Postulate 3: Quantum Operations: Unitary Evolution

Before giving the mathematical definition of quantum operations, it is important to speak a bit about the identity matrix.

It turns out that the identity matrix corresponds to the quantum operation of doing nothing to the system, i.e. leaving the system unchanged.

This will come up a lot when working with quantum operations on quantum systems.

Therefore, it is important to become comfortable with the identity matrix, and its representation in bra-ket form.

Let's start by recalling the 2x2 identity matrix, which we represent as $I$:

$$I= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$

The elements of the identity matrix can be expressed using the computational basis as:

$$\langle 0|I|0\rangle = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = 1,$$ $$\langle 0|I|1\rangle = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = 0,$$ $$\langle 1|I|0\rangle = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = 0,$$ $$\langle 1|I|1\rangle = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = 1.$$

Expressing the identity matrix fully in bra-ket form, using the outer product, gives:

$$|0\rangle \langle 0| + |1\rangle \langle 1| = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I.$$

This is a formula and useful fact that comes up a lot in quantum theory, and is sometimes called the resolution of identity.  

The identity matrix can be written as a sum of the outer products of each basis state with themselves.  

And it is not just for the computational basis that this is true! The identity matrix can be written as a sum of the outer products of each basis state with themselves for any orthonormal basis. For example:  

\begin{align*} ...