In my research, I stumbled on a smooth map: $$\pi_{\rho_0}: B \setminus \{\rho_0\} \to \partial B$$ where $B$ is the open Bloch ball, corresponding to the set of mixed states of a single qubit and $\partial B$ is the Bloch sphere proper, consisting of the set of pured states of a single qubit and $\rho_0 \in B$ is a fixed mixed state of a single qubit. The map is defined as follows.
After fixing $\rho_0 \in B$, you define the image of a mixed state $\rho \in B$, $\rho \neq \rho_0$ as the intersection of the ray starting from $\rho_0$ and passing through $\rho$ with $\partial B$ (the Bloch sphere proper). This can be characterized as follows. It is the unique pure state which can be written as a linear combination
$$a \rho_0 + b \rho$$
where $a, b \in \mathbb{R}$ such that $a + b = 1$ and $b > 1$ (and therefore $a < 0$).
Note that $\pi$ can be viewed as a map
$$ B \times B \setminus \Delta \to \partial B $$
mapping $(\rho_0, \rho)$ to $\pi_{\rho_0}(\rho)$, where $\Delta \subset B \times B$ is the diagonal (i.e. $\Delta$ consists of all pairs $(\rho, \rho)$ such that $\rho \in B$).
Then if $G = \mathrm{SU}(2)$, then $G$ acts on $B \times B$ by $$ g.(\rho_0, \rho) := (g \rho_0 g^{-1}, g \rho g^{-1}). $$ Moreover, $G$ acts on $\partial B$ by $$ g.\nu = g \nu g^{-1}.$$
Then, if I am not mistaken, $\pi$ is $G$-equivariant.
This map $\pi$ seems natural. I know it appears in the literature when $\rho_0 = \frac{1}{2}I$ is at the "origin" of the Bloch ball. But did the more general $\pi_{\rho_0}$, where $\rho_0$ is any fixed mixed state of the qubit, appear in the quantum information theory literature please?
I would like to investigate whether there may be a link between a problem in geometry that I am interested in and quantum information theory.