arXiv subjects
Publications and source records attributed to Xiaofei Qi.
Linear maps preserving pure states of a quantum system of any dimension are characterized. This is then used to establish a structure theorem for linear maps that preserve separable pure states in multipartite systems. As an application, a characterization of separable pure state preserving affine maps is obtained.
We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered.
Let $Φ_{t,π}: M_3({\mathbb C}) \rightarrow M_3({\mathbb C})$ be a linear map defined by $Φ_{t,π}(A)=(3-t)\sum_{i=1}^3E_{ii}AE_{ii}+t\sum_{i=1}^3E_{i,π(i)}AE_{i,π(i)}^†-A$, where $0\leq t\leq 3$ and $π$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{Φ_{t,π}}$ induced by $Φ_{t,π}$ is an optimal entanglement witness if and only if $t=1$ and $π$ is cyclic.
Maps $Φ$ which do not increase the spectrum on complex matrices in a sense that $\Sp(Φ(A)-Φ(B))\subseteq\Sp(A-B)$ are classified.
Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $Φ:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $Φ$ is strong skew commutativity preserving (that is, satisfies $Φ(A)Φ(B)-Φ(B)Φ(A)^*=AB-BA^*$ for all $A,B\in{\mathcal M}$) if and only if there exists some self-adjoint element $Z$ in the center of ${\mathcal M}$ with $Z^2=I$ such that $Φ(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.
In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain new indecomposable optimal witnesses which have no spanning property. These also provide new examples which support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).
In this paper we discuss the fidelity of states in infinite dimensional systems, give an elementary proof of the infinite dimensional version of Uhlmann's theorem, and then, apply it to generalize several properties of the fidelity from finite dimensional case to infinite dimensional case. Some of them are somewhat different from those for finite dimensional case.
Let $\mathcal A$ and $\mathcal B$ be unital rings and $\mathcal M$ be a $(\mathcal A, \mathcal B)$-bimodule, which is faithful as a left $\mathcal A$-module and also as a right $\mathcal B$-module. Let ${\mathcal U}={\rm Tri}(\mathcal A, \mathcal M, \mathcal B)$ be the associated triangular ring. It is shown that every additive generalized Jordan (triple) higher derivation on $\mathcal U$ is a generalized higher derivation.
For any bipartite systems, a universal entanglement witness of rank-4 for pure states is obtained and a class of finite rank entanglement witnesses is constructed. In addition, a method of detecting entanglement of a state only by entries of its density matrix with respect to some product basis is obtained.
In this paper, a class of indecomposable positive finite rank elementary operators of order $(n,n)$ are constructed. This allows us to give a simple necessary and sufficient criterion for separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order $(2,2)$ and get some new mixed entangled states that can not be detected by the positive partial transpose (PPT) criterion and the realignment criterion.
It is shown that, every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $αI+T$ with $α$ a nonnegative number and $T$ a finite rank self-adjoint operator. We also provide two methods of constructing entanglement witness and apply them to obtain some entangled states that cannot be detected by the PPT criterion and the realignment criterion.
The additive (generalized) $ξ$-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumption, that an additive map $L$ is an additive (generalized) Lie derivation if and only if it is the sum of an additive (generalized) derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) $ξ$-Lie derivation with $ξ\not=1$ if and only if it is an additive (generalized) derivation satisfying $L(ξA)=ξL(A)$ for all $A$. These results are then used to characterize additive (generalized) $ξ$-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.