It is interesting that the basis of the tensor product of the spaces can be constructed as the tensor product of each possible combination of the basis of the s

It is interesting that the basis of the tensor product of the spaces can be constructed as the tensor product of each possible combination of the basis of the spaces. Among other things, it makes clear that the dimension of Hilbert space for the composite system is the product of the dimensions of the Hilbert spaces. But this can distract us from the most interesting part, which is that the space spanned by these basis vectors is much larger than the simple cartesian product. 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.)