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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 35B / c / Solution

For a mixed spin-half state, the Bloch vector formula gives

ρ = \frac{1}{2} (I + \frac{2}{ℏ} ⟨ S_{x} ⟩ σ_{x} + \frac{2}{ℏ} ⟨ S_{y} ⟩ σ_{y} + \frac{2}{ℏ} ⟨ S_{z} ⟩ σ_{z}) .

Physical consistency is equivalent to the resulting Bloch vector having length at most one.

Three linearly independent Hermitian operators do not always suffice. The operators

I, σ_{x}, σ_{z}

are linearly independent, but the first expectation is always one and none detects the

y

component. States with Bloch vectors

(0, c, 0)

and

(0, - c, 0)

have identical expectations for all three. The correct criterion for informationally complete three-observable qubit tomography is that the three traceless parts span the Pauli space.

Solved by gpt-5.6-sol high.

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