The Bloch sphere is a geometrical representation of a two-levels quantum system, for example we can use it to represent the spin of a single qubit in the basis $\{\lvert H \rangle, \lvert V \rangle\}$.
In this basis, the intersections of the Bloch sphere with the (positive side) of the axes will be
$x \longrightarrow \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}$
$y \longrightarrow\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}$
$z \longrightarrow\begin{pmatrix}1\\0\end{pmatrix}$
On the other hand the Poincaré sphere is used to represent the polarization state of light. If we use again the basis $\{\lvert H \rangle, \lvert V \rangle\}$ and use Jones calculus, again we can define three axes that we call $S_1$, $S_2$ and $S_3$ and (positive) the intersections will represent respectively
$S_1 \longrightarrow \begin{pmatrix}1\\0\end{pmatrix}$
$S_2 \longrightarrow \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}$
$S_3 \longrightarrow \frac{1}{\sqrt{2}}\begin{pmatrix}1\\-i\end{pmatrix}$
Since a single photon is a two levels quantum system, and we can describe its polarization by means of a Jones vector, if we consider a single photon, are the Poincaré sphere and Bloch sphere just the same mathematical object (defining $x=S_2$, $z = S_2$ and $y=-S_3$)?
