# 布洛赫球面

## 布洛赫球面諸點與純態的對應

${\displaystyle \alpha =\cos \theta \,e^{i\delta },\quad \beta =\sin \theta \,e^{i(\delta +\phi )}\,}$
${\displaystyle \Rightarrow |\psi \rangle =\cos \theta \,e^{i\delta }\,|0\rangle +\sin \theta \,e^{i(\delta +\phi )}\,|1\rangle =e^{i\delta }(\cos \theta \,|0\rangle +\sin \theta \,e^{i\phi }\,|1\rangle )}$

${\displaystyle |\psi \rangle =\cos \theta \,|0\rangle +\sin \theta \,e^{i\phi }\,|1\rangle }$

${\displaystyle 0\leq \theta \leq {\frac {\pi }{2}}\Rightarrow 0\leq 2\theta \leq \pi ,\quad }$
${\displaystyle 0\leq \phi <2\pi }$

${\displaystyle 2\theta \,}$${\displaystyle \phi \,}$的所有分佈在三維空間${\displaystyle \mathbb {R} ^{3}}$中畫出來，就可以得到一個球面，此即布洛赫球面，如同圖1。

${\displaystyle {\begin{matrix}x&=&\sin 2\theta \times \cos \phi \\y&=&\sin 2\theta \times \sin \phi \\z&=&\cos 2\theta \end{matrix}}}$

• ${\displaystyle |0\rangle }$${\displaystyle z_{+}:\,(0,0,1)}$
• ${\displaystyle |1\rangle }$${\displaystyle z_{-}:\,(0,0,-1)}$

## 習慣差異

${\displaystyle {\begin{matrix}x&=&\sin \theta \times \cos \phi \\y&=&\sin \theta \times \sin \phi \\z&=&\cos \theta \end{matrix}}}$

${\displaystyle 0\leq \theta \leq \pi ,\quad 0\leq \phi <2\pi }$

${\displaystyle |\psi \rangle =\cos {\frac {\theta }{2}}\,|0\rangle +\sin {\frac {\theta }{2}}\,e^{i\phi }\,|1\rangle }$

## 布洛赫球與混合態

${\displaystyle {\frac {1}{2}}\mathbf {I} ={\frac {1}{2}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$
${\displaystyle ={\frac {1}{2}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}+{\frac {1}{2}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}}$${\displaystyle ={\frac {1}{2}}|0\rangle \langle 0|+{\frac {1}{2}}|1\rangle \langle 1|={\frac {1}{2}}z_{+}+{\frac {1}{2}}z_{-}}$
${\displaystyle ={\frac {1}{2}}{\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}+{\frac {1}{2}}{\begin{pmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}={\frac {1}{2}}x_{+}+{\frac {1}{2}}x_{-}}$
${\displaystyle ={\frac {1}{2}}{\begin{pmatrix}{\frac {1}{2}}&-{\frac {i}{2}}\\{\frac {i}{2}}&{\frac {1}{2}}\end{pmatrix}}+{\frac {1}{2}}{\begin{pmatrix}{\frac {1}{2}}&{\frac {i}{2}}\\-{\frac {i}{2}}&{\frac {1}{2}}\end{pmatrix}}={\frac {1}{2}}y_{+}+{\frac {1}{2}}y_{-}}$

## 註釋

1. ^ Bloch, Felix. Nuclear induction. Phys. Rev. Oct 1946, 70 (7-8): 460–474. Bibcode:1946PhRv...70..460B. doi:10.1103/physrev.70.460.
2. ^ Nielsen, Michael A.; Chuang, Isaac L. Quantum Computation and Quantum Information. Cambridge University Press. 2004. ISBN 978-0-521-63503-5.
3. ^ 存档副本. [2017-07-25]. （原始内容存档于2015-09-25）.