The Exercise

A qubit is prepared in a pure state $|\psi\rangle$ and sent through the depolarizing channel $\mathcal{E}$ with parameter $p$, a real number with $0 \le p \le 1$. With probability $1-p$ the qubit passes untouched; with probability $p/3$ each it suffers a bit flip $X$, a phase flip $Z$, or the combined flip $Y$. The output $\mathcal{E}(|\psi\rangle\langle\psi|)$ is in general mixed, and its fidelity with the input measures how much of the state survives the noise. The exercise asks for the smallest value of this fidelity over all pure inputs, as a function of $p$.

The library builds the channel as a mixture of conjugations of the qubit by the identity and the three Pauli matrices, and gives the answer in two parts: the fidelity for the input $|0\rangle$, and the minimum over all pure inputs.

A Quantum System

A quantum system $S$ is described by its state space, a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, pure or mixed.^✦ A pure state $|\psi\rangle$ of $S$ is a unit vector in the state space of $S$, determined physically up to a global phase^✦; it gives the state $|\psi\rangle\langle\psi|$, the projector onto $|\psi\rangle$.^✦

The qubit is the two level system whose state space is $\mathbb{C}^2$.^✦ For a bit $i$, the basis state $|i\rangle$ of the qubit is the pure state whose vector is the $i$th standard basis vector of $\mathbb{C}^2$, so that $|0\rangle$ and $|1\rangle$ form the computational basis.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = I$.^✦ Under it a state $\rho$ evolves to $U \rho U^\dagger$.^✦

The Depolarizing Channel

A random unitary channel $\mathcal{C}$ of $S$ is a finite family of unitaries $U_k$ of $S$ together with probabilities $p_k$, nonnegative and summing to one.^✦ It acts on a state $\rho$ of $S$ by averaging the evolved states with these weights: $$\mathcal{C}(\rho) = \sum_k p_k\, U_k \rho U_k^\dagger .$$(1)^✦

On the qubit the unitaries come from the Pauli matrices, $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(2) For a real vector $\vec{n}$ in three dimensions, $\vec{n} \cdot \vec{\sigma}$ is the matrix $n_x X + n_y Y + n_z Z$^✦, and the rotation about $\vec{n}$ by an angle $\theta$ is the matrix $$R_{\vec{n}}(\theta) = \exp\!\left( -\frac{i\theta}{2}\, \vec{n} \cdot \vec{\sigma} \right),$$(3)^✦ written $R_{\hat{n}}(\theta)$ when the axis is a unit vector $\hat{n}$. Acting on $\mathbb{C}^2$, it is a unitary of the qubit.^✦

Let $\hat{x}$, $\hat{y}$, $\hat{z}$ be the unit vectors along the three coordinate axes. The four Pauli conjugations of the qubit are the rotations $R_{\hat{x}}(0) = I$, $R_{\hat{x}}(\pi)$, $R_{\hat{y}}(\pi)$ and $R_{\hat{z}}(\pi)$.^✦ Since $R_{\hat{n}}(\pi) = -i\, \hat{n} \cdot \vec{\sigma}$ and the global phase cancels in $U \rho U^\dagger$, they conjugate a state by $\sigma_0 = I$, $\sigma_1 = X$, $\sigma_2 = Y$ and $\sigma_3 = Z$.

Given weights $p_0, \dots, p_3$, nonnegative and summing to one, the qubit Pauli channel is the random unitary channel that mixes these four conjugations, $\rho \mapsto \sum_i p_i\, \sigma_i \rho \sigma_i$.^✦

The depolarizing channel $\mathcal{E}$ of the qubit with parameter $p$, $0 \le p \le 1$, is the Pauli channel with weights $1-p$, $p/3$, $p/3$, $p/3$: $$\mathcal{E}(\rho) = (1-p)\, \rho + \frac{p}{3} \left( X \rho X + Y \rho Y + Z \rho Z \right).$$(4)^✦

Fidelity

The fidelity of two states $\rho$ and $\sigma$ of a system $S$ is the nonnegative real number $$F(\rho, \sigma) = \operatorname{tr} \sqrt{\sqrt{\rho}\, \sigma \sqrt{\rho}},$$(5)^✦ the trace over the state space of $S$, with each square root of a positive operator taken on its eigenvalues. The fidelity the exercise asks about is $F\bigl(|\psi\rangle\langle\psi|, \mathcal{E}(|\psi\rangle\langle\psi|)\bigr)$, between the projector onto the input and the output of the channel.

The Claim

What Is Checked

The claim is stated over the depolarizing channel (4), a Pauli channel of the qubit, and over the fidelity (5) between the projector onto a pure state and its image under the channel.

The library proves that the weights $1-p$, $p/3$, $p/3$, $p/3$ are nonnegative when $0 \le p \le 1$^✦ and sum to one for every $p$^✦, so the depolarizing channel is a Pauli channel. It proves that the projector onto a unit vector is a density operator, and that conjugating a density operator by a unitary gives a density operator; the output (1) of a random unitary channel, an average of such conjugates, is therefore again a state.