hilbert space - Tensor product vs. Cartesian Product for composite quantum systems

Why are states describing composite systems given by tensor products and not Cartesian products? They both belong in spaces of equal dimension, what are the differences between the two?

Answer

When the Cartesian product is equipped with the "natural" vector space structure, it's usually called the direct sum and denoted by the symbol $\oplus$. As other answers state, the direct sum (Cartesian product) and the tensor product of two vector spaces can be clearly seen to be different by their dimension.

If $\{v_i\}$ and $\{w_i\}$ are basis of $V$ and $W$, we have that $\{v_i\}\cup\{w_j\}$ is a basis of $V\oplus W$ and $\{v_i\otimes w_j\}$ is a basis of $V\otimes W$. Therefore,
$$\operatorname{dim}(V\oplus W)=\operatorname{dim}V+\operatorname{dim}W$$ $$\operatorname{dim}(V\otimes W)= \operatorname{dim}V\cdot\operatorname{dim}W$$

As you can see, the "Cartesian product" behaves more like a sum when dealing with vector spaces whereas the role of a product is adopted by the tensor product.

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Now, to get some intuition about why we should use the tensor product of the spaces of states of two different quantum systems when we want to describe the composite system we can use the following analogy.

Consider two classical systems with a finite number $m$ and $n$ of states $\{s_i\}$ and $\{r_i\}$. When describing the joint system, would we want to have as space of states the union or the Cartesian product of the original sets?

We would like the product, because we expect to have all the possible combined states $\{S_{ij}=(s_i,r_j)\}$ for all $i$, $j$. As you can see, number of elements of the new set is $m\cdot n$.

In the quantum analog of this setting, the two subsystems are described by vector spaces whose basis are $\{s_i\}$ and $\{r_i\}$. In the same way, the composite system will have a basis $\{S_{ij}= (s_i,r_j)\}$ (which we write as $S_{ij}=s_i\otimes r_j$) so it must be the tensor product.

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Note: The key of this argument is the observation that the set whose number of elements is the product is the Cartesian product (classical case), whereas the vector space whose dimension is the product is the tensor product (quantum case).