〈 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.

I’m seeing what you’re saying, but I thought that a tensor product of reps can always be decomposed into a direct sum of its irreps.. or is that just SU(N)?

You are right, but I'm not talking about irreps of SU(N) or spins or anything in particular, just quantum systems in general.