[sigmoid10]

A nice, high school level writeup of how to calculate this product with ordinary vectors, but it totally drops the ball on the necessity and its use in physics. It would probably be best to ignore the entire last paragraph and instead read up on pure vs. mixed quantum states if you actually care about that.

[hackandthink]

It's in the text:

"The tensor product itself captures all ways that basic things can 'interact' with each other!"

Tensor Product is also the way to go when combining classical probabilistic systems.

And you need the tensor product already for pure states in QM.

(mixed states need density matrices)

[sigmoid10]

>The tensor product itself captures all ways that basic things can 'interact' with each other!

In quantum physics, "interacting" usually has a different meaning. So one should use these terms more carefully.

>And you need the tensor product already for pure states in QM.

No. Pure states are just vectors (or more precise: rays) in Hilbert space. The usual inner product is sufficient to work with them. An outer (=tensor) product of these states will just give you a density matrix with tr(ρ^2)=1.

[hackandthink]

"Product states are multipartite quantum states that can be written as a tensor product"

"In the special case of pure states the definition simplifies: a pure state is separable if and only if it is a product state."

https://en.wikipedia.org/wiki/Separable_state

[sigmoid10]

Keyword is can. There's a reason why most university level physics curricula defer quantum density matrices (and by extension tensor algebra) to more advanced classes. There's a lot of mathematical legwork required before you can actually make use of that.