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I am interested in the state:

$\frac{1}{\sqrt{2}} (\left|11\right> + \left|22\right>)$

If I find the density matrix of this, I find the $9 \times 9$ matrix $\rho$.

If I want to find the reduced density matrix of Alice's qubit, I obtain:

$$\rho_{A} = \left|1\right>\left<1\right| + \left|2\right>\left<2\right|$$ this gives us a $3 \times 3$ matrix with 1's in $\rho_{A_{2,2}}$ and $\rho_{A_{3,3}}$

However, this would imply that the eigenvalues of the system are $0,1,1$. How is this possible?

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Maybe you forgot about the coefficient $\frac{1}{\sqrt{2}}$. The correct reduced density matrix is $$\frac12 (\left|1\right>\left<1\right| + \left|2\right>\left<2\right|)$$

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