Detection of d ₁ ⨂ d ₂ dimensional bipartite entangled state: a graph theoretical approach

Braunstein et. al. have started the study of entanglement properties quantum states through graph theoretical approach. Their idea was to start from a simple unweighted $G$ and then they defined state Laplacian $G$. A lot research had already been done using similar idea. We ask here opposite one i.e can we generate density matrix? To investigate this question, constructed unital map $\phi$ such that $\phi(\rho)=L_{\rho}+\rho$, where is described by operator $\rho$. The entries $L_{\rho}$ depends on $\rho$ are taken in way satisfies all Laplacian. This make possible design connected weighted $L_{\rho}$. show characterize with respect its purity showing if determinant matrix $\phi(\rho)-I$ positive represent mixed state. Moreover, partial transpose (PPT) criterion terms spectrum under investigation associated given matrix. Furthermore, derive inequality between minimum eigenvalue weight edges subgraph detect $d_{1} \otimes d_{2}$ dimensional bipartite states. Lastly, illustrated our results few examples.