DeterminingX-chains in graph states

Abstract

Abstract. The representation of graph states in the X-basis as well as the calculation of graph state overlaps can efficiently be performed by using the concept of X-Chains [Phys. Rev. A 92 (1) 012322]. We present a necessary and sufficient criterion for X-chains and show that they can efficiently be determined by Bareiss algorithm. An analytical approach for searching X-chain groups of a graph state is proposed. Furthermore we generalize the concept of X-chains to so-called Euler chains, whose induced subgraphs are Eulerian. This approach helps to determine if a given vertex set is an X-chain and we show how Euler chains can be used in the construction of multipartite Bell inequalities for graph states.

Group structures and representations of graph states

“…A special type of X-resources was defined as X-chains: an X-chain is a subset of vertices for a given graph, such that the product of the stabilizer generators associated with these vertices contains only σ X -Pauli operators. The set of X-chains of a graph state is a group, which can be calculated efficiently [8]. The X-chain groups revealed structures of graph states and showed how to distinguish them by local σ X measurements.…”

“…The X-chains of a graph state |G can be determined efficiently with standard linear algebra tools [8]. As examples, the X-chains of the graph states |S 3 , |S 4 and |C 3 are given in Table II.…”

“…Besides, the X-chains for certain types of graph states (i.e. linear graph states |L n , cycle graph states |C n , complete graph states |K n and star graph states |S n ) are studied in [8]. A MATH-EMATICA package is provided for finding X-chains in general graph states; see Supplemental Material [9].…”

“…The correlation group K G can be directly obtained if one knows the X-chain group. The X-chain group can be searched by a criterion that the cardinality of the intersection of the vertex neighborhood with the X-chain |N v ∩ ξ| should be even for all v ∈ V G [8]. The search of the X-chains of a graph state |G is equivalent to finding the 2-modulus-kernel of the adjacency matrix of the graph G. As this is efficient, the representation of graph states in the X-basis is feasible.…”

“…The X-chain group of a given graph state can be efficiently determined, the search of X-chains in a given graph state will be studied in detail elsewhere [8] and a MATHEMATICA package is available in the Supplemental Material [9]. This paper is organized as follows.…”