Bipartite separability and non-local quantum operations on graphs

Abstract

In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for separability, although there are entangled states with positive partial transpose for which the degree criterion fails. Here we introduce the concept of partially symmetric graphs and degree symmetric graphs by using the well-known concept of partial transposition of a graph and degree criteria, respectively. Thus, we provide classes of bipartite separable states of dimension $m\times n$ arising from partially symmetric graphs. We identify partially asymmetric graphs that lack the property of partial symmetry. We develop a combinatorial procedure to create a partially asymmetric graph from a given partially symmetric graph. We show that this combinatorial operation can act as an entanglement generator for mixed states arising from partially symmetric graphs.

13 pages, version is close to the one published in Physical Review A

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• Efficient Entanglement Measure for Graph States
• Quantum discord of states arising from graphs
• Phase Squeezing of Quantum Hypergraph States
• Seidel switching for weighted multi-digraphs and its quantum perspective
• Condition for zero and non-zero discord in graph Laplacian quantum states
• Multipartite separability of density matrices of graphs
• Detection of $d_1\otimes d_2$ Dimensional Bipartite Entangled State: A Graph Theoretical Approach