純態

區分純態與混態的方法要利用到${\displaystyle tr(\rho )\,}$。${\displaystyle tr(\rho )\,}$表示對矩陣${\displaystyle \rho \,}$取對角線元素和(trace)，將純態和混態做歸一化動作，使得${\displaystyle tr(\rho )\,}$之值皆會是1。

而兩者不同處在於${\displaystyle tr(\rho ^{2})\,}$：歸一化過的純態${\displaystyle tr(\rho ^{2})=tr(\rho )=1\,}$，而歸一化過的混態則${\displaystyle tr(\rho ^{2})<1\,}$，和${\displaystyle tr(\rho )=1\,}$不同，由此得以辨別出純態與混態。

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

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

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

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

量子退相干現象的過程中，與環境的相互作用會讓密度矩陣的非對角線元素(off-diagonal elements)隨時間衰減到0。也就是說在這個例子，隨著時間${\displaystyle t\,}$逐漸增加， 原本純態${\displaystyle \rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}e^{-{\frac {t}{T_{2}}}}\\{\frac {1}{2}}e^{-{\frac {t}{T_{2}}}}&{\frac {1}{2}}\end{pmatrix}}\Rightarrow \rho _{1}(t=0)={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$

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