Tensor products

Tensor products are a way to combine vector spaces. One of the postulates of quantum mechanics is that the state of a qubit is completely described by a unit vector in a Hilbert space. The problem then becomes how to deal with more than one qubit. This is where a tensor product comes in. Each qubit has its own Hilbert space, and to describe many qubits as a system, we need to combine all their Hilbert spaces into one bigger Hilbert space.

Mathematically, that means that if we have a Hilbert space H and another Hilbert space J, we denote their tensor product as:

If H is an h dimensional space and J is a j dimensional space, then the dimension of the combined space M is h ⋅ j. In other words:

Before we go any farther, let's look at the tensor product of two vectors.

The tensor product of vectors

The tensor product of two vectors is denoted in the following way in bra-ket notation. You'll notice that there are four different ways...