〈 Berger | Dillon 〉

From mathematician's point of view entanglement is exactly the fact that not all tensors are decomposable. It totally clicked me once I realized that.

hmm what do you mean? this is the singlet representation, an irreducible representation, of su(2) x su(2)

I mean the following: given a system with Hilbert state X and another with Y, the joint state space is X⨂Y. A state there X & Y are independent would look like a⨂b (a decomposable tensor) for some a∊X and b∊Y, but not all elements of X⨂Y are decomposable.

In general, elements of X⨂Y are linear combinations of decomposable tensors, just like in your post.