Some Characterizations Of Graph States

Graph states are a special class of stabilizer states that are defined based on graphs.And the graph states have been used in many applications in quantum information theory as well as quantum computation,and have been studied extensively.Bell inequality is known as the most profound discovery in science.The violation of the Bell inequality by the quantum entangled state means the nonlocality of this quantum state.Quantum nonlocality is the essential characteristic of quantum mechanics which is different from classical mechanics.It is the foundation of quantum information and quantum computing,and has profound physical significance.Using the theories and methods of operator algebra and matrix theory,some characterizations of graph states are studied.This article is divided into four chapters,the specific content is as follows:The first chapter introduces some research background and status on our main contents,lists some related notations,basic definitions,hypotheses and known theorems.In the second chapter,firstly,we introduce two definitions of graph states.Secondly,we give the detailed proof that the graph state is equivalent under two different definitions.In addition,it is showed that graph state must exist and be unique.Last but not least,the definition and properties of fully separable graph are given.In the third chapter,first of all,we give the CHSH inequality.Then we construct a general Bell inequality for the graph state.Besides,we show that the Bell inequality is maximally violated by graph state corresponding to some given graph.In addition,we obtain the classical bound and maximal quantum violation of Bell inequality are N+|n（i）|-1 and N+（（?）-1）（|n（i）|+1）,respectively.Furthermore,the classical bound and maximal quantum violation of Bell inequality are dependent on the choice of the vertex for most graphs expect for complete graph.They are fixed values,when the graph is a complete graph.In the fourth chapter,the definition of self-testing is given,and then it is proved that the states and measurements which maximally violate the Bell inequality of two-vertex complete graph can self-test.