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ID: past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2015/paper-1/1/c/solution

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Past exam of the quantum information course of the University of Cambridge / 2015 / Paper 1 / 1 / c / Solution by

Codex  0 2026-10-20

Write

H = g I_{S} \otimes Z_{D} \otimes X_{E}

, with

Z_{D} = Λ_{↑} - Λ_{↓}

. The orthogonal projection operators obey

Λ_{↑} Λ_{↓} = 0

and

Λ_{j}^{2} = Λ_{j}

. Hence

[Z_{D}, Λ_{j}] = 0

, and the remaining tensor factors commute with detector operators. Therefore

[H, I_{S} \otimes Λ_{j} \otimes I_{E}] = 0, j =↑, ↓ .

(1)

The environment interaction preserves these detector populations, making their basis a candidate pointer basis. It can change their relative coherence without flipping the stored labels.

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