# 密度矩陣

（重定向自冯诺依曼方程

${\displaystyle {\rho }=\sum _{i}w_{i}|\psi _{i}\rangle \langle \psi _{i}|}$

${\displaystyle \sum _{i}w_{i}=1}$

${\displaystyle \varrho _{ij}=\langle b_{i}|\rho |b_{j}\rangle =\sum _{k}w_{k}\langle b_{i}|\psi _{k}\rangle \langle \psi _{k}|b_{j}\rangle }$

${\displaystyle \langle A\rangle =\sum _{i}w_{i}\langle \psi _{i}|{A}|\psi _{i}\rangle =\sum _{i}\langle b_{i}|{\rho }{A}|b_{i}\rangle =\operatorname {tr} ({\rho }{A})}$

## 純態與混合態

### 數學表述

#### 純態

${\displaystyle \rho \ {\stackrel {def}{=}}\ |\psi \rangle \langle \psi |}$

${\displaystyle \rho ^{\dagger }=(|\psi \rangle \langle \psi |)^{\dagger }=|\psi \rangle \langle \psi |=\rho }$

{\displaystyle {\begin{aligned}{\mathcal {P}}(a_{i})&\ {\stackrel {def}{=}}\ |\langle a_{i}|\psi \rangle |^{2}=\langle a_{i}|\psi \rangle \langle \psi |a_{i}\rangle \\&=\sum _{k}\langle a_{k}|a_{i}\rangle \langle a_{i}|\psi \rangle \langle \psi |a_{k}\rangle \\&=\sum _{k}\langle a_{k}|\Lambda (a_{i})\rho |a_{k}\rangle \\&={\hbox{tr}}(\Lambda (a_{i})\rho )\\\end{aligned}}}

{\displaystyle {\begin{aligned}\langle A\rangle &\ {\stackrel {def}{=}}\ \sum _{i}a_{i}{\mathcal {P}}(a_{i})=\sum _{i}a_{i}\langle a_{i}|\psi \rangle \langle \psi |a_{i}\rangle \\&=\sum _{i}a_{i}\langle a_{i}|\rho |a_{i}\rangle =\sum _{i}\langle a_{i}|A\rho |a_{i}\rangle ={\hbox{tr}}(A\rho )\\\end{aligned}}}

{\displaystyle {\begin{aligned}{\hbox{tr}}(\rho )&={\hbox{tr}}(|\psi \rangle \langle \psi |)=\sum _{i}\langle a_{i}|\psi \rangle \langle \psi |a_{i}\rangle \\&=\sum _{i}\langle \psi |a_{i}\rangle \langle a_{i}|\psi \rangle =\langle \psi |\psi \rangle =1\\\end{aligned}}}

${\displaystyle 0\leq \langle \phi |\rho |\phi \rangle =\langle \phi |\psi \rangle \langle \psi |\phi \rangle =|\langle \phi |\psi \rangle |^{2}\leq 1}$

#### 混合態

${\displaystyle {\rho }\ {\stackrel {def}{=}}\ \sum _{i}w_{i}|\psi _{i}\rangle \langle \psi _{i}|}$

${\displaystyle 0\leq w_{i}\leq 1}$
${\displaystyle \sum _{i}w_{i}=1}$

• 密度算符是自伴算符：${\displaystyle \rho =\rho ^{\dagger }}$
• 密度算符的跡數為1：${\displaystyle {\hbox{tr}}(\rho )=1}$
• 對可觀察量 ${\displaystyle A}$ 做測量得到 ${\displaystyle a_{i}}$ 的機率為 ${\displaystyle {\mathcal {P}}(a_{i})={\hbox{tr}}(\Lambda (a_{i})\rho )}$
• 做實驗測量可觀察量 ${\displaystyle A}$ 獲得的期望值${\displaystyle \langle A\rangle ={\hbox{tr}}(A\rho )}$
• 密度算符是非負算符：${\displaystyle 0\leq \langle \phi |\rho |\phi \rangle \leq 1}$

${\displaystyle \rho =\sum _{i}a_{i}|a_{i}\rangle \langle a_{i}|}$

${\displaystyle a_{i}=a_{i}^{*}}$

${\displaystyle \sum _{i}a_{i}=1}$

### 用密度算符辨認純態與混合態

${\displaystyle \rho \ {\stackrel {def}{=}}\ |\psi \rangle \langle \psi |}$

• ${\displaystyle \rho ^{2}=\rho }$
• ${\displaystyle {\hbox{tr}}(\rho ^{2})={\hbox{tr}}(\rho )=1}$

${\displaystyle {\hbox{tr}}(\rho ^{2})<{\hbox{tr}}(\rho )=1}$

${\displaystyle \varrho ={\begin{bmatrix}0&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&0&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &0\\\end{bmatrix}}}$

${\displaystyle \gamma ={\hbox{tr}}(\rho ^{2})}$

### 連續性本徵態基底

${\displaystyle \varrho (x',x'')=\sum _{i}w_{i}\psi _{i}(x')\psi _{i}^{*}(x'')}$

${\displaystyle \langle A\rangle ={\hbox{tr}}(A\rho )=\int \mathrm {d} x'\int \mathrm {d} x''\langle x'|A|x''\rangle \langle x''|\rho |x'\rangle }$

### 複合系統

${\displaystyle \rho _{A}={\hbox{tr}}_{B}(\rho )}$
${\displaystyle \rho _{B}={\hbox{tr}}_{A}(\rho )}$

${\displaystyle \rho =\rho _{A}\otimes \rho _{B}}$

#### 約化密度算符

${\displaystyle \rho =|\psi \rangle \langle \psi |}$

${\displaystyle \rho _{A}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j}\langle b_{j}|_{B}\left(|\psi \rangle \langle \psi |\right)|b_{j}\rangle _{B}={\hbox{tr}}_{B}(\rho )}$

${\displaystyle \rho _{A}={\frac {1}{2}}{\bigg (}|0\rangle _{A}\langle 0|_{A}+|1\rangle _{A}\langle 1|_{A}{\bigg )}}$

## 範例

### z-軸方向

• 態向量：${\displaystyle |z+\rangle ={\begin{bmatrix}1\\0\end{bmatrix}}}$

• 態向量：${\displaystyle |z-\rangle ={\begin{bmatrix}0\\1\end{bmatrix}}}$

### x-軸方向

• 態向量：${\displaystyle |x+\rangle ={\begin{bmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}\end{bmatrix}}}$

• 態向量：${\displaystyle |x-\rangle ={\begin{bmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{bmatrix}}}$

### y-軸方向

• 態向量：${\displaystyle |y+\rangle ={\begin{bmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {i}{\sqrt {2}}}\end{bmatrix}}}$

• 態向量：${\displaystyle |y-\rangle ={\begin{bmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {i}{\sqrt {2}}}\end{bmatrix}}}$

### 完全隨機粒子束

${\displaystyle \varrho ={\frac {1}{2}}\varrho _{z+}+{\frac {1}{2}}\varrho _{z-}={\frac {1}{2}}\left[{\begin{bmatrix}1&0\\0&0\end{bmatrix}}+{\begin{bmatrix}0&0\\0&1\end{bmatrix}}\right]={\begin{bmatrix}0.5&0\\0&0.5\end{bmatrix}}}$

${\displaystyle \varrho ={\frac {1}{2}}\varrho _{x+}+{\frac {1}{2}}\varrho _{x-}={\frac {1}{2}}\left[{\begin{bmatrix}0.5&0.5\\0.5&0.5\end{bmatrix}}+{\begin{bmatrix}0.5&-0.5\\-0.5&0.5\end{bmatrix}}\right]={\begin{bmatrix}0.5&0\\0&0.5\end{bmatrix}}}$

${\displaystyle \varrho ={\frac {1}{N}}{\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\\\end{bmatrix}}}$

## 热平衡态

${\displaystyle \rho =\sum _{n}\omega _{n}|\psi _{n}\rangle \langle \psi _{n}|}$

## 馮諾伊曼方程式

${\displaystyle \rho (t_{0})=\sum _{i}w_{i}|\psi _{i}(t_{0})\rangle \langle \psi _{i}(t_{0})|}$

${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi _{i}(t)\rangle =H|\psi _{i}(t)\rangle }$

{\displaystyle {\begin{aligned}i\hbar {\frac {\partial }{\partial t}}\rho (t)&=\sum _{i}w_{i}(H|\psi _{i}(t)\rangle \langle \psi _{i}(t)|-|\psi _{i}(t)\rangle \langle \psi _{i}(t)|H)\\&=-[\rho ,H]\\\end{aligned}}}

${\displaystyle {\frac {dA^{(H)}}{dt}}=-\ {\frac {i}{\hbar }}[A^{(H)},H]}$

${\displaystyle \rho (t)=e^{-iHt/\hbar }\rho (0)e^{iHt/\hbar }}$

## 馮諾伊曼熵

${\displaystyle \sigma \ {\stackrel {def}{=}}\ -\mathrm {tr} (\varrho \ln \varrho )}$

${\displaystyle \rho =\sum _{i}a_{i}|a_{i}\rangle \langle a_{i}|}$

${\displaystyle \sigma =-\sum _{i}\varrho _{ii}\ln \varrho _{ii}}$

${\displaystyle \sigma =-\sum _{i}a_{i}\ln a_{i}}$

${\displaystyle \lim _{a\to 0}a\log a=0}$

${\displaystyle 0\log 0=0}$

${\displaystyle \sigma =-\sum _{i}{\frac {1}{N}}\ln {\frac {1}{N}}=\ln N}$

## 註釋

1. ^ 對於本徵態 ${\displaystyle |a_{i}\rangle }$ 的投影算符 ${\displaystyle \Lambda (a_{i})}$ ，假若作用於量子態 ${\displaystyle |\psi \rangle }$ ，則會得到 ${\displaystyle |a_{i}\rangle }$ 與對應機率幅的乘積：
${\displaystyle \Lambda (a_{i})|\psi \rangle =|a_{i}\rangle \langle a_{i}|\psi \rangle =c_{i}|a_{i}\rangle }$
其中，${\displaystyle c_{i}}$ 是在本徵態 ${\displaystyle |a_{i}\rangle }$ 裏找到 ${\displaystyle |\psi \rangle }$機率幅
2. ^ 給定兩個規範正交基 ${\displaystyle \{|a_{i}\rangle \},\{|b_{i}\rangle \}}$ ，對於任意算符 ${\displaystyle W}$
${\displaystyle \operatorname {tr} (W)=\sum _{i}\langle a_{i}|W|a_{i}\rangle =\sum _{i,j}\langle a_{i}|b_{j}\rangle \langle b_{j}|W|a_{i}\rangle =\sum _{i,j}\langle b_{j}|W|a_{i}\rangle \langle a_{i}|b_{j}\rangle =\sum _{j}\langle b_{j}|W|b_{j}\rangle }$
因此，對於不同的規範正交基，跡數是個不變量。
3. 量子退相干裏，約化密度算符代表的是反常混合物，它不能被視為處於某個未知的純態；它是依賴環境與系統之間的相互作用使得所有的非對角元素趨於零，實際而言，這些非對角元素所表現的量子相干性已被遷移至環境，只有從整個密度算符才能查覺到這量子相干性的存在。[6]:48-49
4. ^ 在薛丁格繪景裏，純態隨著時間而演化的形式為
${\displaystyle |\psi _{i}(t)\rangle =e^{-iH(t-t_{0})}|\psi _{i}(t_{0})\rangle }$
因此，密度算符與時間無關：
{\displaystyle {\begin{aligned}\rho (t)&=\sum _{i}w_{i}|\psi _{i}(t)\rangle \langle \psi _{i}(t)|\\&=\sum _{i}w_{i}\left(|\psi _{i}(t_{0})\rangle e^{iH(t-t_{0})}e^{-iH(t-t_{0})}\langle \psi _{i}(t_{0})|\right)\\&=\sum _{i}w_{i}\left(|\psi _{i}(t_{0})\rangle \langle \psi _{i}(t_{0})|\right)\\\end{aligned}}}
採用薛丁格繪景來計算密度算符這動作很合理，因為密度算符是由薛丁格左矢與薛丁格右矢共同組成，而這兩個向量都是隨著時間流逝而演進。
5. ^ 矩陣對數（logarithm of a matrix）也是矩陣；後者的矩陣指數等於前者。這是純對數的推廣。這運算是矩陣指數的反函數。並不是所有矩陣都有對數，有些矩陣有很多個對數。

## 參考资料

1. ^ von Neumann, John, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik, Göttinger Nachrichten, 1927, 1: 245–272
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10. ^ {S. VanEnk, "Mixed states and pure states," [Online Note]. University of Oregon. Available: https://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf页面存档备份，存于互联网档案馆） [Accessed: September 25, 2023]}
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