# 純態

（重定向自混合態

${\displaystyle {\begin{bmatrix}0.5&0\\0&0.5\\\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}}$

## 量子力學

${\displaystyle S=|\Psi \rangle }$

${\displaystyle S=\rho =|\Psi \rangle \langle \Psi |}$

${\displaystyle S=\rho =\Sigma _{i}c_{i}|\Psi _{i}\rangle \langle \Psi _{i}|,\Sigma _{i}c_{i}=1}$

### 區分純態與混態

#### 舉例

${\displaystyle \rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$ 為純態，${\displaystyle \rho _{2}={\begin{pmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{pmatrix}}}$ 為混態

${\displaystyle \Rightarrow tr(\rho _{1})=tr(\rho _{2})={\frac {1}{2}}+{\frac {1}{2}}=1}$

${\displaystyle \rho _{1}^{2}=\rho _{1}*\rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$ ${\displaystyle \rho _{2}^{2}=\rho _{2}*\rho _{2}={\begin{pmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{pmatrix}}}$

${\displaystyle \Rightarrow tr(\rho _{1}^{2})=tr(\rho _{1})={\frac {1}{2}}+{\frac {1}{2}}=1}$ ${\displaystyle tr(\rho _{2}^{2})={\frac {1}{4}}+{\frac {1}{4}}={\frac {1}{2}}\neq tr(\rho _{2})=1}$

${\displaystyle {\begin{matrix}{}^{t\rightarrow \infty }\\\to \\{}\\{}\\{}\\{}\end{matrix}}\quad \,}$ 混態${\displaystyle \rho _{2}={\begin{pmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{pmatrix}}}$