# 密度矩陣

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

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

${\rho} = \sum_i w_i | \psi_i \rangle \langle \psi_i |$

$\sum_i w_i =1$

$\varrho_{ij}=\lang b_i|\rho| b_j\rang= \sum_k w_k\lang b_i | \psi_k \rangle \langle \psi_k |b_j\rang$

$\langle A \rangle = \sum_i w_i \langle \psi_i | {A} | \psi_i \rangle = \sum_i \langle \psi_i | {\rho}{A} | \psi_i \rangle = \operatorname{tr}({\rho}{A})$

## 純態與混合態

### 數學表述

#### 純態

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

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

\begin{align}\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 \\ & =\operatorname{tr}(\Lambda(a_i)\rho) \\ \end{align}

\begin{align}\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=\operatorname{tr}(A\rho) \\ \end{align}

\begin{align}\operatorname{tr}(\rho) & =\operatorname{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{align}

$0\le\lang\phi|\rho|\phi\rang=\lang\phi|\psi \rangle\langle\psi|\phi\rang=|\lang\phi|\psi \rangle|^2 \le 1$

#### 混合態

${\rho} \ \stackrel{def}{=}\ \sum_i w_i | \psi_i \rangle \langle \psi_i |$

$0\le w_i \le 1$
$\sum_i w_i =1$

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

$\rho=\sum_i a_i | a_i \rangle \langle a_i |$

$a_i = a_i ^*$

$\sum_i a_i =1$

### 用密度算符辨認純態

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

• $\rho^2=\rho$
• $tr(\rho^2)=tr(\rho)=1$

$tr(\rho^2)

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

### 連續性本徵態基底

$\varrho(x',x'')=\sum_i w_i\psi_i(x')\psi_i^*(x'')$

$\lang A\rang=\operatorname{tr}(A\rho)=\int \mathrm{d}x' \int \mathrm{d}x'' \lang x'|A|x''\rang\lang x''|\rho|x'\rang$

### 複合系統

$\rho_A=\operatorname{tr}_B\rho$
$\rho_B=\operatorname{tr}_A\rho$

$\rho=\rho_{A}\otimes\rho_{B}$

## 範例

### z-軸方向

• 態向量：$|z+\rang = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$

• 態向量：$|z-\rang = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

### x-軸方向

• 態向量：$|x+ \rangle = \begin{pmatrix} \frac{1}{\sqrt2} \\ \frac{1}{\sqrt2} \end{pmatrix}$

• 態向量：$|x- \rangle = \begin{pmatrix} \frac{1}{\sqrt2} \\ -\frac{1}{\sqrt2} \end{pmatrix}$

### y-軸方向

• 態向量：$|y+ \rangle = \begin{pmatrix} \frac{1}{\sqrt2} \\ \frac{i}{\sqrt2} \end{pmatrix}$

• 態向量：$|y- \rangle = \begin{pmatrix} \frac{1}{\sqrt2} \\ -\frac{i}{\sqrt2} \end{pmatrix}$

### 完全隨機粒子束

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

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

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

## 馮諾伊曼方程式

$\rho(t_0) =\sum_i w_i| \psi_i(t_0) \rangle\langle\psi_i(t_0)|$

$i\hbar\frac{\partial}{\partial t}|\psi_i(t)\rang=H|\psi_i(t)\rang$

\begin{align}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{align}

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

$\rho(t) = e^{-i H t/\hbar} \rho(0) e^{i H t/\hbar}$

## 馮諾伊曼熵

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

$\rho=\sum_i a_i | a_i \rangle \langle a_i |$

$\sigma = -\sum_i \varrho_{ii} \ln \varrho_{ii}$

$\sigma = -\sum_i a_i \ln a_i$

$\lim_{a \to 0} a \log a = 0$

$0 \log 0 = 0$

$\sigma = -\sum_i \frac{1}{N}\ln\frac{1}{N}=\ln N$

## 註釋

1. ^ 對於本徵態 $|a_i\rang$ 的投影算符 $\Lambda(a_i)$ ，假若作用於量子態 $|\psi\rangle$ ，則會得到 $|a_i\rang$ 與對應機率幅的乘積：
$\Lambda(a_i)|\psi\rangle=|a_i\rangle\langle a_i|\psi\rangle=c_i|a_i\rangle$
其中，$c_i$ 是在本徵態 $|a_i\rang$ 裏找到 $|\psi\rangle$機率幅
2. ^ 給定兩個規範正交基 $\{|a_i \rangle\}, \{|b_i \rangle\}$ ，對於任意算符 $W$
$\operatorname{tr}(W)=\sum_i \lang a_i|W|a_i\rang=\sum_{i,j} \lang a_i|b_j\rang \lang b_j|W|a_i\rang=\sum_{i,j} \lang b_j|W|a_i\rang\lang a_i|b_j\rang=\sum_{j} \lang b_j|W|b_j\rang$
因此，對於不同的規範正交基，跡數是個不變量。
3. ^ 3.0 3.1 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
$|\psi_i(t)\rang=e^{-iH(t-t_0)}|\psi_i(t_0)\rang$
因此，密度算符與時間無關：
\begin{align}\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{align}
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。

## 參考文獻

1. ^ von Neumann, John, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik, Göttinger Nachrichten, 1927, 1: 245–272
2. ^ 2.0 2.1 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. ^ 4.0 4.1 4.2 4.3 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. ^ 6.0 6.1 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. ^ 8.0 8.1 8.2 8.3 Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914
9. ^ Breuer, Heinz; Petruccione, Francesco, The theory of open quantum systems, 110, ISBN 9780198520634
10. ^ Schwabl, Franz, Statistical mechanics, 16, 2002, ISBN 9783540431633
11. ^ 11.0 11.1 Bengtsson, Ingemar; Zyczkowski, Karol. Geometry of Quantum States: An Introduction to Quantum Entanglement 1st.
12. ^ 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"
13. ^ 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