The Exercise

A qubit with density matrix $\rho$ undergoes phase damping with parameter $\lambda$, between $0$ and $1$, when the populations $\langle 0|\rho|0\rangle$ and $\langle 1|\rho|1\rangle$ stay as they are and the coherences $\langle 0|\rho|1\rangle$ and $\langle 1|\rho|0\rangle$ are multiplied by $\sqrt{1-\lambda}$. The exercise produces this process with a circuit. A second qubit, the environment, is prepared in $|0\rangle$ and rotated about the $y$ axis through an angle $\theta$, by the rotation $R_y(\theta)$, only when the system is in $|1\rangle$; the environment is then discarded. The exercise asks to show that, for a suitable angle, the circuit acts on the system as phase damping.

A Qubit and Its Environment

The system and the environment are qubits with the same state space $H$ and computational basis $|0\rangle, |1\rangle$. On the product $H \otimes H$ the system is the first factor and the environment the second, and $\operatorname{tr}_E$ is the partial trace over the environment. The library states the exercise as an identity between two maps on the operators on $H$: the phase damping channel and the map that the circuit induces on the system.

The projector $P_i$ onto the basis vector $|i\rangle$, for $i$ in $\{0,1\}$, is the rank one operator $|i\rangle\langle i|$ on $H$.^✦

Phase Damping

The operation elements $E_0$ and $E_1$ of phase damping with a real parameter $\lambda$ are the operators on $H$ $$E_0 = P_0 + \sqrt{1-\lambda}\,P_1,$$(1) $$E_1 = \sqrt{\lambda}\,P_1.$$(2) Amplitude damping of a qubit with damping probability $\gamma$, a real number between $0$ and $1$, is a channel whose first operation element is $P_0 + \sqrt{1-\gamma}\,P_1$ on $H$.

The phase damping channel $\mathcal{E}$ with parameter $\lambda$ is the map on operators on $H$ given by $$\mathcal{E}(\rho) = E_0\,\rho\,E_0^\dagger + E_1\,\rho\,E_1^\dagger.$$(3)^✦

The Circuit

The generator $G$ of rotations about the $y$ axis is the operator $|1\rangle\langle 0| - |0\rangle\langle 1|$ on $H$.^✦ The rotation $R_y(\theta)$ through the angle $\theta$ about the $y$ axis is the operator on $H$ given by the following, where $I$ is the identity operator on $H$, $$R_y(\theta) = \cos\tfrac{\theta}{2}\, I + \sin\tfrac{\theta}{2}\, G.$$(4)^✦ With the Pauli operator $Y = -i|0\rangle\langle 1| + i|1\rangle\langle 0|$ on $H$ one has $G = -iY$, so $R_y(\theta)$ is $e^{-i\theta Y/2}$.

The controlled rotation $\mathrm{C}R_y(\theta)$, with the system as control and the environment as target, is the operator on $H \otimes H$ $$\mathrm{C}R_y(\theta) = P_0 \otimes I + P_1 \otimes R_y(\theta).$$(5)^✦

The dilation $V$ of the circuit with angle $\theta$ is the linear map from $H$ to $H \otimes H$ that prepares the environment in $|0\rangle$ and applies the controlled rotation: for every $|\psi\rangle$ in $H$, $$V|\psi\rangle = \mathrm{C}R_y(\theta)\,\big(|\psi\rangle \otimes |0\rangle\big).$$(6)^✦ On the basis it sends $|0\rangle$ to $|0\rangle \otimes |0\rangle$ and $|1\rangle$ to $|1\rangle \otimes \big(\cos\tfrac{\theta}{2}|0\rangle + \sin\tfrac{\theta}{2}|1\rangle\big)$.

The circuit operation $\mathcal{E}_\theta$ is the map on operators on $H$ that conjugates by $V$ and traces out the environment: $$\mathcal{E}_\theta(\rho) = \operatorname{tr}_E\big(V\rho V^\dagger\big).$$(7)^✦

The Claim

What Is Checked

The claim is stated over the phase damping channel (3), with its elements (1) and (2), and over the circuit operation (7), with the dilation (6). These rest on the projectors $P_0$ and $P_1$, on the rotation (4) with its generator $G$, and on the controlled rotation (5).