# Photon as a qubit

I am studying Quantum Key Distribution which relies on photonic qubits, the photon.

Indeed, this one is viewed as a qubit, that is a two state system, just like the spin of an electron, an atom with ground/excited energy levels, ...

I wonder however what are the two states at stake for the photon.

I usually hear about the photon polarization. But polarization rather sounds for me as a classical property : that of the electromagnetic wave which can be either rectilinear, circular or elliptic polarized.

What is the property used for using photons as qubits ?

I know that a photon has a spin-1, which means its spin has three degrees of freedom, but one of them is forbidden (the degree of freedom parallel to its direction propagation).

Do the two remaining degree of freedom act like polarization ? Indeed, if you make a beam of polarized light go through two orthogonal classical filters, there will be extinction. But on the Bloch sphere, states $$|0\rangle$$ and $$|1\rangle$$ are not orthogonal in 3D space, as one is the $$180^°$$ rotation of the other.

How can I explain clearly what is the photon polarization ?

## 3 Answers

Polarization is photon spin. The two helicity eigenstates are left and right handed circular polarization, which are linear combinations of linear polarization.

The aforementioned rotation on the Bloch sphere is not a rotation in physical space.

A photon has both spin and spatial/temporal properties. You can encode a qubit using any pair of states which are orthogonal in the Hilbert-space sense. Since there is an infinite set of spatial/temporal states available (often called modes) there is an infinite set of ways of encoding a qubit that way.

The spin of the photon is associated with a spin-1 spin state, but only the $$+1$$ and $$-1$$ states are available for the component of spin in any chosen direction (this restriction arises because the photon has a null energy-momentum four-vector; you have to take it on trust if you don't want to study the particle physics/field theory related to that). As a result there are just two orthogonal quantum states so the spin state space is just big enough to store one qubit.

The space can be spanned by the spin states right- and left-circularly polarized, and also by the spin states corresponding to linear polarization in two directions orthogonal in ordinary 3D space (e.g. $$x$$ and $$y$$ directions if the photon is propagating along $$z$$). This fact may lead to confusion when you look at the Bloch sphere picture, because 'linearly polarized along $$x$$ direction' is not the same as 'spin-up along $$x$$ direction'.

If $$|0\rangle$$ is right circular polarization and $$|1\rangle$$ is left circular polarization then $$|x\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$$ and $$|y\rangle = (|0\rangle - |1\rangle)/\sqrt{2}$$ are linearly polarized states. Notice that they are mutually orthogonal.

Because you really have a spin-1 particle, not spin half, the interpretation of the Bloch sphere picture is in some respects different than for a spin half particle.

• How do you demonstrate that the photon spin is 1 ? and how do you show that the 0 state is forbidden because photon has no mass at rest ? If you are aware of courses explaining this, I would be interested. Commented Dec 20, 2023 at 15:02
• @deb2014 experimental way is to measure it. Theoretical way is to argue it is an excitation of certain type of field. The measurement could e.g. be excite an atom between two states known to differ by 1 unit of angular momentum. Commented Dec 20, 2023 at 20:09

In the case of subatomic particles, there is clearly a unity between the orientation of the magnetic dipole of the particle and its spin. Ultimately, the spin was explained on the experimental basis of the influence of a magnetic field on the fine and hyperfine structure of the emission spectrum. I therefore find it somewhat problematic to speak of a spin of the photon without reflection.

How can I explain clearly what is the photon polarization?

A photon can be manipulated at a polarisation grating in such a way that its field components are aligned in any desired direction. After passing through a slit of suitable dimensions, the magnetic and electric field components of the photon are clearly aligned, namely parallel and 90° to the grid.

What is the property used for using photons as qubits?

The alignment of its field components is very clear and unambiguous. Both the magnetic and the electric field components can be used for engineering purposes, but since both are spatially orthogonally related, you cannot encode or extract more than one bit from them.

TL;DR
The spin of a photon would be located somewhere else entirely. The orthogonal arrangement of two vectors (which embody the magnetic and electric field components) has exactly two possible arrangements. If the magnetic vector points upwards, the electric vector can point to the left or to the right. As both field components oscillate +/- with each other, after half a period the M vector is pointing downwards and the E vector is rotated 90° again.

As we have a direction of propagation and look at our vectors in this direction without any change, the term helicity of these vector states could be used for this connection.
TL;DR