Terminology: what do $|i\rangle$ and $|\mbox{-}i\rangle$ represent?

Question

$|0⟩$ and $|1⟩$ are usually referred as the computational basis. $|+⟩$ and $|-⟩$, the polar basis.

What about $|i\rangle$ and $|\mbox{-}i\rangle$?

And collectively? Orthonormal states?

References are welcomed!

Answer 1

In my opinion the nature of these states becomes quite clear when we look at it from an optics angle. We can identify the computational basis states with the vertical and horizontal polarization directions: $$ |0\rangle \sim |\updownarrow\,\rangle \qquad |1\rangle \sim |\leftrightarrow\,\rangle $$ The superposition states then correspond to diagonally polarized light: $$ |+\rangle \sim |⤢\,\rangle \qquad |-\rangle \sim |⤡\,\rangle $$

Now, the superposition states that have an $i$ do actually correspond to circularly polarized light: $$ |+i\rangle \sim |\circlearrowright\,\rangle \qquad |-i\rangle \sim |\circlearrowleft\,\rangle $$ Which also explains the labels $R$ for right and $L$ for left in @Z..'s post.

This correspondence is explained by the fact that circularly polarized light is created by superposing vertical light with horizontal light that has a $\pi/2$ phase difference. This phase difference is exactly $\mathrm{e}^{i \pi/2}=i$.

Answer 2

Quirk refers to the $\frac{1}{\sqrt{2}}|0\rangle + \frac{i}{\sqrt{2}}|1\rangle$ state as $|i\rangle$ and to the $\frac{1}{\sqrt{2}}|0\rangle - \frac{i}{\sqrt{2}}|1\rangle$ state as $|-i\rangle$:

When I implemented this it just seemed like a natural choice at the time. I didn't get it from a textbook or a paper.

Answer 3

This is another reference.

$|i\rangle$ and $|\mbox{-}i\rangle$ are two orthogonal y-basis states. In the above link they are called $|R\rangle$ and $|L\rangle$.

$$|i\rangle = \frac{1}{\sqrt{2}}\left[ \begin{array}{c} 1 \\ i \end{array} \right] \;\; , \;\; |\mbox{-}i\rangle = \frac{1}{\sqrt{2}}\left[ \begin{array}{c} 1 \\ -i \end{array} \right]$$

You can simply check the orthonormality by using the definition of inner product space $\mathbb{C}^2$, $\langle v | w\rangle =\sum(v_i^{*} w_i)$, and Kronecker delta function.

$$\langle i|i\rangle = [1.1 + (-i).i]/2 = 1$$

$$\langle i|\mbox{-}i\rangle = [1.1 + (-i).(-i)]/2 = 0$$