Does this point-projection of a mixed state onto a pure state appear in the quantum information theory literature?

Question

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.

Markus Heinrich · Answer

I am not aware of a direct application of this projection. However, maybe the following geometrical construction is nevertheless of interest to you.
Similar ray constructions appear in resource theories, namely in the definition of robustness and generalised robustness monotones.
Consider a convex set of states $mathcal{F}$ which we call the free states in the resource theory. Let us denote by $mathcal{S}$ the convex set of all states. Then, given a state $rho$ we define the following functions
$$
R(rho) = infbig{ tgeq 0 ; | ; rho = (1+t)sigma_0 - t sigma_1 text{ for } sigma_0,sigma_1inmathcal{F} big},
$$
$$
GR(rho) = infbig{ tgeq 0 ; | ; rho = (1+t)sigma_0 - t sigma_1  text{ for } sigma_0inmathcal{F}, sigma_1inmathcal{S} big}.
$$
In the definitions, we optimise over all rays going through $mathcal{F}$ which include $rho$.
If $rho$ is pure, than $rho$ is the projection of $sigma_1$ w.r.t to $sigma_0$ under your map.
The monotones have the following geometrical interpretation:

For $R$: $sigma_0$ is the closest state in $mathcal{F}$ to $rho$ measured w.r.t. to the length of the ray segment which lies in $mathcal{F}$ (the diameter of $mathcal{F}$ in this direction)
For $GR$: similar, but compared to the distance of $sigma_0$ to the opposite boundary of $mathcal{S}$.

The generalised robustness is equivalent to what is called the max-relative entropy
$$
mathcal{D}_mathrm{max}(rho) := inf_{sigmainmathcal{F}} D_mathrm{max}(rho||sigma) = logleft( 1 + GR(rho) right)
$$
where $D_mathrm{max}(rho||sigma)=log inf{lambdageq 0 , | , rho leq lambdasigma}$.
After your remark, I changed $min$ to $inf$. However, $mathcal{F}$ is  usually compact.
Here is a review on resource theories which appeared recently in RMP. The above functions and more appear in Sec. VI.
Chitambar and Goul: "Quantum Resource Theories". https://arxiv.org/abs/1806.06107

Does this point-projection of a mixed state onto a pure state appear in the quantum information theory literature?

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