Possible error in the Purification example on Github

Hey,
sorry for going back to this topic, but the question seems relevant here.

In the example of the Purification the initial maximally mixed state is:

  rho = MPO(s, "Id") ./ √2

Can it be that there is a mistake ? because the Tr(\rho)=4. On the other hand is you remove the \sqrt2 and replace it with 2 you will get Tr(\rho)=1 . but the norm will change so I am kind of confused. Maybe someone knows what is the reason, and explanation for that ?

Another question I have on this topic is if I want to change the last site so it won’t be a maximally mixed, but something else.

I want the last site will be just |\psi \rangle=\frac{1}{\sqrt2}(|0 \rangle +|1 \rangle) and \rho[end]=|\psi \rangle \langle \psi|.
I thought that we only need to take the maximally mixed state and chance the tensor elements for that site to be 0.5 in each element. Is that correct ?