Direct product vs. tensor product

Suppose you have two vector spaces. How can you combine them into another vector space?

The simplest way is to use a direct product. For example, suppose I have vector space A with Cartesian coordinates a1 and a2 and vector space B with Cartesian coordinates b1, b2, and b3. The direct product of A and B is a new vector space with coordinates a1, a2, b1, b2, and b3. Basically, the idea is to add the dimensions of A and B together. Notice that A has 2 dimensions, B has 3 dimensions, and their direct product has 5 dimensions. Pictorially we might imagine a direct product as taking the union of vector spaces A and B. Then we get a new vector space where you label the elements of the set by a set of coordinates in A (a1, a2) and a set of coordinates in B (b1, b2, b3).

A slightly more interesting thing we can do is to take a tensor product. Let's call the basis vectors of A, ^a1 and ^a2, and similarly the basis vectors of B, ^b1, ^b2, and ^b3. The tensor product of A and B is a new vector space with basis vector space with basis vectors ^a1 x ^b1, ^a1 x ^b2, ^a1 x ^b3, ^a2 x ^b1, ^a2 x ^b2, and ^a2 x ^b3, where "x" connotes tensor product. So the tensor product is kind of like multiplying two vector spaces. The tensor product space of A and B has 6 = 2 x 3 dimensions.

In physics, a simple example of direct and tensor products is spin-1/2 particles. A single spin-1/2 particle is a direct product of |&uarr> and |&darr> -- dimension 2. However, the Hilbert space of 2 spin-1/2 particles is a tensor product of two single spin-1/2 Hilbert spaces -- dimension 4 = 2 x 2. We see that in general that a tensor product space is larger than its corresponding direct product space. For example, the direct product of 3 spin-1/2 particles has dimension 6 = 2 + 2 + 2, but the tensor product of 3 spin-1/2 particles has dimension 8 = 2 x 2 x 2. That means that there are vectors in the tensor product space that can't be mapped to the corresponding direct product space. We know that spins combine in tensor products because we observe "entanglement." A direct product of 2 spin-1/2 particles would have states like |&uarr&uarr> but not Bell states like |&uarr&darr> - |&darr&uarr>.