tailieunhanh - Báo cáo toán học: "Multipartite separability of Laplacian matrices of graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Multipartite separability of Laplacian matrices of graphs. | Multipartite separability of Laplacian matrices of graphs Chai Wah Wu IBM T. J. Watson Research Center P. O. Box 704 Yorktown Heights NY 10598 USA cwwu@ chaiwahwu@ Submitted Sep 4 2008 Accepted Apr 29 2009 Published May 15 2009 Mathematics Subject Classifications 81P15 81P68 Abstract 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 Introduction The objects of study in this paper are density matrices of quantum mechanics. Density matrices are used to describe the state of a quantum system and are fundamental mathematical constructs in quantum mechanics. They play a key role in the design and analysis of quantum computing and information systems 1 . Definition 1 A complex matrix A is a density matrix if it is Hermitian positive semidef-inite and has unit trace. Remark In this paper we will often use the following simple fact tr A A is a density matrix if A is Hermitian positive semidefinite and has a strictly positive trace. We will refer to trA A as a normalization of A. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R61 1 Definition 2 A complex matrix A is row diagonally dominant if An 2j i Aij I for all