The Exercise

A qubit is sent through a depolarizing channel with parameter $p$, $0 \le p \le 1$: with probability $p$ its state is replaced by the maximally mixed state, and with probability $1-p$ it passes unchanged. Two states $\rho$ and $\sigma$ of the qubit, with Bloch vectors $\vec r$ and $\vec s$, are sent in. The exercise asks for the trace distance between the two output states, worked out from the Bloch vectors, and for a proof that when $p > 0$ the channel brings any two distinct states strictly closer together.

A Quantum System

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ The qudit with $d$ levels is the system whose state space is $\mathbb{C}^d$ with its standard inner product.^✦ The qubit is the qudit with two levels, the system with state space $\mathbb{C}^2$.^✦

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ The maximally mixed state of a system $S$ with a nonzero state space of dimension $d$ is the state of $S$ whose operator is $I/d$, with $I$ the identity.^✦

Bloch Vectors

The Pauli matrices 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) The Pauli vector is $\vec\sigma = (X, Y, Z)$; for a real vector $\vec n = (n_1, n_2, n_3)$, the matrix $\vec n \cdot \vec\sigma = n_1 X + n_2 Y + n_3 Z$.

The Bloch matrix of a real vector $\vec r$ is $\tfrac12(I + \vec r \cdot \vec\sigma)$. For $\vec r$ of length $|\vec r| = \sqrt{r_1^2 + r_2^2 + r_3^2} \le 1$, the Bloch state $\rho_{\vec r}$ of $\vec r$ is the state of the qubit whose operator on $\mathbb{C}^2$ is this matrix: $$\rho_{\vec r} = \tfrac12 \left( I + \vec r \cdot \vec\sigma \right).$$(2)^✦

Trace Distance

The trace distance between two states $\rho$ and $\sigma$ of $S$ is $$D(\rho, \sigma) = \tfrac12 \operatorname{tr} |\rho - \sigma|,$$(3)^✦ Here the trace norm is the sum of the absolute values of the eigenvalues.

A map $f$ on the states of $S$ is strictly contractive when it strictly decreases the trace distance between any two distinct states.^✦

The Depolarizing Channel

For a qudit with $d \ge 1$ levels and a probability $p$ with $0 \le p \le 1$, the depolarizing channel is the map on the states of the qudit that sends each state to its mixture with the maximally mixed state, $\rho$ with weight $1-p$ and $I/d$ with weight $p$.^✦ The depolarizing channel $\mathcal{E}$ of the qubit is the case $d = 2$: $$\mathcal{E}(\rho) = (1-p)\,\rho + p\,\frac{I}{2}.$$(4)^✦

The Claim

What Is Checked

The three claims are stated over the depolarizing channel (4) on the states of the qubit and the trace distance (3); the first is stated for the Bloch states (2), and the third over the definition of a strictly contractive map.

The library proves that each of these constructions yields a state. The Bloch matrix of $\vec r$ is positive exactly when $|\vec r| \le 1$, and its trace is one, so $\rho_{\vec r}$ is a state of the qubit. The operator $I/d$ is positive with unit trace, and the state space $\mathbb{C}^d$ of a qudit with $d \ge 1$ levels contains a nonzero vector^✦, so the qudit has a maximally mixed state. The output of the depolarizing channel is a convex combination of two states and therefore again a state.