$C$ to denote the field of complex numbers.
$i=\sqrt{-1}\in C$ to denote the imaginary unit.(虚数单位)
Complex number $c$:
The complex conjugate (复数共轭) of $c\in C$:
$c^*=a-ib$
The absolute value(绝对值) of $c\in C$ :
$\left|c\right|=\sqrt{c^*c}=\sqrt{a^2+b^2}$
In quantum physics / information, the quantum state can be in superposition(叠加态) state with complex number.
The superposition state is generally written in a complex vector：“bra-ket” notation (“Dirac” notation)
Defination of bra-ket notation:
Ket, denoted $|·\rang$, represents a d-dimensional column vector in the complex vector space $C^d$ . (The dimension $d$ is usually left implicit in the notation.)
Bra, denoted $\lang·|$, is a d-dimensional row vector equal to the complex conjugate of the corresponding ket.
The “ket” vector :
$|v\rangle=v=\begin{pmatrix}a_1 \\a_2 \\a_3 \\\vdots\\a_n\end{pmatrix}$ ($v=\begin{pmatrix}a_1 \\a_2 \\a_3 \\\vdots\\a_n\end{pmatrix}$）
The “bra” vector :
$\langle v|=(v^)^T=((|v\rangle)^)^T=\begin{pmatrix} a_1^* & a_2^* & a_3^* & \cdots & a_n^* \end{pmatrix}$
The “bra” of a vector is its conjugate transpose.
The relation of the ket & bra vector:
$\forall|u\rang\in C^d~~,~~|u\rang^\dagger=(|u\rang^*)^T=\lang u|$
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$*$: entry-wise conjugate
$T$: transpose
$\dagger$: “dagggan” to represent “conjugate transpose” (共轭转量，转量共轭)
$A^+=(A^*)^T$
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