General properties on reduced density matrix with assumption on the global density matrix

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 ?

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