The Exercise

A qubit that loses energy to its environment, such as an atom decaying from its excited state $|1\rangle$ to its ground state $|0\rangle$, undergoes amplitude damping. The channel $\mathcal{E}_{AD}$ that describes it has one parameter $\gamma$ between $0$ and $1$, the probability that an excited qubit decays. The exercise is to write the qubit’s state as a $2 \times 2$ matrix in the computational basis and to compute, entry by entry, the matrix of its image under $\mathcal{E}_{AD}$.

Amplitude Damping

Let $H$ be the state space of a qubit and $|0\rangle$, $|1\rangle$ its computational basis. For $i$ in $\{0,1\}$, the projector $P_i$ onto $|i\rangle$ is the operator $|i\rangle\langle i|$ on $H$.^✦ The lowering operator $\sigma_-$ on $H$ is the operator $|0\rangle\langle 1|$, which takes $|1\rangle$ to $|0\rangle$ and $|0\rangle$ to zero.^✦

The operation elements $E_0$ and $E_1$ of amplitude damping with parameter $\gamma$ are the operators on $H$ $$E_0 = P_0 + \sqrt{1-\gamma}\,P_1, \qquad E_1 = \sqrt{\gamma}\,\sigma_- .$$(1) The amplitude damping channel $\mathcal{E}_{AD}$ with parameter $\gamma$ is the map that takes an operator $\rho$ on $H$ to $$\mathcal{E}_{AD}(\rho) = E_0\, \rho\, E_0^\dagger + E_1\, \rho\, E_1^\dagger .$$(2)^✦

The Claim

For an operator $\rho$ on $H$, write $\rho_{ij} = \langle i|\rho|j\rangle$ for its entries in the computational basis. The claim is made for every operator of unit trace, $\rho_{00} + \rho_{11} = 1$, a density matrix among them.

What Is Checked

The claim is stated over the amplitude damping channel $\mathcal{E}_{AD}$ of (2), its operation elements $E_0$ and $E_1$ of (1), made of the projectors $P_0$, $P_1$ and the lowering operator $\sigma_-$, and the entries of an operator in the computational basis.