The Exercise

A quantum operation $\mathcal{E}$ on a qubit is unital if it maps the identity operator to itself, $\mathcal{E}(I) = I$. A unital operation leaves the maximally mixed state $I/2$ where it is. The exercise asks to show that two of the standard noise channels on a qubit, the depolarizing channel and the phase damping channel, are unital, and that the amplitude damping channel is not.

A Qubit

The library states the three facts together, for a qubit with a fixed computational basis. Let $H$ be the state space of the qubit and $|0\rangle$, $|1\rangle$ its computational basis, with $|0\rangle$ the ground state and $|1\rangle$ the excited state. For $i$ in $\{0,1\}$, the projector $P_i$ onto the basis vector $|i\rangle$ is the rank one operator $|i\rangle\langle i|$ on $H$.^✦ The lowering operator $\sigma_-$ of the qubit is the operator $|0\rangle\langle 1|$ on $H$.^✦

Three Noise Channels

The amplitude damping channel $\mathcal{E}_{AD}$ with damping probability $\gamma$ is the quantum operation that sends a linear operator $\rho$ on $H$ to $$\mathcal{E}_{AD}(\rho) = E_0 \rho E_0^\dagger + E_1 \rho E_1^\dagger ,$$(1) where the dagger denotes the adjoint and the operation elements $E_0$ and $E_1$ are the operators on $H$ $$E_0 = P_0 + \sqrt{1-\gamma}\, P_1 , \qquad E_1 = \sqrt{\gamma}\, \sigma_- .$$(2)^✦ The element $E_1$ describes the loss of one quantum of energy.

The phase damping channel $\mathcal{E}_{PD}$ with parameter $\lambda$ is the quantum operation that sends a linear operator $\rho$ on $H$ to $$\mathcal{E}_{PD}(\rho) = F_0 \rho F_0^\dagger + F_1 \rho F_1^\dagger , \qquad F_0 = P_0 + \sqrt{1-\lambda}\, P_1 , \quad F_1 = \sqrt{\lambda}\, P_1 ,$$(3)^✦ with operation elements $F_0$ and $F_1$.

The Pauli matrices $X$, $Y$ and $Z$ are the two by two complex matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} , \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .$$(4) The depolarizing channel $\mathcal{E}_{\mathrm{dep}}$ with parameter $p$ is the map on two by two complex matrices $\rho$ given by $$\mathcal{E}_{\mathrm{dep}}(\rho) = \Bigl(1 - \frac{3p}{4}\Bigr) \rho + \frac{p}{4} \bigl( X \rho X + Y \rho Y + Z \rho Z \bigr) .$$(5)^✦

The Claim

What Is Checked

The claim is stated over the three channels (1), (3) and (5).