Single Qubits and Product States

We should point out that product states are also used to describe the states of a single qubit (physical object) if that qubit has several distinct dynamical properties—properties that can be manipulated and change with time. In tech speak, such dynamical properties are called degrees of freedom. A specific example will be helpful. Suppose the qubit is a photon. One of the dynamical properties is its linear polarization. We have seen how we can manipulate the linear polarization state by using polarizing filters, for example. But the photon has other dynamical properties: the direction or path in which it is traveling and its energy, which is proportional to the oscillation frequency of the corresponding classical wave. You may recall that a green photon has more energy than a red photon.

Taking those dynamical properties into account, we write the state $\ket S$ of the photon as

$$\ket S = \ket{\text{polarization}}\ket{\text{direction of propagation}}\ket{\text{energy}}.$$

Each part of the product state might be a superposition state, as we have seen for polarization. The overall state space for the qubit is then a tensor product state space.

If a particular dynamical property does not change in the situation we have set up, we can drop that part of the product state and ignore that part of the product state space. That is often, but not always, the case with the photon’s energy.

Let’s return to our old friend the polarizing beam splitter, shown in the figure below: