2.6.2 Polarization

Now that we are mathematically familiar with qubit states and operations, it would be nice to connect them with something physical.

One of the simplest ways of representing a qubit physically is by polarization of light. Light is an electromagnetic wave that propagates through space in a straight line. This wave oscillates in a direction perpendicular to one in which it travels. Note that there are several possible such directions – a wave that travels forward can oscillate from left to right or from top to bottom. These horizontal and vertical modes of oscillation can be used to represent the two basis states of a qubit:

|↔⟩=(10),|↕⟩=(01).|{\leftrightarrow}\rangle=\begin{pmatrix}1\\ 0\end{pmatrix},\qquad|{\updownarrow}\rangle=\begin{pmatrix}0\\ 1\end{pmatrix}.

More generally, we can use a wave that oscillates at an angle θ\theta with the horizontal axis to represent the state

|ψ⁢(θ)⟩=cos⁡θ⁢|↔⟩+sin⁡θ⁢|↕⟩=(cos⁡θsin⁡θ).\left|\psi(\theta)\right\rangle=\cos\theta\,|{\leftrightarrow}\rangle+\sin% \theta\,|{\updownarrow}\rangle=\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}.

For example, diagonally polarized light that oscillates at 45∘45^{\circ} angle between the vertical and horizontal directions represents the state |ψ⁢(π/4)⟩=|+⟩\left|\psi(\pi/4)\right\rangle=\left|+\right\rangle. Note that in this representation the direction of oscillation of the electromagnetic wave agrees with the direction of the vector we used in Fig. 2.1 of Section 2.1.2 to represent a qubit state on a circle.

To prepare one of these states, we can simply pass a beam of light through a polarizer, like the one in your sunglasses or in 3D glasses at the cinema. A polarizer lets through only some part of the wave – that whose direction of oscillation is compatible with the direction of the polarizer. To prepare the state |ψ⁢(θ)⟩\left|\psi(\theta)\right\rangle, we can simply tilt the polarizer at angle θ\theta from the horizontal axis. For example, Fig. 2.8 depicts how to prepare the states |0⟩\left|0\right\rangle, |1⟩\left|1\right\rangle, and |+⟩\left|+\right\rangle.

Figure 2.8: Horizontally, vertically, and diagonally polarized light can be used to represent the qubit states |0⟩\left|0\right\rangle, |1⟩\left|1\right\rangle, and |+⟩\left|+\right\rangle.

An interesting feature of representing qubit states by polarized light is that the states |ψ⁢(θ)⟩\left|\psi(\theta)\right\rangle and |ψ⁢(θ+π)⟩\left|\psi(\theta+\pi)\right\rangle are prepared using the same procedure – tilting the polarizer at an angle θ\theta. This means that these two states must be identical! Hence polarization gives an intuitive explanation for why the states |ψ⟩\left|\psi\right\rangle and −|ψ⟩-\left|\psi\right\rangle should be indistinguishable (see 2.7).

Figure 2.9: The amount of horizontal polarization in light can be determined by passing it through a horizontal polarizer and then measuring its brightness. For horizontally, vertically, and diagonally polarized light this results in 100%100\%, 0%0\%, and 50%50\% brightness, which coincide with the probabilities of observing outcome 0 when measuring the states |0⟩\left|0\right\rangle, |1⟩\left|1\right\rangle, and |+⟩\left|+\right\rangle.

Another nice feature of viewing qubits as polarized light is that we can easily visualize measurement. Assume we want to measure the state |ψ⁢(θ)⟩\left|\psi(\theta)\right\rangle to determine the probability of the outcome 0. If the state is provided to us as a beam of light, polarized at angle θ\theta, we can simply pass it through a horizontal polarizer and see how much light gets through – if the brightness has decreased to 70%70\%, the probability of outcome 0 is 70%70\%. In particular, if the input beam was horizontally polarized, all of it will get through, while if it was vertically polarized, none of it will get through. Passing a diagonally polarized beam of light through a horizontal polarizer will result in a 50%50\% decrease in brightness (see Fig. 2.9).

Exercise 2.9 (Polarization experiment).

If you happen to have a pair of polarized sunglasses at home, you can put them on and take a look at the screen of your phone or computer. Usually screens emit polarized light (whose direction of polarization depends on device). When you tilt your head sideways, you should see the screen becoming lighter or darker. Can you explain why this is the case?

Solution. Tilting your head changes the angle between the polarizer in your sunglasses and the direction in which the electromagnetic light waves emitted by your screen oscillate. Since the amount of light that can pass through a polarizer depends on this angle, the screen will appear either brighter or darker. Similarly, changing the angle θ\theta will change the probability that measuring the state |ψ⁢(θ)⟩\left|\psi(\theta)\right\rangle will produce the outcome 0.

Polarization of light is just one example of how a qubit could be implemented in laboratory. Another example is the location of the light-carrying particle called photon – since a photon behaves according to the laws of quantum mechanics, it can simultaneously be at two locations in superposition. If we call these locations 0 and 11, the photon’s state corresponds to a qubit. There are many other options: the current in a superconducting circuit can simultaneously flow in both directions, an electron can simultaneously occupy two orbitals around an atom, and so on. In short, any quantum mechanical system that can be in two distinct states can also be in their superposition, hence it can potentially be used as a physical representation of a qubit.