A special configuration of graph state stabilizers, which contains only Pauli ${\ensuremath{\sigma}}_{X}$ operators, is studied. The vertex sets $\ensuremath{\xi}$ associated with such configurations are defined as what we call
X chains of graph states. The
X chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the
X basis via the so-called
X-chain factorization diagram. We show that graph states with different
X-chain groups can have different probability distributions of
X-measurement outcomes, which allows one to distinguish certain graph states with
X measurements. We provide an approach to find the Schmidt decomposition of graph states in the
X basis. The existence of
X chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of
X chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via
X chains, while its computational complexity with other known methods increases exponentially.
A special configuration of graph state stabilizers, which contains only Pauli
σX operators, is studied. The vertex sets
ξ associated with such configurations are defined as the X-chains of graph states. The X-chains of a general graph state can be determined efficiently. With the help of X-chains, one obtains the explicit representation of graph states in the X-basis via the so-called X-chain factorization diagram. We show that graph states with different X-chains can have different probability distributions of X-measurement outcomes, which allows to distinguish certain graph states with X-measurements. We provide an approach to find the Schmidt decomposition of graph states in the X-basis. The existence of X-chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all these applications, the difficulty of the task decreases with increasing number of X-chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X-chains, while its computational complexity with other known methods increases exponentially.
We investigate the Bell-type inequalities of graph states. In this paper, Bell-type inequalities can be derived based on two kinds of the associated subgraphs of the graph states. First, the star subgraphs lead to the maximal violation of the modified Seevinck-Svetlichny inequalities. Second, cycle subgraphs lead to maximal violation of Bell-type inequalities. As a result, once the associated graph of a graph state is given, the corresponding Bell operators can be immediatedly determined using stabilizing generators. In the above Bell-type inequalities, two measurement settings for each party are required.
Abstract In this paper, we use graph theory to study the distinguishability of lattice states under local operations and classical communication (LOCC) in C p 2 ⨂ C p 2 . Firstly, we present that for the basis of lattice unitary matrices, there are (p 2 + 1)(p + 1) distinct maximal commuting sets in p 2-dimensional system. Secondly, for any set of lattice states, we can obtain a graph G which consists of some k-complete subgraphs. Using the graph G, we show that the set can be distinguished by one-way LOCC if n ( p + 1 ) − ∑ k = 1 p + 1 s k ( k − 1 ) < ( p 2 + 1 ) ( p + 1 ) , where n is the number of vertices of a graph G and s k is the number of k-complete subgraphs. Moreover, we also proved that this conclusion is necessary and sufficient in C 2 2 ⨂ C 2 2 . For C p 2 ⨂ C p 2 system, our result can cover the results in (2015, Phys. Rev. A 92, 042 320); (2020, Sci. China-Phys. Mech. Astron. 63, 280 312).
We present a general theory for the construction of witnesses that detect genuine multipartite entanglement in graph states. First, we present explicit witnesses for all graph states of up to six qubits which are better than all criteria so far. Therefore, lower fidelities are required in experiments that aim at the preparation of graph states. Building on these results, we develop analytical methods to construct two different types of entanglement witnesses for general graph states. For many classes of states, these operators exhibit white noise tolerances that converge to 1 when increasing the number of particles. We illustrate our approach for states such as the linear and the 2D cluster state. Finally, we study an entanglement monotone motivated by our approach for graph states.