# What is a "spatially encoded qubit"?

I am studying the photonic CNOT (https://arxiv.org/abs/quant-ph/0403062v1).

I am not understanding what is a spatially encoded qubit? How the setup in fig. 1(b) can create this spatially encoded qubit?

Thank you

## 1 Answer

A "qubit/qudit" is in general just a degree of freedom of a certain dimension.

So, for example, a single photon can have several degrees of freedom. Its polarisation is one that is often considered, but its position is another degree of freedom that is often used to encode information. If a photon (or other particle) can be in one of $$N$$ possible positions, then that provides you an $$N$$-dimensional degree of freedom. If it can be in one of two possible spatial positions, then it amounts to a qubit.

In the scheme you're referring to, you have a photon which can be in two different polarisation states, and in two different position states. In other words, the photon is the physical substratum for a two-qubit system. One qubit is provided by the polarisation, and the other by the position. Something like a polarising beamsplitter (PBS) effectively acts as a CNOT between these two qubits, as it affects the position conditionally on the polarisation.