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The Electronic Journal of Combinatorics

Separability of Density Matrices of Graphs for Multipartite Systems

Chen Xie, Hui Zhao, Zhi‐Xi Wang

Published Nov 22, 2013• 5 citations
Publisher

Abstract

We investigate separability of Laplacian matrices of graphs when seen as density matrices. This is a family of quantum states with many combinatorial properties. We firstly show that the well-known matrix realignment criterion can be used to test separability of this type of quantum states. The criterion can be interpreted as novel graph-theoretic idea. Then, we prove that the density matrix of the tensor product of N graphs is N-separable. However, the converse is not necessarily true. Additionally, we derive a sufficient condition for N-partite entanglement in star graphs and propose a necessary and sufficient condition for separability of nearest point graphs.

DOI
10.37236/3092

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We investigate separability of Laplacian matrices of graphs when seen as density matrices. This is a family of quantum states with many combinatorial properties. We firstly show that the well-known matrix realignment criterion can be used to test separability of this type of quantum states. The criterion can be interpreted as novel graph-theoretic idea. Then, we prove that the density matrix of the tensor product of N graphs is N-separable. However, the converse is not necessarily true. Additionally, we derive a sufficient condition for N-partite entanglement in star graphs and propose a necessary and sufficient condition for separability of nearest point graphs.
The density matrix of a graph is the combinatorial laplacian matrix of a graph normalized to have unit trace. In this paper we generalize the entanglement properties of mixed density matrices from combinatorial laplacian matrices of graphs discussed in Braunstein et al. [Annals of Combinatorics, 10 (2006) 291] to tripartite states. Then we prove that the degree condition defined in Braunstein et al. [Phys. Rev. A, 73 (2006) 012320] is sufficient and necessary for the tripartite separability of the density matrix of a nearest point graph.
Recently, Braunstein et al. [1] 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 [1]. Finally, we extend some of the results in [1,2] to the multipartite case and show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, categorical, etc.) is multipartite separable.
We reconsider density matrices of graphs as defined in quant-ph/0406165. The density matrix of a graph is the combinatorial Laplacian of the graph normalized to have unit trace. We describe a simple combinatorial condition (the ``degree condition'') to test the separability of density matrices of graphs. The condition is directly related to the Peres-Horodecki partial transposition condition. We prove that the degree condition is necessary for separability, and we conjecture that it is also sufficient. We prove special cases of the conjecture involving nearest-point graphs and perfect matchings. We observe that the degree condition appears to have a value beyond the density matrices of graphs. In fact, we point out that circulant density matrices and other matrices constructed from groups always satisfy the condition and indeed are separable with respect to any split. We isolate a number of problems and delineate further generalizations.
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.
We give a new separability criterion, a necessary condition for separability of N-partite quantum states. The criterion is based on the Bloch representation of a N-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N-partite quantum states in H=Hd1⊗Hd2⊗⋯⊗HdN.\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.H=Hd1​⊗Hd2​⊗⋯⊗HdN​. The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the NNN-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and sufficient condition for separability of a class of N-qubit states which includes N-qubit PPT states.