One can try to be more clever in defining a tensor product.

That is, however, something particular to the case of tensor product.

Higher "weights" then just correspond to taking additional tensor products with this space in the range.

This is a special case of a topological tensor product.

This is the application of the more general tensor product applied to matrices.

The state of the composite system is then described by the following tensor product:

In general, the tensor product of complete spaces is not complete again.

This is just a tensor product of three qubits, and different from cloning a state.

We have a tensor product for the combined state of both systems.

This construction is related to repeated tensor products of two spaces.