Can All 4-D column matrices be given as tensor product of 2-D column matrices? I am familiar with entanglement concept. But it feels bit weird to me that all possibilities of a system in a $4$-dimensional  vector  space cannot be given as tensor  product of  two $2$-dimensional vector spaces for eg. 
$$ A=\left( \begin{array}{ccc}
1  \\
0  \\
0  \\
1  \end{array} \right)$$ 
can't be given as tensor product of two $2$x$1$ matrices.As I see two $2$x$1$ matrices are independent and still tensor product of them is not spanning the entire $4$x$1$ vector space.
Am I missing something or making a blunder ? If I am correct then why in classical theory every 4 bit of information can be given as two $2$ bits of information but the same is not true for qubits in quantum mechanics ?  
 A: Looking at the Hilbert spaces themselves, we indeed find the puzzling equality
$$\mathbb{C}^2\times\mathbb{C}^2 = \mathbb{C}^2\otimes\mathbb{C}^2 = \mathbb{C}^4$$
so the tensor product of the qubit spaces is equal to the pairs of non-entangled states. Or so it would seem.
The superficial equality is wrong in the physical context, because the isomorphism between $\mathbb{C}^2\times\mathbb{C}^2$ and $\mathbb{C}^2\otimes\mathbb{C}^2$ would not be given by the map $v\times w = (v,w)\mapsto v\otimes w$, but by a different one. Indeed, the map
$$\mathbb{C}^2\times\mathbb{C}^2\to\mathbb{C}^2\otimes\mathbb{C}^2, (v_1,v_2,w_1,w_2)\mapsto v\otimes w = (v_1w_1,v_1w_2,v_2w_1,v_2w_2) $$
is not an isomorphism, since vectors like $(1,0,0,1)$ do not lie in its image. So, although the spaces of non-entangled and all states are abstractly isomorphic, they are not in a way that would indicate all states are entangled.
Passing to the projective Hilbert space takes care of that for us (removes the superfical equality), and then we get the Segre embedding of the non-entangled states into the total space:
$$ S^2 \times S^2 \to P\mathbb{C}^4$$
where $S^2 = P\mathbb{C}^2$ is the well-known Bloch sphere. This map is not bijective, but only a "part" of the larger Hopf fibration
$$ S^3\to S^7\to S^4$$ 
that can be used to describe the full two-qubit system. (For details on this Hopf fibration, see "Geometry of entangled states, Bloch spheres and Hopf fibrations" by R. Mosseri and R. Dandoloff)
