Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 33 / 3 / a

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- Solution a

Solution

Choose

∣0 ⟩

as the ground state and

∣1 ⟩

as the excited state. The relevant quantum channel is the amplitude damping channel. Its Kraus operators are

K_{0} = (10 0 1 - p), K_{1} = (00 p 0) = p ∣0 ⟩ ⟨ 1∣.

They satisfy

K_{0}^{†} K_{0} + K_{1}^{†} K_{1} = I

. The

K_{0}

branch corresponds to no emitted photon: the ground-state amplitude is unchanged and the excited-state amplitude is attenuated. The

K_{1}

branch corresponds to photon emission and transition to the ground state. These are unnormalized branch states; the squared norm or matrix trace of each branch gives its probability.

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