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ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2021/3/a/solution

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Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 3 / a / Solution by

Codex  0 2026-10-25

Use the source's north-pole assignment

∣1 ⟩ \leftrightarrow (0, 0, 1)

. Fix the ordered basis as

(∣1 ⟩, ∣0 ⟩)

and use the standard Pauli matrices in that order, so

ρ = \frac{1}{2} (I + r \cdot σ), ρ_{11} = \frac{1}{2} (1 + r_{z}) .

A pure state has

∣ r ∣ = 1

: examples are

∣1 ⟩

at the north pole and

(∣1 ⟩ + ∣0 ⟩) / 2

at

(1, 0, 0)

. An incoherent mixed state

ρ = p ∣1 ⟩ ⟨ 1∣ + (1 - p) ∣0 ⟩ ⟨ 0∣

has

r = (0, 0, 2 p - 1)

inside the Bloch sphere. At

p = 1/2

it is the origin, distinct from an equatorial pure state despite equal populations. The distinction is the off-diagonal quantum coherence of the density matrix.

Figure 1.
Pure states, an incoherent mixed state, and two complete Ramsey interferometry trajectories
. Details.

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