 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019/4/a/ii/solution

Bob knows the initial amplitudes and the measurement basis but not its result, so he averages the two conditional states:

ρ_{B} = ∣ α ∣^{2} ∣0 ⟩ ⟨ 0∣ + ∣ β ∣^{2} ∣1 ⟩ ⟨ 1∣ = (∣ α ∣^{2} 0 0 ∣ β ∣^{2}) .

The initial matrix had off-diagonal entries

α β^{*}

and

α^{*} β

, which the unread measurement removes. Alice's description is conditioned on an available record; Bob's is the nonselective projective measurement mixture. Averaging Alice's records gives exactly Bob's predictions, and communicating the result lets him make the same conditional update. This is a difference of conditioning information, not two inconsistent physical state assignments.

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