Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 65 / 2

The $+1$ measurement effect of the Pauli operator ${\sigma }_x$ is $P_{+}=(I+{\sigma }_x)/2$. The Born rule and the Bloch vector representation giveThe other two Bloch vector components do not enter this projective measurement probability.

In the computational basis, writeThe Kraus representation of the amplitude damping channel givesReading off its Bloch vector gives the Bloch-vector map of amplitude damping:Here $0\le p\le 1$. The shift in $s_z$ shows that this quantum channel is not unital unless $p=0$; its fixed ground state lies at the north pole, rather than the center of the Bloch sphere.

Iterating the Bloch-vector map of amplitude damping givesThereforeThe repeated amplitude damping limit describes irreversible relaxation of the excited state $|1⟩$ into the ground state $|0⟩$, with energy transferred to the environment and no compensating thermal excitation. At $p=1$, the ground state is reached in one action. At the endpoint $p=0$, the quantum channel is the identity and the initial Bloch vector remains unchanged; the ground-state conclusion requires nonzero damping.

For Kraus operators $A_j$, the Kraus formula for entanglement fidelity isIndeed each Kraus contribution to the purified-state overlap is $|⟨\Psi |(I\otimes A_j)|\Psi ⟩{|}^2$, and the expectation in a purification is $\mathrm{Tr}(\rho A_j)$. In this case,The entanglement fidelity of amplitude damping is thusFor $p=0$, this is one. For a ground-state input it is one for every $p$, whereas for an excited-state input it is $1-p$. These checks agree with the physical relaxation mechanism.