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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2019/4/d/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 4 / d / Solution by

Codex  0 2026-10-18

The opposite

x

-axis pure states have projectors

(I \pm σ_{x}) /2

. Averaging them gives

ρ_{C} = \frac{1}{2} \frac{I + σ _{x}}{2} + \frac{1}{2} \frac{I - σ _{x}}{2} = \frac{I}{2} = ρ_{B} .

(1)

Therefore C and B cannot be distinguished by any measurement on the atoms alone. Every POVM outcome has the same probability

Tr (Eρ)

for the two preparations; this is the operational equivalence of different quantum state ensembles. Accessible preparation labels or additional correlations would be extra information. C can be distinguished from A by the

x

-axis spin measurement in part (c).

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