 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2015/5/a/i/solution

The normalized input density operator is

ρ = (∣ α ∣^{2} α^{*} β α β^{*} ∣ β ∣^{2}) .

P_{j} = ∣ j ⟩ ⟨ j ∣

. Discarding its recorded outcome gives the dephasing channel

ρ^{'} = P_{0} ρ P_{0} + P_{1} ρ P_{1} = ∣ α ∣^{2} ∣0 ⟩ ⟨ 0∣ + ∣ β ∣^{2} ∣1 ⟩ ⟨ 1∣.

The original off-diagonal coherence disappears. If the outcome is retained instead, the conditional state is

∣0 ⟩

∣1 ⟩

and the record carries information about which occurred. Thus discarding the record is an essential part of this channel specification.

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