论文标题
$ d_ {1} \ otimes d_ {2}的检测
Detection of $d_{1}\otimes d_{2}$ Dimensional Bipartite Entangled State: A Graph Theoretical Approach
论文作者
论文摘要
Braunstein等。 al。已经通过图理论方法开始研究量子状态的纠缠特性。他们的想法是从一个简单的未加权图$ g $开始，然后他们从图$ g $的拉普拉斯（Laplacian）定义了量子状态。已经使用类似的想法已经完成了许多研究。我们在这里询问相反的一个，即我们可以从密度矩阵中生成图吗？为了调查这个问题，我们构建了一个Unital Map $ ϕ $，以便$ ϕ（ρ）=l_ρ+ρ$，其中量子状态由密度运算符$ρ$描述。 $l_ρ$的条目取决于量子状态$ρ$的条目，并且条目以$l_ρ$满足Laplacian的所有属性的方式进行。这使得从Laplacian $L_ρ$设计一个简单的连接加权图。我们表明，构造的Unital Map $ ϕ $表征了量子状态相对于其纯度的特征，表明如果矩阵$ ϕ（ρ）-i $的决定因素为正，则量子状态$ρ$代表混合状态。此外，我们根据所研究的密度基质的光谱和与给定密度基质相关的拉普拉斯元素的光谱研究了阳性部分转置（PPT）标准。此外，我们得出了密度矩阵的最小特征值与简单加权图的连接子图的边缘的重量之间的不平等，以检测$ d_ {1} \ otimes d_ {2} \ otimes d_ {2} $ d_ {2} $ dimensional dimensional bipartite量子态。最后，我们以很少的例子说明了结果。
Braunstein et. al. have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph $G$ and then they have defined the quantum state from the Laplacian of the graph $G$. A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map $ϕ$ such that $ϕ(ρ)=L_ρ+ρ$, where the quantum state is described by the density operator $ρ$. The entries of $L_ρ$ depends on the entries of the quantum state $ρ$ and the entries are taken in such a way that $L_ρ$ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian $L_ρ$. We show that the constructed unital map $ϕ$ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix $ϕ(ρ)-I$ is positive then the quantum state $ρ$ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of $d_{1} \otimes d_{2}$ dimensional bipartite quantum states. Lastly, We have illustrated our results with few examples.
