The Exercise

A qubit $Q$ is coupled to a second qubit $E$, its environment, prepared in the state $|0\rangle$. The two pass through the circuit of Figure 8.13 of the book. First $Q$ controls a rotation of $E$ through an angle $\theta$ about the $\hat{y}$ axis; then a controlled not (CNOT) gate, with $E$ as its control, flips $Q$. Afterwards $E$ is discarded. The exercise is to show that $Q$ has undergone amplitude damping, with damping parameter $\gamma = \sin^2(\theta/2)$.

Operators on a Qubit

Let $H$ be the state space of a qubit, with computational basis $|0\rangle$, $|1\rangle$. For $i$ in $\{0,1\}$, the projector $P_i$ onto $|i\rangle$ is the operator $|i\rangle\langle i|$ on $H$.^✦ The bit flip, or Pauli operator $X$, is the operator $|0\rangle\langle 1| + |1\rangle\langle 0|$ on $H$; it exchanges the two basis states.^✦ The lowering operator $\sigma_-$ on $H$ is $|0\rangle\langle 1|$, which takes $|1\rangle$ to $|0\rangle$ and annihilates $|0\rangle$.^✦

The rotation generator $G$ is the operator on $H$ $$G = |1\rangle\langle 0| - |0\rangle\langle 1| .$$(1)^✦ The rotation $R_y(\theta)$ through an angle $\theta$ about the $\hat{y}$ axis is the operator on $H$ $$R_y(\theta) = \cos\frac{\theta}{2}\, I + \sin\frac{\theta}{2}\, G = \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\[2pt] \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix},$$(2)^✦ where $I$ is the identity on $H$ and the matrix is taken in the computational basis.

Amplitude Damping

Amplitude damping describes a qubit that decays from $|1\rangle$ to $|0\rangle$ with probability $\gamma$. The operation elements of amplitude damping with parameter $\gamma$ are the operators $E_0$ and $E_1$ on $H$ given by $$E_0 = P_0 + \sqrt{1-\gamma}\, P_1 , \qquad E_1 = \sqrt{\gamma}\, \sigma_- .$$(3) The amplitude damping channel $\mathcal{E}_{\mathrm{AD},\gamma}$ is the map on operators on $H$ given by $$\mathcal{E}_{\mathrm{AD},\gamma}(\rho) = E_0\, \rho\, E_0^\dagger + E_1\, \rho\, E_1^\dagger .$$(4)^✦

The Circuit

The circuit acts on the state space $H \otimes H$ of the two qubits, with $Q$ in the first factor and $E$ in the second. The controlled rotation $CR_y(\theta)$ is the operator on $H \otimes H$ that rotates $E$ by $R_y(\theta)$ when $Q$ is in $|1\rangle$: $$CR_y(\theta) = P_0 \otimes I + P_1 \otimes R_y(\theta) .$$(5)^✦ The controlled not $\mathrm{CNOT}$ on $H \otimes H$ is the operator that flips $Q$ when $E$ is in $|1\rangle$ and leaves it unchanged when $E$ is in $|0\rangle$: $$\mathrm{CNOT} = I \otimes P_0 + X \otimes P_1 .$$(6) The circuit $U(\theta)$ is the controlled rotation followed by the CNOT: $$U(\theta) = \mathrm{CNOT}\; CR_y(\theta) .$$(7)^✦

The dilation $V(\theta)$ of the circuit is the linear map from $H$ to $H \otimes H$ that prepares $E$ in $|0\rangle$ and then applies $U(\theta)$: $$V(\theta)\, |\psi\rangle = U(\theta) \bigl( |\psi\rangle \otimes |0\rangle \bigr) \qquad \text{for every } |\psi\rangle \text{ in } H .$$(8)^✦ The channel $\mathcal{E}_\theta$ of the circuit sends an operator $\rho$ on $H$ through the dilation and discards the environment: $$\mathcal{E}_\theta(\rho) = \mathrm{tr}_E \bigl( V(\theta)\, \rho\, V(\theta)^\dagger \bigr) ,$$(9)^✦ where $\mathrm{tr}_E$ is the partial trace over the second factor.

The Claim

On the computational basis the dilation acts as $$V(\theta)|0\rangle = |0\rangle|0\rangle , \qquad V(\theta)|1\rangle = \cos\frac{\theta}{2}\, |1\rangle|0\rangle + \sin\frac{\theta}{2}\, |0\rangle|1\rangle ,$$(10) so tracing out $E$ leaves $Q$ with the operators $P_0 + \cos\frac{\theta}{2}\, P_1$ and $\sin\frac{\theta}{2}\, \sigma_-$, one for each basis state of the environment. For $0 \le \theta \le \pi$ the half angle lies between $0$ and $\pi/2$, where cosine and sine are nonnegative; then $\cos\frac{\theta}{2} = \sqrt{1-\gamma}$ and $\sin\frac{\theta}{2} = \sqrt{\gamma}$ for $\gamma = \sin^2\frac{\theta}{2}$.

What Is Checked

The claim is stated over two channels: the channel $\mathcal{E}_\theta$ of the circuit (9), built through the dilation (8) on the circuit (7) and its gates (5) and (6), and the amplitude damping channel (4) with its operation elements (3). What the library proves about these constructions is the equality (11) of the two.