[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.