Determining X-chains in graph states

TL;DR

This paper introduces a new criterion for identifying X-chains in graph states, providing an efficient method using the Bareiss algorithm, and extends the concept to Euler chains for constructing Bell inequalities.

Contribution

It presents a necessary and sufficient criterion for X-chains, an efficient determination method, and generalizes to Euler chains for advanced quantum state analysis.

Findings

Efficient determination of X-chains using Bareiss algorithm.

Analytical approach for searching X-chain groups.

Euler chains help construct multipartite Bell inequalities.

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.