1. 狄拉克符號
1.1 基矢
\(|0 \rang = \binom{1}{0}\) \(|1\rang = \binom{0}{1}\)
\(\lang 0| = (1~0)\) \(\lang 1| = (0~1)\)
1.2 態矢
\(| \psi \rangle = a|0\rangle + b|1\rangle\)
\(\lang \psi| = |\psi \rang ^{\dagger} = \Big( \left(|\psi \rang \right)^* \Big)^T = \Big( \left(|\psi \rang \right)^T \Big)^* = a^* \lang0| + b^*\lang1|\)
共軛，轉置順序可以互換.
2. 內積
\(\lang \varphi | \psi \rang = \lang \varphi || \psi \rang = | \varphi \rang^ \dagger |\psi\rang\) ，也可以表示為 \(( | \varphi \rang, |\psi \rang)\)
內積的性質：
(1) \((\cdot~, \cdot)\) 對第二個變量是線性的，即
\((|v\rang, \sum_{i}\lambda_i|w_i\rang) = \sum_i \lambda_i(|v\rang, |w_i\rang)\)
(2) \(\lang \varphi| \psi \rang = \Big(\lang \psi| \varphi \rang \Big)^*\)
推導如下：
\(\lang \varphi | \psi \rang = \lang \varphi|| \psi \rang = | \varphi\rang^ \dagger |\psi\rang\)
\(\lang \psi| \varphi \rang = \lang \psi|| \varphi \rang = | \psi\rang^ \dagger |\varphi\rang\)
\(\Big(\lang \psi| \varphi \rang \Big)^* = \Big(| \psi\rang^ \dagger |\varphi\rang \Big)^* = |\psi\rang^T |\varphi\rang^* = (|\varphi\rang^*)^T |\psi\rang = | \varphi\rang^ \dagger |\psi\rang\).
(3) 內積非負，當且僅當\(|v\rang=0\) 時取等號.
\(\lang i|j \rang = \delta_{ij} = \left\{\begin{matrix} 1~~~i=j \\ 0~~~~i\neq j \end{matrix}\right.\)
參考文獻
[1] 馬瑞霖. 量子密碼通信[M]. 北京：科學出版社，2006.
[2] [英]尼爾森，庄著. 量子計算和量子信息(一)--量子計算部分[M]. 趙千川譯. 北京：清華大學出版社，2009.
[3] [法]Emmanuel Desurvire. Classical and Quantum Information Theory[M]. 北京：科學出版社，2013.