I know tensor products. I know SQL joins. I see superficial similarities between the two, but no unifying underlying principle.
The world has had quite enough of nonsense thrown around because it's "obvious". Put your money where your runaway mouth is.
In the cross join case, all the elements are pairs of elements from the original tables. In the Hilbert space of a composite system only the separable states can be written as the tensor product of states of the subsystems.
"Observation 3.6.5: Interesting two-qubit states. Not every 4-dimensional vector can be written as a Kronecker product of two 2-dimension vectors, e.g., you can have a two-qubit state |Ψ⟩ such that:
|Ψ⟩ = ̸= |ψ⟩|φ⟩, for any one-qubit state |ψ⟩ and |φ⟩
These types of states (called entangled states) are very intriguing and play a fundamental role in quantum mechanics."
But I have to stress that I do not intend this to be a deep insight, just an interesting (IMHO) observation.
Focusing on superficial similarities is a two-edged sword: anchoring on a familiar concept can help or make things more difficult. And it's guaranteed to annoy people! (I'm mostly getting over it, but I still dislike the use of the word "tensor" to refer to multi-dimensional arrays. Tensor has a meaning in geometry and tensor algebra is not about doing linear algebra on 2-d slices of a larger-dimensional object.)
You are now beyond the limits of my understanding. I never grokked category theory.
SQL join is a Cartesian product. The Tensor product is analogous to a Cartesian product except that the inputs and outputs are ordered tuples instead of sets. That's all I was getting at.
As for HN, a lot of people are going to be self-educated in advanced topics. That means there's going to be holes in people's knowledge, especially as it relates to alternative theoretical treatments.