The Exercise

A qubit in the state $\rho$ is measured in the computational basis, and the state is kept only when the outcome is $0$. Without renormalisation it becomes $P_0 \rho P_0^\dagger$, where $P_0$ is the projector onto $|0\rangle$. The exercise asks to show two things about this operation: it does not preserve the trace, and it is not an affine map of the Bloch sphere. For any real $3 \times 3$ matrix $M$ and real vector $\vec{c}$ there is a state whose image has a Bloch vector different from $M\vec{r} + \vec{c}$, with $\vec{r}$ the Bloch vector of that state, the real vector with $\rho = \tfrac{1}{2}(I + \vec{r} \cdot \vec{\sigma})$, where $I$ is the identity and $\vec{\sigma}$ the vector of the three Pauli matrices.

The Bloch Picture

The Pauli matrices $X$, $Y$ and $Z$ are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) For a real vector $\vec{r} = (r_x, r_y, r_z)$ in three dimensions and the Pauli vector $\vec{\sigma} = (X, Y, Z)$, the matrix $\vec{r} \cdot \vec{\sigma}$ is the complex $2 \times 2$ matrix $$\vec{r} \cdot \vec{\sigma} = r_x X + r_y Y + r_z Z.$$(2)^✦

The Bloch matrix $\rho_{\vec{r}}$ of a real vector $\vec{r}$ in three dimensions is the complex $2 \times 2$ matrix, with $I$ the $2 \times 2$ identity, $$\rho_{\vec{r}} = \tfrac{1}{2} \left( I + \vec{r} \cdot \vec{\sigma} \right).$$(3)^✦

The Projection

For $i$ in $\{0, 1\}$, the computational basis ket $|i\rangle$ is the column vector of $\mathbb{C}^2$ with entry $1$ in position $i$ and $0$ in the other, so that $|0\rangle = (1, 0)$ and $|1\rangle = (0, 1)$.^✦ For $i$ and $j$ in $\{0, 1\}$, the outer product $|i\rangle\langle j|$ is the complex $2 \times 2$ matrix obtained as the column $|i\rangle$ times the row $\langle j|$, whose entries are the complex conjugates of those of $|j\rangle$.^✦

The projector $P_0$ onto $|0\rangle$ is the outer product $$P_0 = |0\rangle\langle 0| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.$$(4)^✦ The projection $\mathcal{E}$ onto $|0\rangle$ is the map on complex $2 \times 2$ matrices, states and others alike, that sends $A$ to $$\mathcal{E}(A) = P_0 A P_0^\dagger.$$(5)^✦

The Claim

What Is Checked

The claim is stated over the projection $\mathcal{E}$ of (5), built from the projector $P_0$ of (4), and over the Bloch matrices (3), built from the Pauli matrices (1). About these constructions the library proves both parts of the theorem: $\mathcal{E}$ does not preserve the trace, and on the Bloch vectors of the unit ball it acts as no affine map.