The Exercise

A state $\rho$ of a qubit is a density operator, and it is written $\rho = (I + \vec{r} \cdot \vec{\sigma})/2$, with $\vec{\sigma}$ the vector of Pauli matrices and $\vec{r}$ its Bloch vector, a point of the unit ball in $\mathbb{R}^3$. A trace preserving quantum operation on the qubit acts on Bloch vectors as an affine map, $$\vec{r} \mapsto M \vec{r} + \vec{c},$$(1) where $M$ is a real $3 \times 3$ matrix and $\vec{c}$ a vector of $\mathbb{R}^3$. The exercise bounds the number of points one has to follow to know the map: if the images of the points of a finite set determine $M$ and $\vec{c}$, there are at least four of them.^✦

Points That Fix an Affine Map

The library poses the question for every affine map of $\mathbb{R}^3$, with any real matrix and any translation, and for any finite set of points of $\mathbb{R}^3$, inside the Bloch ball or not. A finite set $P$ of points of $\mathbb{R}^3$ specifies affine Bloch maps if any two affine maps that agree at every point of $P$ have the same matrix and the same translation: for all real $3 \times 3$ matrices $M_1, M_2$ and all vectors $\vec{c}_1, \vec{c}_2$ of $\mathbb{R}^3$, $$M_1 \vec{r} + \vec{c}_1 = M_2 \vec{r} + \vec{c}_2 \ \text{ for every } \vec{r} \in P \qquad \Longrightarrow \qquad M_1 = M_2 \ \text{ and } \ \vec{c}_1 = \vec{c}_2 .$$(2)^✦

The Claim

What Is Checked

The claim is stated over the sets of points that specify affine Bloch maps, the condition (2). What the library proves about such a set is the bound (3) on its size.