在20世紀90年代中期，量子資訊的研究引起廣泛的關注與研究，在這段期間Shor所提出了量子因子分解算法，證明了量子計算機比古典計算機在信息的處理上有更巨大的潛力。除此之外，量子資訊包含很多新興的領域，如量子計算、量子糾纏、量子搜索算法等，而我們認為量子資訊的應用最根本的物理是量子糾纏，因此我們以量子糾纏為主要的研究方向。

In the 1990’s, the research of quantum information attracts many people’s attention. In this period of time, Shor find a new method to demonstrate that a quantum computer could factor very large numbers super-efficiently. The method also shows that quantum computer has more potential than classical computer. Beside, quantum information contains many different new fields, such as quantum computation, quantum entanglement, quantum searching, etc. We believe the most fundamental physics of the applications of quantum information is quantum entanglement. In order to understand the physical meaning of entanglement, we choose entanglement as the goal of our thesis.

致謝 I
摘要 II
ABSTRACT III
CONTENTS IV
LIST OF FIGURES VI

CHAPTER 1 INTRODUCTION 1
1.1 HISTORY AND MOTIVATION 1
1.2 STRUCTURE OF THE THESIS 2

CHAPTER 2 QUANTUM ENTANGLEMENT 3
2.1 BACKGROUNDS 3
2.1.1 Axioms of Quantum Mechanics 3
2.1.2 The Qubit 3
2.1.3 The Density Matrix 4
2.1.4 The Bloch Sphere 6
2.1.5 Schmidt Decomposition 8
2.2 EPR PARADOX 10
2.3 HIDDEN VARIABLE 11
2.4 BELL INEQUALITY 11
2.5 QUANTUM ENTANGLEMENT 13
2.6 APPLICATIONS OF ENTANGLEMENT 14
2.6.1 Superdense coding 14
2.6.2 Quantum teleportation 15

CHAPTER 3 DETECTIONS FOR QUANTUM ENTANGLEMENT 18
3.1 INTRODUCTION 18
3.2 CHSH INEQUALITY 18
3.3 GENEVA INEQUALITY 20
3.4 QUANTUM STATE TOMOGRAPHY 23
3.5 HEFEI INEQUALITY 26

CHAPTER 4 WITNESS CRITERIA 29
4.1 CORRELATION CRITERIA 29
4.2 WITNESS FOR GHZ STATE 30
4.2.1 The GHZ State 30
4.2.2 Witness for N-qubit GHZ State 31
4.3 WITNESS FOR CLUSTER STATE 34
4.3.1 The Cluster state 34
4.3.2 Witness for N-qubit Cluster State 36
4.4 WITNESS FOR GRAPH STATE 39
4.4.1 The graph state 39
4.4.2 Witness for N-qubit graph State 40

CHAPTER 5 SUMMARY 41
5.1 CONCLUSION 41
5.2 FURTHER WORK 41

APPENDIX 43
A.1 CREATING ENTANGLED STATE 43
A.2 COMPUTING PROGRAMS 45

REFERENCE 49

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