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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

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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.

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No, this is provably impossible using unitary evolution. You expressly need some form of projective measurement (possibly followed by a unitary gate) to create such a coherence. This can be shown using the resource theory of quantum coherence.

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    $\begingroup$ There are other quantum operations possible besides unitary evolution and projective measurement, which could allow for the "purification" of a mixed state. $\endgroup$ – tparker Oct 1 '18 at 19:30
  • $\begingroup$ (Also, projective measurements don't necessarily fully disentangle an entangled state in the case of degeneracy. So projective measurement is neither necessary nor sufficient for disentangling an entangled state.) $\endgroup$ – tparker Oct 1 '18 at 21:00
  • $\begingroup$ @tparker Hence the hedged language with a reference to the rigorous theory. There are indeed other, more complicated operations which can achieve this, but they can all be decomposed as a (partial) projective measurement followed by a unitary, and the ability to perform them implies the ability to perform (at least partial) projective measurements. In other words, projective measurements really are the key, and your corrections, while not wrong, boil down to unnecessary technicalities. $\endgroup$ – Emilio Pisanty Oct 2 '18 at 7:27
  • $\begingroup$ @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. $\endgroup$ – Jitendra Oct 4 '18 at 20:23

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