纯态

对于密度矩阵${\displaystyle \rho }$表述的量子态，若其不能写作纯态的密度矩阵（其中${\displaystyle N=1}$且${\displaystyle c_{1}=1}$），则称作混态。

区分纯态与混态的方法要利用到${\displaystyle tr(\rho )\,}$。${\displaystyle tr(\rho )\,}$表示对矩阵${\displaystyle \rho \,}$取对角线元素和(trace)，将纯态和混态做归一化动作，使得${\displaystyle tr(\rho )\,}$之值皆会是1。

而两者不同处在于${\displaystyle tr(\rho ^{2})\,}$：归一化过的纯态${\displaystyle tr(\rho ^{2})=tr(\rho )=1\,}$，而归一化过的混态则${\displaystyle tr(\rho ^{2})<1\,}$，和${\displaystyle tr(\rho )=1\,}$不同，由此得以辨别出纯态与混态。

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

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

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

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

量子退相干现象的过程中，与环境的相互作用会让密度矩阵的非对角线元素(off-diagonal elements)随时间衰减到0。也就是说在这个例子，随着时间${\displaystyle t\,}$逐渐增加，原本纯态，

${\displaystyle \rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}e^{-{\frac {t}{T_{2}}}}\\{\frac {1}{2}}e^{-{\frac {t}{T_{2}}}}&{\frac {1}{2}}\end{pmatrix}}\Rightarrow \rho _{1}(t=0)={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$

演化为混态，

${\displaystyle {\overset {t\rightarrow \infty }{\to }}\rho _{2}={\begin{pmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{pmatrix}}}$