Decoherence

A process by which Coherences are lost.

Warning

This process is dependent on the choice of basis. For example:

Let ${\hat{\rho }}_1=|+⟩⟨+|$, where $|+⟩=\frac{|0⟩+|1⟩}{\sqrt{2}}$. Then ${\hat{\rho }}_1$ is represented in the $\{|0⟩,|1⟩\}$ basis as $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$

Let ${\hat{\rho }}_2=\frac{|+⟩⟨+|+|-⟩⟨-|}{2}$, where $|\pm ⟩=\frac{|0⟩\pm |1⟩}{\sqrt{2}}$. Then ${\hat{\rho }}_2$ is represented in the $\{|0⟩,|1⟩\}$ basis as $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$. The calculatulation for this is unclear. Help me ❓