Orthonormality

In this section, we will look at the concepts of the norm and orthogonality to come up with orthonormality.

The norm

We can define a metric on our vector spaces called the norm and denote it this way, ∥x∥, where x is the vector on which the norm is being measured. In two- and three-dimensional Euclidean spaces, it is often called the length of a vector, but in higher dimensions, we use the term norm. It gives us a way to measure vectors.

We define the norm using our inner product from the previous section, like so:

As always, let's look at an example. What is the norm of the vector |x⟩ here?

Well, let's work it out:

As you can see, the norm, ∥x∥, of |x⟩ is the square root of 29.

Normalization and unit vectors

Oftentimes, especially in quantum computing, we will want to represent our vectors by something called a unit vector. The word unit refers...