# Difference between Poincaré and Bloch sphere for single photons

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 $$y=-S_3$$)?

• I am not an expert, but yes, I believe that this is correct. Both the Poincare sphere and the Bloch sphere are the same mathematical object, which in turn can also be considered "the same" as $\mathbb{CP}^1$ the 1-(complex-)dimensional complex projective space, which can also be identified with the Riemann spehre en.wikipedia.org/wiki/Complex_projective_space . You might find this youtube playlist discussing both the Bloch spehre and the Poincare sphere in the more general context of spinors helpful: youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs Commented Jan 16, 2023 at 4:09