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

Energy Functionals for the Parabolic Monge-Ampere Equation

We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.Comment: 12 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

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

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

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

Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP) to reconstruct such faithful representations. While those SDP models are capable of producing high quality configuration numerically, they suffer two major drawbacks. One is that there exist no theoretically guaranteed bounds on the quality of the configuration. The other is that they are slow in computation when the data points are beyond moderate size. In this paper, we propose a convex optimization model of Euclidean distance matrices. We establish a non-asymptotic error bound for the random graph model with sub-Gaussian noise, and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low-rank matrix up to a logarithmic factor. Our results partially explain why MVU and MVE often work well. Moreover, we develop a fast inexact accelerated proximal gradient method. Numerical experiments show that the model can produce configurations of high quality on large data points that the SDP approach would struggle to cope with.Comment: 44 pages, 10 figures, 1 tabl

qq-Deformed Chern Characters for Quantum Groups SUq(N)SU_{q}(N)

In this paper, we introduce an $N\times N$ matrix $\epsilon^{a\bar{b}}$ in the quantum groups $SU_{q}(N)$ to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on $SU_{q}(N)$, such as the $q$-deformed structure constant ${\bf C}_{IJ}^{~K}$ and the $q$-deformed transposition operator $\Lambda$. From the $q$-gauge covariant condition we define the generalized $q$-deformed Killing form and the $m$-th $q$-deformed Chern class $P_{m}$ for the quantum groups $SU_{q}(N)$. Some useful relations of the generalized $q$-deformed Killing form are presented. In terms of the $q$-deformed homotopy operator we are able to compute the $q$-deformed Chern-Simons $Q_{2m-1}$ by the condition $dQ_{2m-1}=P_{m}$, Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gauge covariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived