Fixed Points Associated to Power of Normal Completely Positive Maps*

Let be a normal completely positive map with Kraus operators . An operator X is said to be a fixed point of , if . Let be the fixed points set of . In this paper, fixed points of are considered for , where means j-power of . We obtain that and for integral when A is self-adjoint and commutable. Moreover, holds under certain condition.

Completely Positive Definite Maps Over Topological-algebras

In this paper, we discuss completely positive definite maps over topological algebras. A Schwarz type inequality for n-positive definite maps, and the Stinespring representation theorem for completely positive definite maps over topological algebras are given.

A Classification of Completely Positive Maps

This paper concerns classifying completely positive maps between certain C*-algebras. Several invariants for classifying completely positive maps are constructed. It is proved that one of them is isomorphic to the Ext-group of C*-algebra extensions in special circumstances. Furthermore, this invariant induces a functor from C*-algebras to abelian groups which is split-exact.

ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS

For n≥3,we construct a class{Wn,π1,π2}of n^(2)×n^(2) hermitian matrices by the permutation pairs and show that,for a pair{π1,π2}of permutations on(1,2,…,n),Wn,π1,π2 is an entanglement witness of the n⊗n system if{π1,π2}has the property(C).Recall that a pair{π1,π2}of permutations of(1,2,…,n)has the property(C)if,for each i,one can obtain a permutation of(1,…,i−1,i+1,…,n)from(π1(1),…,π1(i−1),π1(i+1),…,π1(n))and(π2(1),…,π2(i−1),π2(i+1),…,π2(n)).We further prove that Wn,π1,π2 is not comparable with Wn,π,which is the entanglement witness constructed from a single permutationπ;Wn,π1,π2 is decomposable ifπ1π2=id orπ21=π22=id.For the low dimensional cases n∈{3,4},we give a sufficient and necessary condition onπ1,π2 for Wn,π1,π2 to be an entanglement witness.We also show that,for n∈{3,4},Wn,π1,π2 is decomposable if and only ifπ1π2=id orπ21=π22=id;W3,π1,π2 is optimal if and only if(π1,π2)=(π,π2),whereπ=(2,3,1).As applications,some entanglement criteria for states and some decomposability criteria for positive maps are established.

Lower Bound of Concurrence Based on Generalized Positive Maps

We study the concurrence of arbitrary dimensional bipartite quantum systems. By using a positive but not completely positive map, we present an anaJytical lower bound of concurrence. Detailed examples are used to show that our bound can detect entanglement better and can improve the well known existing lower bounds.

New Developments on Operator Reverse AM-GM Inequality for Positive Linear Maps

In this research article,we shall give some reverse Arithmetic-Geometric mean inequalities for unital positive linear maps on Hilbert space operators under some different conditions.Our results are sharper and more precise as compared to some recent published results.Moreover,we shall present refinements of the Lin conjecture.

Positively Expansive Differentiable Maps

We study the space of positively expansive differentiable maps of a compact connected C ∞ Riemannian manifold without boundary. It is proved that （i） the C1-interior of the set of positively expansive differentiable maps coincides with the set of expanding maps, and （ii） Cl-generically, a differentiable map is positively expansive if and only if it is expanding.

Complementary Inequalities to Improved AM-GM Inequality

Following an idea of Lin, we prove that if A and B are two positive operators such that 0 〈 mI 〈 A 〈 m＇I ≤ M＇I ≤B 〈 MI, then Ф^2（A＋B/2）≤K^2（h）/（1＋（logM＇/m＇）^2/8）^2Ф^2（A#B）,and Ф^2（A＋B/2）≤K^2（h）/（1＋（logM＇/m＇）^2/8）^2（Ф（A）#Ф（B））^2,where K（h） = （h＋1）2 /4h and h = M and Ф is a positive unital linear map.

SPECTRUM-PRESERVING ELEMENTARY OPERATORS ON B(X)

Characterizations for elementary operators of length 2 to be invertibility-preserving,spectrum-preserving or spectrum-compressing are obtained.