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I am trying to calculate the density matrix for following mixed state $|\Omega\rangle$:

$|\Omega\rangle=\frac{1}{2}(|\Psi\rangle+|\Phi\rangle)$
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|u\rangle+|d\rangle)$
$|\Phi\rangle=\frac{1}{\sqrt{2}}(|u\rangle-|d\rangle)$

using two different paths but the answers I get are different. The first result seem to be correct and makes sense to me, but I can not arrive at the same result using the second method! Can you please tell me where I am making the mistake?

  1. $\rho=\sum_{1}^{k}p_{k}\rho_{k}$
    $\rho_{\Psi}=|\Psi\rangle\langle\Psi|= \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}= \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} $
    $\rho_{\Phi}=|\Phi\rangle\langle\Phi|= \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{-1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{pmatrix}= \begin{pmatrix} \frac{1}{2} & \frac{-1}{2} \\ \frac{-1}{2} & \frac{1}{2} \end{pmatrix} $
    $\rho=\frac{1}{2}\left( \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} + \begin{pmatrix} \frac{1}{2} & \frac{-1}{2} \\ \frac{-1}{2} & \frac{1}{2} \end{pmatrix}\right)= \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} $

  2. $|\Omega\rangle=\frac{1}{2}(|\Psi\rangle+|\Phi\rangle)= \frac{1}{2}(\frac{(|u\rangle+|d\rangle)}{\sqrt{2}}+\frac{(|u\rangle-|d\rangle)}{\sqrt{2}})=\frac{2|u\rangle}{2\sqrt{2}}=\frac{|u\rangle}{\sqrt{2}}$
    $\rho=|\Omega\rangle\langle\Omega|= \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} \\ 0 \end{pmatrix}= \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & 0 \end{pmatrix} $

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The second approach is not a valid approach because your final $\rho$ is not a pure state (and thus cannot be represented as a pure state). The fact that only pure states can be represented as vectors (bras/kets or row/column vectors) is the reason we need something more complex such as the density matrix in the first place.

$\rho_\Omega$ is an incoherent mixture of coherent states so can only be calculated via a procedure such as what you did in #1, i.e. as a convex sum of other density matrices (which may or may not themselves be a coherent/pure states).

In your second approach, the key misconception was the belief that you can write an incoherent mixture as some pure state, $\left|\Omega\right\rangle$. This would only work if the mixture was a coherent mixture of states (i.e. a quantum superposition state). Note an indication that this wasn't correct is the fact that $\left|\Omega\right\rangle$ isn't a valid/normalized state (i.e. $\langle\Omega|\Omega\rangle = 2 \ne 1$).

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