What is the difference between the "density matrix combination" and the "tensor product" methods?

Question

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?

flippiefanus · Accepted Answer

Starting with a pure state
$$ rho_n = |psi_nranglelanglepsi_n| , $$
one can form a mixed state by adding several such pure states
$$ rho = sum_n P_n rho_n , $$
where $P_n$ represents probabilities, such that
$$ sum_n P_n = 1 . $$
What this means is that there is some ignorance so that the best information is represented by having different pure states that can be observed with different probabilities. If the pure state represent a single particle, then the mixed state is still that of a single particle.
A tensor product represents multiple particles. I $rho_1$ and $rho_2$ are respectively single particle states, then
$$ rho=rho_1otimesrho_2 $$
represents a two-particle state. In this case, the respective one particle states can either be mixed or pure.
Hope this explanation clarifies the difference.

What is the difference between the "density matrix combination" and the "tensor product" methods?

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