# 密度矩阵

## 维基百科，自由的百科全书

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

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

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

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

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

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

## 纯态与混合态

### 数学表述

#### 纯态

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

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

{\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))}

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

#### 混合态

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

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

• 密度算符是自伴算符：${\displaystyle \rho =\rho ^{\dagger ))$
• 密度算符的迹数为1：${\displaystyle {\hbox{tr))(\rho )=1}$
• 对可观察量 ${\displaystyle A}$ 做测量得到 ${\displaystyle a_{i))$ 的概率为 ${\displaystyle {\mathcal {P))(a_{i})={\hbox{tr))(\Lambda (a_{i})\rho )}$
• 做实验测量可观察量 ${\displaystyle A}$ 获得的期望值${\displaystyle \langle A\rangle ={\hbox{tr))(A\rho )}$
• 密度算符是非负算符：${\displaystyle 0\leq \langle \phi |\rho |\phi \rangle \leq 1}$

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

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

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

### 用密度算符辨认纯态与混合态

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

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

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

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

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

### 连续性本征态基底

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

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

### 复合系统

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

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

#### 约化密度算符

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

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

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

## 范例

### z-轴方向

• 态矢量：${\displaystyle |z+\rangle ={\begin{bmatrix}1\\0\end{bmatrix))}$

• 态矢量：${\displaystyle |z-\rangle ={\begin{bmatrix}0\\1\end{bmatrix))}$

### x-轴方向

• 态矢量：${\displaystyle |x+\rangle ={\begin{bmatrix}{\frac {1}{\sqrt {2))}\\{\frac {1}{\sqrt {2))}\end{bmatrix))}$

• 态矢量：${\displaystyle |x-\rangle ={\begin{bmatrix}{\frac {1}{\sqrt {2))}\\-{\frac {1}{\sqrt {2))}\end{bmatrix))}$

### y-轴方向

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

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

### 完全随机粒子束

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

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

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

## 冯诺伊曼方程

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

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

{\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))}

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

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

## 冯诺伊曼熵

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

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

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

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

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

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

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

## 注释

1. ^ 对于本征态 ${\displaystyle |a_{i}\rangle }$ 的投影算符 ${\displaystyle \Lambda (a_{i})}$ ，假若作用于量子态 ${\displaystyle |\psi \rangle }$ ，则会得到 ${\displaystyle |a_{i}\rangle }$ 与对应概率幅的乘积：
${\displaystyle \Lambda (a_{i})|\psi \rangle =|a_{i}\rangle \langle a_{i}|\psi \rangle =c_{i}|a_{i}\rangle }$
其中，${\displaystyle c_{i))$ 是在本征态 ${\displaystyle |a_{i}\rangle }$ 里找到 ${\displaystyle |\psi \rangle }$概率幅
2. ^ 给定两个规范正交基 ${\displaystyle \{|a_{i}\rangle \},\{|b_{i}\rangle \))$ ，对于任意算符 ${\displaystyle W}$
${\displaystyle \operatorname {tr} (W)=\sum _{i}\langle a_{i}|W|a_{i}\rangle =\sum _{i,j}\langle a_{i}|b_{j}\rangle \langle b_{j}|W|a_{i}\rangle =\sum _{i,j}\langle b_{j}|W|a_{i}\rangle \langle a_{i}|b_{j}\rangle =\sum _{j}\langle b_{j}|W|b_{j}\rangle }$
因此，对于不同的规范正交基，迹数是个不变量。
3. 量子退相干里，约化密度算符代表的是反常混合物，它不能被视为处于某个未知的纯态；它是依赖环境与系统之间的相互作用使得所有的非对角元素趋于零，实际而言，这些非对角元素所表现的量子相干性已被迁移至环境，只有从整个密度算符才能查觉到这量子相干性的存在。[6]:48-49
4. ^ 在薛定谔绘景里，纯态随着时间而演化的形式为
${\displaystyle |\psi _{i}(t)\rangle =e^{-iH(t-t_{0})}|\psi _{i}(t_{0})\rangle }$
因此，密度算符与时间无关：
{\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))}
采用薛定谔绘景来计算密度算符这动作很合理，因为密度算符是由薛定谔左矢与薛定谔右矢共同组成，而这两个矢量都是随着时间流逝而演进。
5. ^ 矩阵对数（logarithm of a matrix）也是矩阵；后者的矩阵指数等于前者。这是纯对数的推广。这运算是矩阵指数的反函数。并不是所有矩阵都有对数，有些矩阵有很多个对数。

## 参考资料

1. ^ von Neumann, John, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik, Göttinger Nachrichten, 1927, 1: 245–272
2. Ballentine, Leslie. Quantum Mechanics: A Modern Development 2nd, illustrated, reprint. World Scientific. 1998. ISBN 9789810241056.
3. ^ Fano, Ugo, Description of States in Quantum Mechanics by Density Matrix and Operator Techniques, Reviews of Modern Physics, 1957, 29: 74–93, Bibcode:1957RvMP...29...74F, doi:10.1103/RevModPhys.29.74.
4. Laloe, Franck, Do We Really Understand Quantum Mechanics, Cambridge University Press, 2012, ISBN 978-1-107-02501-1
5. ^ Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, 2004, ISBN 0-13-111892-7
6. Maximilian A. Schlosshauer. Decoherence: And the Quantum-To-Classical Transition. Springer Science & Business Media. 1 January 2007. ISBN 978-3-540-35773-5.
7. ^ Bernard d' Espagnat. Conceptual Foundations of Quantum Mechanics. Advanced Book Program, Perseus Books. 1999. ISBN 978-0-7382-0104-7.
8. Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914
9. ^ Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Mathematical Proceedings of the Cambridge Philosophical Society. 2008, 26 (3): 376. Bibcode:1930PCPS...26..376D. doi:10.1017/S0305004100016108.
10. ^ Breuer, Heinz; Petruccione, Francesco, The theory of open quantum systems: 110, ISBN 9780198520634
11. ^ Schwabl, Franz, Statistical mechanics: 16, 2002, ISBN 9783540431633
12. Bengtsson, Ingemar; Zyczkowski, Karol. Geometry of Quantum States: An Introduction to Quantum Entanglement 1st.
13. ^ Nielsen, Michael; Chuang, Isaac, Quantum Computation and Quantum Information, Cambridge University Press, 2000, ISBN 978-0-521-63503-5. Chapter 11: Entropy and information, Theorem 11.9, "Projective measurements cannot decrease entropy"
14. ^ Everett, Hugh, The Theory of the Universal Wavefunction (1956) Appendix I. "Monotone decrease of information for stochastic processes", The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press: 128–129, 1973, ISBN 978-0-691-08131-1