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L.A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The new separability criterion naturally generalizes the reduction separability criterion introduced independently in previous work of [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999)] and [N.J. Cerf, C. Adami and R.M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L.A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.Comment: 17 pages, 2 figure

Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.Comment: 17 pages, to appear in JF

On the separability of graphs

Recently, Cicalese and Milanič introduced a graph-theoretic concept called separability. A graph is said to be k-separable if any two non-adjacent vertices can be separated by the removal of at most k vertices. The separability of a graph G is the least k for which G is k-separable. In this paper, we investigate this concept under the following three aspects. First, we characterize the graphs for which in any non-complete connected induced subgraph the connectivity equals the separability, so-called separability-perfect graphs. We list the minimal forbidden induced subgraphs of this condition and derive a complete description of the separability-perfect graphs.We then turn our attention to graphs for which the separability is given locally by the maximum intersection of the neighborhoods of any two non-adjacent vertices. We prove that all (house,hole)-free graphs fulfill this property – a class properly including the chordal graphs and the distance-hereditary graphs. We conclude that the separability can be computed in O(m∆) time for such graphs.In the last part we introduce the concept of edge-separability, in analogy to edge-connectivity, and prove that the class of k-edge-separable graphs is closed under topological minors for any k. We explicitly give the forbidden topological minors of the k-edge-separable graphs for each 0 ≤ k ≤ 3

On the separability of graphs

Recently, Cicalese and Milanič introduced a graph-theoretic concept called separability. A graph is said to be k-separable if any two non-adjacent vertices can be separated by the removal of at most k vertices. The separability of a graph G is the least k for which G is k-separable. In this paper, we investigate this concept under the following three aspects. First, we characterize the graphs for which in any non-complete connected induced subgraph the connectivity equals the separability, so-called separability-perfect graphs. We list the minimal forbidden induced subgraphs of this condition and derive a complete description of the separability-perfect graphs.We then turn our attention to graphs for which the separability is given locally by the maximum intersection of the neighborhoods of any two non-adjacent vertices. We prove that all (house,hole)-free graphs fulfill this property ? a class properly including the chordal graphs and the distance-hereditary graphs. We conclude that the separability can be computed in O(m?) time for such graphs.In the last part we introduce the concept of edge-separability, in analogy to edge-connectivity, and prove that the class of k-edge-separable graphs is closed under topological minors for any k. We explicitly give the forbidden topological minors of the k-edge-separable graphs for each 0 ≤ k ≤ 3

Separability and Fourier representations of density matrices

Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for $d$-dimensional spaces, and the resulting set of unitary matrices $S(d)$ is a basis for $d\times d$ matrices. If $N=d_{1}\times d_{2}\times...\times d_{b}$ and H^{[ N]}=\bigotimes H^{% [ d_{k}]}, we give a sufficient condition for separability of a density matrix $\rho$ relative to the $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of $\rho >.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space $H^{[ N]}$% . It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime $p$ and $n>1$ the generalized Werner density matrix $W^{[ p^{n}]}(s)$ is fully separable if and only if $s\leq (1+p^{n-1}) ^{-1}$