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Let's consider $\mathcal{H_1} \otimes \mathcal{H_2}$ the space of the problem.

I call $\rho$ a density matrix of the full space and :

$\rho_1=Tr_2(\rho)$ the reduced density matrix in $\mathcal{H_1}$

I have two quite general questions. For me the assumptions I make are true but I didn't find a way to prove it rigorously, so maybe it is not. I am thus looking for an answer that tells me if indeed those properties are true or not. And if possible a proof or a link that prove them.

First question :

If $\rho$ is a pure but entangled state, does it imply that $\rho_1$ is a mixt state ? I tried with examples and it is always the case but I can't prove it rigorously.

Second question :

If $\rho$ is a mixt state, does it necesseraly imply that $\rho_1$ will also be a mixt state ?

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    $\begingroup$ You can prove (i) using Schmidt decomposition. For (ii) you have a counter example : $|\psi \rangle\langle \psi | \otimes \rho_{mixed}^{}$. $\endgroup$ – Sunyam Oct 11 '18 at 16:31

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