Three Basis Sets: 0°, 60°, and −60°

Let’s apply what we have learned to the change of basis for linearly polarized photons, which we will need in the third last chapter. And in fact, the change of basis states for photons is an operation commonly used in many quantum information processes.

We will consider three state-space sets of basis vectors, one with the $|\text{hlp}⟩$ along the direction labeled $0\degree$ and the other two with the corresponding basis vectors rotated by $60\degree$ and $-60\degree$ from the initial direction $0\degree$. We will label the basis vectors as follows.

• Basis vectors $\ket{\text{hlp}}_{0\degree}$ and $|\text{vlp}⟩_{0\degree}$.
• Basis vectors $\ket{\text{hlp}}_{60\degree}$ and $|\text{vlp}⟩_{60\degree}$.
• Basis vectors $\ket{\text{hlp}}_{-60\degree}$ and $|\text{vlp}⟩_{-60\degree}$.

We can use the following equation to relate the various basis vectors:

$$\begin{align*} \ket{q} &= \cos\theta~\ket{x} + \sin\theta~\ket{y}\\ \ket{r} &= -\sin\theta~\ket{x} + \cos\theta~ \ket{y}. \end{align*}$$

For example,

$$\begin{align*} \ket{\text{hlp}}_{60\degree} &= \cos 60\degree \ket{\text{hlp}}_{0\degree} + \sin 60\degree \ket{\text{vlp}}_{0\degree} &= \frac{1}{2} \ket{\text{hlp}}_{0\degree} + \frac{\sqrt{3}}{2} \ket{\text{vlp}}_{0\degree}\\ \ket{\text{vlp}}_{60\degree} &= -\sin 60\degree \ket{\text{hlp}}_{0\degree} + \cos 60\degree \ket{\text{vlp}}_{0\degree} &= -\frac{\sqrt{3}}{2} \ket{\text{hlp}}_{0\degree} + \frac{1}{2} \ket{\text{vlp}}_{0\degree}. \end{align*}$$