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The photon spin-1 has two states, $\pm\hbar$, just like the spin qubit ($\pm\frac{\hbar}{2}$).

From a quantum information point of view, they can encode the same amount of data.

However, I am confused about the usual qubit representation:

\begin{equation} |q\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\gamma}\sin\frac{\theta}{2}|1\rangle. \tag{1} \end{equation}

The $\frac{1}{2}$ factor comes from the spin operators:

\begin{equation} S_{x,y,z} = \frac{\hbar}{2}\sigma_{x,y,z} \end{equation} where $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices.

They imply that a full rotation needs $\theta=4\pi$ and a bit flip (from 0 to 1) needs a $2\pi$ rotation.

But for the photon, a full rotation (for polarization) only need $2\pi$.

So, is the usual qubit representation (1) valid for photons? can the photon spin state be represented on the Bloch sphere? I sometimes heard that the right representation for the photon is the Poincaré sphere.

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  • $\begingroup$ Can you write an equation like (1) for the Poincaré sphere? I do not think so... $\endgroup$
    – Mauricio
    Commented Mar 15 at 12:51

2 Answers 2

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We can take any two quantum states, be they the spin of a spin 1/2 particle, or the polarisation of a photons or even things like a single excitation shared across two atoms, and represent superpositions of the two states via the Bloch sphere. However, what those angles physically mean doesn’t have to correspond to physical rotation at all: if I interpolate a superposition between an excitation on atom A and an excitation on atom B I can represent that as a rotation of the Bloch sphere, but clearly no physical rotation is going on. It just so happens that for spin 1/2 particles the dynamics of their spin indeed has a basis of two states, and the Bloch sphere in this case describes actual physical rotations.

In the case of photon polarisation it’s a case where it’s not really a direct representation of a physical rotation by the parameters on the Bloch sphere.

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  • $\begingroup$ Do you mean that Bloch sphere is valid for photon qubit representation (an actually valid for any 2 quantum levels system) but that rotation on he Bloch sphere do not correspond to a physical (polarization) rotation of the photon ? If I want the physical rotation it has to be on the Poincaré sphere ? $\endgroup$
    – deb2014
    Commented Mar 20 at 9:08
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Yes the Poincare sphere (representing optical polarization on the transverse plane) and the Bloch sphere (representing qubits) are mathematically equivalent. They are both the configuration spaces of spin half modulo a global phase. The reason why it works for optics is because it comes from restricting the three dimensional space to just the two transverse dimensions for paraxial propagation. The polarization of photons are often used to represent qubits in experimental implementations of quantum information systems.

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  • $\begingroup$ I do not quite understand the sentence "restricting the three dimensional space to just the two transverse dimensions for paraxial propagation". Yes, one can draw a bijection between bloch and Poincaré spheres, but does it mean I can use the Bloch sphere for photons (as a qubit) ? It is still striking for me because of the rotation angles which are not the same (between spin qubit and photons) $\endgroup$
    – deb2014
    Commented Mar 15 at 9:44

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