Detection of $d_{1}\otimes d_{2}$ Dimensional Bipartite Entangled State: A Graph Theoretical Approach

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 $\phi$ such that $\phi(\rho)=L_{\rho}+\rho$, where the quantum state is described by the density operator $\rho$. The entries of $L_{\rho}$ depends on the entries of the quantum state $\rho$ and the entries are taken in such a way that $L_{\rho}$ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian $L_{\rho}$. We show that the constructed unital map $\phi$ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix $\phi(\rho)-I$ is positive then the quantum state $\rho$ 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.