我們使用量子力學的性質來處理資訊，量子密碼便是利用測不準原理及粒子間的糾纏性質來達到保密及傳訊的效果。量子密碼目前可以分為兩種情況，第一種是量子金鑰共享，第二種則是秘密共享。第一種情況是Alice傳訊給Bob，並在傳遞的訊息上加密。第二種情況則是有三個人，共享一個包含三個粒子的糾纏態，每個人對自己擁有的粒子做測量，則只有在蒐集其中兩個人測量的結果後，才能推測第三個人測量到的狀態為何，所以第三個粒子的狀態為另兩個人所共享。
此篇論文將Phys. Rev. A.59, 1829【18】的二態系統，推廣到多態、三粒子及多粒子系統。我們使用的是經富立葉轉換過後的基底，加以線性組合後，所構成d組基底。和Phys. Rev. A.59, 1829【18】一樣，讓參與秘密共享協定的人，任意的在上述d組基底中，隨意選取一組測量自己的粒子。經由計算我們發現在d=4的系統下，有最高的機率發現竊聽者，所以在多態系統下操作可增加協定的安全性。

We use quantum mechanics to handle our information，such as Quantum Cryptography。Quantum Cryptography can send information and keep the transmission secret by using uncertainty principal and entanglement between particles。

We can divide Quantum Cryptography into two parts。One of them is Quantum Key Distribution （QKD），the other is Quantum Secret Sharing （QSS）。QKD is the protocol between Alice and Bob，Alice is the person who sends messages to Bob and hopes to keep them in secret。QSS is the protocol between three people ，each of them holds a qubit from the initial three qubits entangled state。Every one measures his or her own particle。Only when we have information about the measuring results of two among them do we have ideas about the measuring result of the third person。
As a result，we can say that the quantum state of the third particle is shared by others。

This paper develops Phys. Rev. A.59, 1829【18】 which uses the two level 、three-partite system to multilevel、three-partite and multipartite system 。We measure in d sets of bases which are linearly combined by the Fourier Transform bases。The same as the Phys. Rev. A.59, 1829 【18】 QSS protocol ，people participating in the QSS protocol measure their own particle individually in one of the d sets bases。After calculation，we find it has the largest opportunities to detect the eavesdropper in d=4 system。

Therefore，we can have safer QSS protocols through increasing the level of quantum states。

【1】R. Feynman, “Simulating Physics with computers”, S. International Journal of Theoretical Physics 21, p.467-488 (1982).

【2】P. W. Shor, “Algorithms for quantum computation: discrete logarithm and factoring”, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, p.124-134 (1994).

【3】S. Wiesner, “Conjugate coding”, Sigact News 15, pp.78-88（1983）.

【4】Charles H. Bennett and Gilles Brassard, “Quantum cryptography: Public key and coin tossing” , Proceedings of IEEE International Conference on Computers, System and Signal Processing, Bangalore, India, p.175-179 (1984).

【5】A. Ekert, “Quantum cryptography based on Bell’s theorem”, Phys. Rev. Lett. 67, pp.661-663 (1991).

【6】C. H. Bennett, “Quantum cryptography using two nonorthogonal states”, Phys. Rev. Lett. 68,3121-3124 （1992）.

【7】J. S. Bell, Physics 1,195；J. F. Clauser, M. A. Horne,A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories”, Phys. Rev. Lett. 23, 880 （1969）.

【8】A. Einstein, B. Podolsky and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Phys. Rev. 47, 777（1935）.

【9】D. Bohm, “Quantum Theory”, Prentice Hall, Englewood Cliffs, NJ (1951).

【10】C.H. Bennett, G. Brassard and N.D. Mermin, “Quantum cryptography without Bell’s theorem”, Phys. Rev. Lett. 68, 557 (1992).

【11】H . Bechmann -Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets”, Phys. Rev. A 61, 062308 (2000).

【12】H. Bechmann-Pasquinucci and Asher Peres,“Quantum Cryptography with 3-State Systems”, Phys. Rev. Lett. 85, 3313 (2000).

【13】Nicolas J. Cerf , Mohamed Bourennane , Anders Karlsson and Nicolas Gisin,“Security of Quantum Key Distribution Using d-Level Systems”, Phys. Rev. Lett. 88, 127902 (2002).

【14】Samuel L. Braunstein and Peter Van Loock, “Quantum information with continuous variables”, quant-ph/0410100 (2004).

【15】T. C. Ralph,“Continuous variable quantum cryptography”, Phys. Rev. A 61, 010303 （2000）.

【16】M. D. Reid,“Quantum cryptography with a predetermined key, using continuous-variable Einstein-Podolsky-Rosen correlations”, Phys. Rev. A 62, 062308 (2000).

【17】D. Bruss,“Optimal Eavesdropping in Quantum Cryptography with Six States”, Phys. Rev. Lett. 81, 3018-3021 (1998).

【18】M. Hillery, Vladimír Bužek and André Berthiaume,“Quantum secret sharing”, Phys. Rev. A 59, 1829 (1999).

【19】R. Cleve, D. Gottesman and Hoi-Kwong Lo,“How to Share a Quantum Secret”, Phys. Rev. Lett. 83, 648 (1999).

【20】Saber Bagherinezhand and Vahid Karimipour, “Quantum secret sharing based on reusable Greenberger-Horne-Zeilinger states as secure carriers”, Phys. Rev. A 67, 044302 (2003).

【21】Vahid Karimipour, Alireza Bahraminasab and Saber Bagherinezhand, “Entanglement swapping of generalized cat states and secret sharing”, Phys. Rev. A 65, 042320 (2002).

【22】Li-Yi Hsu and Che-Ming Li,“Quantum secret sharing using product states”, Phys. Rev. A 71, 022321 (2005).

【23】Guo-Ping Guo and Guang-Can, “Quantum secret sharing without entanglement” Phy. Lett. A 310 (2003).

【24】Yu-Ao Chen, An-Ning Zhang, Zhi Zhao, Xiao-Qi Zhou, Chao-Yang Lu, Cheng-Zhi Peng, Tao Yang, and Jian-Wei Pan, “Experimental Quantum Secret Sharing and Third-Man Quantum Cryptography”,Phys. Rev. Lett. 95, 200502 (2005).

【25】Hideomi Nihira and C. R. Stroud, Jr.,“Robust multipartite multilevel quantum protocols”, Phys. Rev. A 72, 022337 (2005).

【26】Zhang-yin Wang and Zhan-jun Zhang, “Three-party qutrit- state sharing”, quant-ph/0607187 (2006).

【27】Adan Cabello,“N-Particle N-Level Singlet States: Some Properties and Applications”, Phys. Rev. Lett. 89, 100402 （2002）.

【28】Zhan-jun Zhang, Yong Li, and Zhong-xiao Man,“Multiparty quantum secret sharing”, Phys. Rev. A 71, 044301 （2005）.

【29】Li Xiao, Gui Lu Long, Fu-Guo Deng, and Jian-Wei Pan,“Efficient multiparty quantum-secret-sharing schemes”, Phys. Rev. A 69, 052307 (2004).

【30】Tomas Tyc and Barry C. Sanders,“How to share a continuous-variable quantum secret by optical interferometry”, quant-ph/0107074 (2001).

【31】Wang-Jung Kim and Seung-Hyun Cha, Journal of the Korean Physical Society, Vol. 48, No. 6 (2006).

【32】A. Karlsson、M. Koashi and N. Imoto,“Quantum entanglement for secret sharing and secret splitting”, Phys. Rev. A 59, 162 (1999).