Recently, Braunstein et al. introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in Braunstein et al. Finally, we show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, lexicographical, etc.) is multipartite separable, extending analogous results for bipartite and tripartite separability.
[1]
C. Wu.
Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices
,
2005,
quant-ph/0508163.
[2]
S. Severini,et al.
The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States
,
2004,
quant-ph/0406165.
[3]
Simone Severini,et al.
Some families of density matrices for which separability is easily tested (10 pages)
,
2006
.
[4]
Zhen Wang,et al.
The Tripartite Separability of Density Matrices of Graphs
,
2007,
Electron. J. Comb..
[5]
Thierry Paul,et al.
Quantum computation and quantum information
,
2007,
Mathematical Structures in Computer Science.