文中簡單介紹了Clauser-Horne-Shimony-Holt
（CHSH） 不等式，且詳細推導了CHSH期望值的極
大值。此推導把求極大值的問題變成矩陣的本徵值
（eigenvalues)問題，即
2 1 max 2 l l + = CHSH
而其中λ1和λ2是某個本徵值為｛1, D, D｝矩陣
的兩個最大本徵值，其中D=4|det c|2≦1且c是物理態
的展開係數，即|ψ>=cij|ij>。我們發現<CHSH>極大值
和自旋方向的關係式。而我們的討論適用於當物理態
不是貝爾態（Bell State）時，仍可以連續演化過去。
然後我們代入古典關聯（Classically
correlated）的密度矩陣（Density matrix）和貝爾
態，得到與其他人一致的結果，即
古典關係密度矩陣 <CHSH>≦2
純量子態 2≦<CHSH>≦2√2
最後我們簡短討論了Werner 密度矩陣。

In this thesis, a short review on the Clauser-Horne-Shimony-Holt (CHSH) Inequality is given.
The maximization of the expectation value of the CHSH operator for pure bipartite qubit
systems is derived explicitly.
The derivation reduces the maximization of hCHSHi to an eigenvalue problem. We show that
hCHSHimax = 2p 1 + 2
where 1and 2 are the greatest two eigenvalues of a certain matrix with eigenvalues {1,D,D}
where D 4 | det c |2 1 and c is the coefficient of the expansion of the states, i.e. | i =
Pi,j cij |iji. We discover the relation between the maximum of hCHSHi and the directions for spin measurement ˆr and ˆq, and these values can be continuously deformed even when the state is not a Bell State.
We then specialize to pure classically correlated density matrices and also Bell State and arrive at results consistent with previous conclusions, i.e.
For classically correlated density matrices hCHSHi 2 ,
For pure quantum states 2 hCHSHi 2p2.
We also briefly discuss the case of Werner density matrix.

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