Is there a way to write p1 |v1> + p2 |v2> if it is not a pure state but a mixture?
If it is a true classical ensemble then no. However, if it's something that you got from tracing over unobserved degrees of freedom then yes. The representation is then the original pure state on the larger space.
My question is about notations.
How to write correctly such a mixture?
Mixtures are hermitian operators or matrices. You can construct them from state vectors by taking a convex sum of their outer products. So for example
\rho = a^* a \left|\psi\rangle \langle\psi\right| + b^* b \left|\phi\rangle \langle\phi\right|
I did not realize that i could write it in this way.
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