Operators

In this section, we will consider linear operators. We first described these in Chapter 5, Transforming Space with Matrices. To reiterate, linear operators are linear transformations that map vectors from and to the same vector space. They are represented by square matrices. For just this section, I will put a "hat" or caret on the top of operators and use just the uppercase letter for matrices, as I want to be deliberate when referencing operators.

For instance, let's look at the X̂ operator that transforms the zero and one states:

Now, let's come up with a matrix that represents this operator. The question becomes, which basis will we use? Let's use the computational basis, which is |0⟩ and |1⟩. I will denote this set of basis vectors by the letter C. So, the X̂ operator in the C basis is represented by:

Now, I want to come up with a matrix representation of X̂ in the |+⟩, |-⟩ basis...