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# 纯态

## 维基百科，自由的百科全书

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

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

## 量子力学

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

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

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

### 区分纯态与混态

#### 举例

${\displaystyle \rho _{1}={\begin{pmatrix}{\frac {1}{2))&{\frac {1}{2))\\{\frac {1}{2))&{\frac {1}{2))\end{pmatrix))}$为纯态，${\displaystyle \rho _{2}={\begin{pmatrix}{\frac {1}{2))&0\\0&{\frac {1}{2))\end{pmatrix))}$为混态

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

${\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}$

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

## 泛函分析

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