# Is it possible to create coherence in a two-level system starting from the maximally mixed state?

I know how to create coherence between two states if I have a pure state. For example if the system is in pure state $$|1\rangle = \begin{bmatrix}1\\0 \\\end{bmatrix}$$ then density matrix is given by $$\rho_o = \begin{bmatrix}1 & 0\\0 & 0\\\end{bmatrix}$$. To create coherence we can apply $$\sqrt{NOT} = \frac{1}{2}\begin{bmatrix}1+i & 1-i\\1-i & 1+i\\\end{bmatrix}$$ gate and the density matrix becomes $$\rho = \sqrt{NOT}\rho_o\sqrt{NOT'} = \frac{1}{2}\begin{bmatrix}1 & 1\\1 & 1\\\end{bmatrix}$$. The $$\sqrt{NOT}$$ gate is equivalent to applying a $$\frac{\pi}{2}$$ pulse and we can see a superposition and coherence is created in the system.

I was wondering if one can create coherence if we start from a mixed state $$\rho_o = \frac{1}{2}\begin{bmatrix}1 & 0\\0 & 1\\\end{bmatrix}$$. Is there some quantum gate or any pulse sequence to achieve coherence in system starting from mixed state?

EDIT- Using Unitary Transformations only

If you apply any unitary $$U$$ to the maximally mixed state $$\rho_0$$, you will get out...
$$U \rho_0 U^\dagger = U U^\dagger \rho_0 = \rho_0$$
again. $$\rho_0$$ is a multiple of the identity matrix.
• @EmilioPisanty- I think your argument is only true for maximally mixed state because $\rho=U\rho_oU' = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & -i\\-i & 1\end{bmatrix}\frac{1}{a+b}\begin{bmatrix}a & 0\\0 & b\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1 & i\\i & 1\end{bmatrix} = \frac{1}{2(a+b)}\begin{bmatrix}a+b & i(a-b)\\-i(a-b) & a+b\end{bmatrix}$. Some coherence is created after starting from partially mixed state. – Jitendra Oct 4 '18 at 20:23