The Exercise

The exercise is about the bit flip channel on a qubit, which leaves the state alone with probability $p$ and applies the Pauli matrix $X$ with probability $1-p$, so that a state $\rho$ becomes $X \rho X$. The channel never increases the trace distance between two states: it is contractive.^✦ Yet it is not strictly contractive, for some pair of distinct states comes out of it no closer than it went in.^✦ When the flip has nonzero probability, the states the channel leaves unchanged are exactly those whose Bloch vector lies on the $x$ axis.^✦

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ The qubit is the quantum system whose state space is $\mathbb{C}^2$.^✦

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) For a real vector $\vec n = (n_x, n_y, n_z)$ the matrix $\vec n \cdot \vec\sigma$ is $n_x X + n_y Y + n_z Z$^✦, with the matrices (1), and the Bloch matrix of a real vector $\vec r$ is $\frac12 (I + \vec r \cdot \vec\sigma)$^✦. When $\vec r$ lies in the closed unit ball, $r_x^2 + r_y^2 + r_z^2 \le 1$, the Bloch state $\rho_{\vec r}$ with Bloch vector $\vec r$ is the state of the qubit whose matrix is the Bloch matrix of $\vec r$: $$\rho_{\vec r} = \tfrac12 \left( I + r_x X + r_y Y + r_z Z \right).$$(2)^✦

The Bit Flip Channel

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = I$^✦, and it takes a state $\rho$ of $S$ to $U \rho U^\dagger$^✦. A random unitary channel of $S$ is a finite family of unitaries $U_k$ of $S$ together with probabilities $p_k$, nonnegative and summing to one^✦; it takes a state $\rho$ to the average of the conjugated states, $$\rho \mapsto \sum_k p_k\, U_k \rho\, U_k^\dagger .$$(3)^✦

For a real vector $\vec n$ in three dimensions and an angle $\theta$, the rotation $R_{\vec n}(\theta)$ is the unitary of the qubit with matrix $$R_{\vec n}(\theta) = \exp\left( -\frac{i\theta}{2}\, \vec n \cdot \vec\sigma \right).$$(4)^✦

A Pauli channel of the qubit, for probabilities $q_0, q_1, q_2, q_3$, is the random unitary channel (3) that applies $R_{\hat x}(0) = I$, $R_{\hat x}(\pi)$, $R_{\hat y}(\pi)$ and $R_{\hat z}(\pi)$ with these probabilities, so that $\rho \mapsto q_0 \rho + q_1 X \rho X + q_2 Y \rho Y + q_3 Z \rho Z$.^✦

The bit flip channel $\mathcal{E}_p$, for $0 \le p \le 1$, is the Pauli channel with weights $(p,\, 1-p,\, 0,\, 0)$: $$\mathcal{E}_p(\rho) = p\, \rho + (1-p)\, X \rho X .$$(5)^✦

Trace Distance

The trace distance $D(\rho, \sigma)$ between two states $\rho$ and $\sigma$ of $S$ is half the trace norm of their difference: $$D(\rho, \sigma) = \tfrac12 \operatorname{tr} \lvert \rho - \sigma \rvert .$$(6)^✦ A map $f$ on the states of $S$ is strictly contractive when it brings any two distinct states strictly closer together: $$D\bigl(f(\rho), f(\sigma)\bigr) < D(\rho, \sigma) \qquad \text{whenever } \rho \neq \sigma .$$(7)^✦

The Claim

What Is Checked

The claim is stated over the bit flip channel (5) on the qubit, the Bloch states (2), the trace distance (6) and the strict contraction (7) defined from it.

The library proves of these constructions that the weights $(p, 1-p, 0, 0)$ are nonnegative for $0 \le p \le 1$^✦ and sum to one^✦, and that the matrix (4) is unitary for every $\vec n$ and $\theta$. It proves that a state conjugated by a unitary, and the average (3) of such states, is again a state, and that the Bloch matrix of a vector in the closed unit ball is positive with unit trace, so that $\rho_{\vec r}$ is a state.