量子位元

定義

$|\psi \rangle = \alpha |0 \rangle + \beta |1 \rangle ; \quad \alpha, \beta \in \mathbb{C}$

$|\alpha |^2 + |\beta |^2 = 1 \,$

$1 = \langle \psi |\psi \rangle = (\alpha |0 \rangle + \beta |1 \rangle)^{\dagger} (\alpha |0 \rangle + \beta |1 \rangle) = (\alpha^* \langle 0| + \beta^* \langle 1| ) (\alpha |0 \rangle + \beta |1 \rangle)$
$= \alpha^* \alpha \langle 0|0 \rangle + \alpha^* \beta \langle 0|1 \rangle + \beta^* \alpha \langle 1|0 \rangle + \beta^* \beta \langle 1|1 \rangle$
$= |\alpha |^2 + |\beta |^2 \,$，即要求總機率要是1。

按方向所採的諸多表示法

z方向

$z_- = |1 \rangle\langle 1| = \begin{pmatrix} 0 \\ 1 \end{pmatrix} * \begin{pmatrix} 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$

x方向

$x_- = |x_- \rangle\langle x_-| = \begin{pmatrix} \frac{1}{\sqrt2} \\ -\frac{1}{\sqrt2} \end{pmatrix} * \begin{pmatrix} \frac{1}{\sqrt2} & -\frac{1}{\sqrt2} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} \end{pmatrix}$

y方向

$y_- = |y_- \rangle\langle y_-| = \begin{pmatrix} \frac{1}{\sqrt2} \\ -\frac{i}{\sqrt2} \end{pmatrix} * \begin{pmatrix} \frac{1}{\sqrt2} & \frac{i}{\sqrt2} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{i}{2} \\ -\frac{i}{2} & \frac{1}{2} \end{pmatrix}$

参考文献

1. ^ MA Nielsen, IL Chuang. Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000).