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Quantum Information Processing

Entanglement and separability of graph Laplacian quantum states

Anoopa Joshi, Parvinder Singh, Atul Kumar

Published Apr 1, 2022• 1 citation
Publisher
DOI
10.1007/s11128-022-03483-z

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This article delves into an analysis of the intrinsic entanglement and separability feature in quantum states as depicted by graph Laplacian. We show that the presence or absence of edges in the graph plays a pivotal role in defining the entanglement or separability of these states. We propose a set of criteria for ascertaining the separability of quantum states comprising nnn-qubit within a composite Hilbert space, indicated as H=H1⊗H2⊗⋯⊗HnH=H_1 \otimes H_2 \otimes \dots \otimes H_nH=H1​⊗H2​⊗⋯⊗Hn​. This determination is achieved through a combination of unitary operators, neighbourhood sets, and equivalence relations.
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.