量子糾纏是只有在量子力學的框架中才會出現的現象。目前已有許多以量子糾纏為基礎的量子科技藍圖，量子傳輸及超密編碼都利用了二方量子糾纏的特殊的性質來增加通訊的效率。二方量子糾纏的定義及量化目前已經被完備地研究，但多方量子糾纏的定義和量化尚未被研究完全。目前仍有許多量化多方量子糾纏的研究，部分的研究將量子態糾纏的程度定義為其與可分離態的距離，另一部份的研究將量子糾纏的程度定義為子系統混合態的混合程度。
在本論文中，我們將在量子純態的框架下用Schmidt分解和Wallach-Meyer測度討論量化三方量子糾纏的問題。我們將量子態分成幾種類別，並以這兩種方式猜測可能是較佳三方量子糾纏態的幾種量子態，同時討論Scmidt分解及Wallach-Meyer測度用於量化多方量子糾纏的優缺點。

Quantum entanglement is a fundamental phenomenon in quantum mechanics. Several applications are based on quantum entanglement. For example, both quantum teleportation and superdense coding utilize bipartite quantum entanglement to transmit information. Currently the definition and quantification of bipartite quantum entanglement are complete. However, the concept of multipartite quantum entanglement is still not clear. Many researchers are investigating entanglement in multipartite systems. There are literatures defining entanglement by the distance between the target state and the nearest separable state. Also there are researchers saying that multipartite entanglement is defined by the degree of mixing after tracing out subsystems.
In this thesis, we investigate the tripartite entanglement by two-layer Schmidt decomposition and Wallach-Meyer measure in the case of pure states. That is, our approach does not introduce density operators and the partial trace operation.

1 Introduction 5
2 Basic Concept of Quantum Mechanics 7
2.1 Quantum States, Measurement, and Quantum Gates . . . . . . . . . . 7
2.2 Composite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Multipartite Entanglement Measure 11
3.1 Bipartite Quantum Entanglement and Non-Locality . . . . . . . . . . 11
3.2 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Wallach-Meyer Measure . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Schmidt Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Characterizing Quantum Entanglement 19
4.1 Investigate Wallach-Meyer Measure . . . . . . . . . . . . . . . . . . 19
4.2 Tripartite Separable Quantum States . . . . . . . . . . . . . . . . . . 21
4.3 Discussion on Entangled States . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Superposition of 2 basis states . . . . . . . . . . . . . . . . . 26
4.3.2 Superposition of 3 basis states . . . . . . . . . . . . . . . . . 29
4.3.3 Superposition of 4 basis states . . . . . . . . . . . . . . . . . 32
5 Conclusion 42
Bibliography 43

[1] A. Einstein, B. Podolsky, and N. Rosen, ”Can quantum-mechanical description of
physical reality be considered complete?” Physical Review, vol. 47, pp. 777–780,
May, 1935.
[2] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information,
Cambridge University Press, 2000
[3] D. A. Meyer and N. R. Wallach , ”Global entanglement in multiparticle systems,”
J. of Math. Phys., vol. 43, pp. 4273-4278, Aug., 2002.
[4] A. J. Scott, ”Multipartite entanglement, quantum-error-correcting codes, and entangling
power of quantum evolutions,” Phys. Rev. A, vol. 69, 052330, May, 2004.
[5] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters ,
”Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-
Rosen channels,” Phys. Rev. Lett., vol. 70, pp. 1895-1899, Mar., 1993.