Strong skew commutativity preserving maps on von Neumann algebras

Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $\Phi:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $\Phi$ is strong skew commutativity preserving (that is, satisfies $\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^*=AB-BA^*$ 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 $\Phi(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.Comment: 16 page

Detecting entanglement of states by entries of their density matrices

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.Comment: 14 page

A characterization of optimal entanglement witnesses

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).Comment: 1

Positive finite rank elementary operators and characterizing entanglement of states

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.Comment: 26 page

Linear maps preserving separability of pure states

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.Comment: 16 page

Fidelity of states in infinite dimensional quantum systems

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.Comment: 12 page

Optimality of a class of entanglement witnesses for 3⊗33\otimes 3 systems

Let $\Phi_{t,\pi}: M_3({\mathbb C}) \rightarrow M_3({\mathbb C})$ be a linear map defined by $\Phi_{t,\pi}(A)=(3-t)\sum_{i=1}^3E_{ii}AE_{ii}+t\sum_{i=1}^3E_{i,\pi(i)}AE_{i,\pi(i)}^\dag-A$, where $0\leq t\leq 3$ and $\pi$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{\Phi_{t,\pi}}$ induced by $\Phi_{t,\pi}$ is an optimal entanglement witness if and only if $t=1$ and $\pi$ is cyclic.Comment: 12 page

Coherence measures and optimal conversion for coherent states

We discuss a general strategy to construct coherence measures. One can build an important class of coherence measures which cover the relative entropy measure for pure states, the $l_1$-norm measure for pure states and the $\alpha$-entropy measure. The optimal conversion of coherent states under incoherent operations is presented which sheds some light on the coherence of a single copy of a pure state.Comment: in Quantum Information & Computation 201

Lie ring isomorphisms between nest algebras on Banach spaces

Let ${\mathcal N}$ and ${\mathcal M}$ be nests on Banach spaces $X$ and $Y$ over the (real or complex) field $\mathbb F$ and let \mbox{\rm Alg}{\mathcal N} and \mbox{\rm Alg}{\mathcal M} be the associated nest algebras, respectively. It is shown that a map $\Phi:{\rm Alg}{\mathcal N}\rightarrow{\rm Alg}{\mathcal M}$ is a Lie ring isomorphism (i.e., $\Phi$ is additive, Lie multiplicative and bijective) if and only if $\Phi$ has the form $\Phi(A) = TAT^{-1} + h(A)I$ for all A\in \mbox{\rm Alg}{\mathcal N} or $\Phi(A)=-TA^*T^{-1}+h(A)I$ for all A\in \mbox{\rm Alg}{\mathcal N}, where $h$ is an additive functional vanishing on all commutators and $T$ is an invertible bounded linear or conjugate linear operator when $\dim X=\infty$; $T$ is a bijective $\tau$-linear transformation for some field automorphism $\tau$ of $\mathbb F$ when $\dim X<\infty$.Comment: 27 page

Characterizing derivations for any nest algebras on Banach spaces by their behaviors at an injective operator

Let ${\mathcal N}$ be a nest on a complex Banach space $X$ and let \mbox{ Alg}{\mathcal N} be the associated nest algebra. We say that an operator Z\in \mbox{ Alg}{\mathcal N} is an all-derivable point of \mbox{ Alg}{\mathcal N} if every linear map $\delta$ from \mbox{ Alg}{\mathcal N} into itself derivable at $Z$ (i.e. $\delta$ satisfies $\delta(A)B+A\delta(B)=\delta(Z)$ for any A,B \in \mbox{ Alg}{\mathcal N} with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in \mbox{Alg}{\mathcal N} are all-derivable points of \mbox{Alg}{\mathcal N} without any additional assumption on the nest.Comment: 20 page