Types of photon qubit encoding

How many types of qubit encoding on photons exist nowadays? I know only two:

• Encoding on polarization: $$\lvert \Psi \rangle = \alpha \lvert H \rangle + \beta \lvert V \rangle$$ $$\lvert H \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{H}(\mathbf{k}) \lvert 0 \rangle_\text{Vacuum}$$ $$\lvert V \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{V}(\mathbf{k}) \lvert 0 \rangle_\text{Vacuum}$$

• Time-bin: $$\lvert \Psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle$$ $$\lvert 0 \rangle = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -z/c}{\delta t_{ph}}\right) e^{-i w_0 (t-z/c)} \hat{a}^\dagger(z) \lvert 0 \rangle_\text{Vacuum}$$ $$\lvert 1 \rangle = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -z/c+\tau}{\delta t_{ph}}\right) e^{-i w_0 (t-z/c+\tau)} \hat{a}^\dagger(z) \lvert 0 \rangle_\text{Vacuum}$$

Is there anything else?

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No photon vs. one photon. – Norbert Schuch Apr 1 '13 at 21:51
Spin up vs. spin down is also quite popular. – Vašek Potoček Nov 11 '13 at 2:11

In general, any two orthogonal modes of light provide a way of constructing a two-level quantum system that can serve as a qubit. The possibilities are immense! Of course, not every option will be viable in practice. Besides the cases you mention, I know of at least a few others that have been considered and employed in the lab:

1. Dual-rail: A qubit is encoded in the occupation of one single photon in either of two orthogonal modes, that we label A and B. In your notation, the logical qubit states are $$|0\rangle=|0\rangle_A|1\rangle_B$$ $$|1\rangle=|1\rangle_A|0\rangle_B$$

See this paper for a good review.

2. Orbital angular momentum: A photons in modes of different optical angular momentum can serve as basis states of a qubit. If $\ell$ is the angular momentum quantum number, we have for example

$$|0\rangle=|1\rangle_{l=0}$$ $$|1\rangle=|1\rangle_{l=1}$$

See this paper for a good review.

3. Occupation number: As stated in the comment above, given one single mode, qubit basis states can be defined in terms of the photon occupation number. The state $|0\rangle$ corresponds to no photons in the mode and $|1\rangle$ to one photon in the mode.

If you are interested in learning about general implementations of quantum information processing with light, there are many other interesting examples, of which I will provide only a few.

There is an entire paradigm for continuous-variable quantum computation with light, as reviewed in this paper. People have demonstrated entanglement between time and energy in photons. Finally, non-orthogonal coherent states have been used in quantum key distribution protocols.

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