The Exercise

A qubit, the principal system, interacts with a second qubit, its environment, through a single controlled NOT gate: the principal qubit is the control and the environment the target. Before the gate the environment is prepared in a mixture of the states $|\pm\rangle = (|0\rangle \pm |1\rangle)/\sqrt{2}$ , and after it the environment is discarded. The exercise is to show that the operation left on the principal qubit is a phase flip channel, and that with the weight of $|-\rangle$ chosen from a damping parameter $\lambda$ it is phase damping with that parameter.

A Qubit and Its Environment

Both qubits have the state space $H$ , with computational basis $|0\rangle, |1\rangle$ , and the pair has the state space $H \otimes H$ , principal system first. For $i$ in $\{0,1\}$ , the projector $P_i$ onto $|i\rangle$ is the rank one operator $|i\rangle\langle i|$ on $H$ .^✦ The bit flip $X$ is the operator $|0\rangle\langle 1| + |1\rangle\langle 0|$ on $H$ , which exchanges the two basis states, and the Pauli operator $Z$ on $H$ is $P_0 - P_1$ . The projectors onto $|+\rangle$ and $|-\rangle$ are the operators $|+\rangle\langle +| = \tfrac{1}{2}\,(I + X), \qquad |-\rangle\langle -| = \tfrac{1}{2}\,(I - X)$ (1) on $H$ , with $I$ the identity.

The controlled flip $U$ is the operator on $H \otimes H$ $U = P_0 \otimes I + P_1 \otimes X,$ (2)^✦ the controlled NOT in which the principal qubit controls a bit flip of the environment.

For a real number $p$ , the environment state $\sigma_p$ is the operator on $H$ $\sigma_p = (1-p)\,|+\rangle\langle +| + p\,|-\rangle\langle -|;$ (3)^✦ for $0 \le p \le 1$ it is a state.

The Circuit and Phase Damping

For a real number $p$ , the circuit model $\mathcal{E}_p$ of phase damping is the map that takes an operator $\rho$ on $H$ to the operator on $H$ obtained by preparing the environment in $\sigma_p$ , applying $U$ and tracing the environment out: $\mathcal{E}_p(\rho) = \operatorname{tr}_E\!\left[\, U\,(\rho \otimes \sigma_p)\,U^\dagger \,\right],$ (4)^✦ where $\operatorname{tr}_E$ is the partial trace over the second factor.

The phase flip channel $\mathcal{F}_p$ with flip probability $p$ is the map on operators on $H$ $\mathcal{F}_p(\rho) = (1-p)\,\rho + p\,Z \rho Z.$ (5)^✦

For a phase damping parameter $\lambda$ , the no flip probability $\alpha$ is the real number $\alpha = \tfrac{1}{2}\left(1 + \sqrt{1-\lambda}\,\right),$ (6) and the operation elements $\tilde E_0, \tilde E_1$ of phase damping with parameter $\lambda$ are the two operators on $H$ $\tilde E_0 = \sqrt{\alpha}\; I, \qquad \tilde E_1 = \sqrt{1-\alpha}\; Z.$ (7)

The Claim

Theorem 1. Let $p$ be a real number. For every operator $\rho$ on $H$ , the circuit model with the environment in $\sigma_p$ acts as the phase flip channel with flip probability $p$ : $\mathcal{E}_p(\rho) = \mathcal{F}_p(\rho).$ (8)^✦

Theorem 2. Let $\lambda \ge 0$ be a phase damping parameter and $\alpha$ its no flip probability. For every operator $\rho$ on $H$ , the circuit model with the environment in $\sigma_{1-\alpha}$ is phase damping with parameter $\lambda$ , in the operation elements $\tilde E_k$ : $\mathcal{E}_{1-\alpha}(\rho) = \sum_{k=0}^{1} \tilde E_k\, \rho\, \tilde E_k^\dagger.$ (9)^✦

What Is Checked

Both theorems are stated over the constructions above: on one side the controlled flip $U$ and the environment state $\sigma_p$ , which together give the circuit model $\mathcal{E}_p$ ; on the other the phase flip channel $\mathcal{F}_p$ and the operation elements $\tilde E_k$ with their no flip probability $\alpha$ .