I have been studying this lecture from Scott Aaronson: https://www.scottaaronson.com/democritus/lec9.html
In the section "Mixed states", he says you can find the combined state by using the density matrix, and summing. I understand that part.
Further down, in the section "Real vs Complex Numbers", he says if you know the state of two quantum systems, you can find their "combined" state by using the tensor product.
I don't understand when to use the density matrix combination method or the tensor product method. What is the difference between what these two methods calculate?