本論文中，我們將介紹糾纏態的發展歷史，以及如何利用糾纏態來完成一些有趣的應用。糾纏態最早是出現在EPR的假想實驗中，是EPR為了質疑量子力學的完備性而提出的。之後，Bohm重新詮釋EPR的假想實驗，以隱變量及局域性的觀點與量子力學的機率統計詮釋作區分。為了分辨何種觀點較正確，Bell針對糾纏態給出一不等式，顯示出兩種詮釋的衝突並提供實驗方面驗證的依據。由於原本的Bell不等式在實驗上難以實現，CHSH等人便改寫Bell不等式並提出改良實驗的方案。最後，由Aspect團隊根據CHSH的方案完成決定性的實驗，並證實微觀世界確實存有糾纏態。既然糾纏態確實存在，下一步便是找出測量糾纏程度的方法。這裡我們介紹一種方法，是利用熵的概念來量化粒子糾纏的程度。至於糾纏態的應用，大致可分為量子計算、量子密碼及量子遠程傳送三種。量子計算可以解決大數的因數分解難題，也可讓我們用較少的步驟完成搜索資料的工作，大幅減少古典計算所需的時間。量子密碼可讓我們檢測出竊聽者是否存在，以確保傳訊時的安全性。量子遠程傳送讓我們可以傳送粒子給接收者，但不需直接傳送該粒子。

In this thesis, we introduce the history of entanglement development, and some interesting applications of it. Entanglement appeared for the first time in the EPR’s Gedankenexperiment, which argued that quantum mechanics was not a complete theory. Afterwards, Bohm reinterpreted EPR’s Gedankenexperiment with hidden-variable and locality in order to distinguish from the probability interpretation. For finding out which viewpoint was more correct, Bell inferred an inequality of entanglement to show the conflicts of these interpretations and provide a basis for experiments. Because it was difficult to test Bell’s inequality in experiment, CHSH rewrote the Bell’s inequality and proposed a better experimental scheme. Aspect’s team completed the experiment, which was based by CHSH’s scheme, and proved the entanglement existed. Since entanglement existed truly, the next step to find out the methods of measuring the degree of entanglement. We herein introduce a method which quantifies the degree of entanglement with the idea of entropy. There are three kinds of entanglement application, which are the quantum computing, the quantum cryptography and the quantum teleportation. The quantum computing can solve the large number factoring problem and it takes less steps completing the searching, therefore significantly decreasing the time that classical computing needs. The quantum cryptography let us find out whether the eavesdropper exists or not, so we can secure the secrecy of communication. The quantum teleportation allows us to transmit particles to the receiver, instead of transmitting the particles directly.

致謝  .........................................i
中文摘要  .....................................ii
英文摘要  .....................................iii
目錄  .........................................iv
第1章 序論  ...................................1
第2章 糾纏態之簡介  ...........................8
2-1 糾纏態的提出  .............................9
2-2 Schrödinger與Bohm的詮釋  ..................15
2-3 Bell不等式和CHSH不等式  ...................20
2-4 CHSH方案和Aspect的實驗證明  ...............26
2-5 糾纏態的量測  .............................30
附錄A 密度矩陣  ...............................34
第3章 量子計算  ...............................36
3-1 大數的因數分解－Shor的演算法  .............37
3-2 快速的搜索系統－Grover的演算法  ...........47
附錄B 公開碼系統－RSA系統  ....................52
第4章 糾纏態於其他方面的應用  .................55
4-0 量子不可克隆原理  .........................56
4-1 量子密碼  .................................57
4-2 量子遠程傳送  .............................65
第5章 結論  ...................................69
參考文獻  .....................................74

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