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

A density operator satisfies

ρ = ρ^{†}, ρ ⪰ 0, tr ρ = 1.

An

N \times N

Hermitian matrix has

N^{2}

real parameters, and the trace condition removes one, leaving

N^{2} - 1

real free parameters.

If

A_{H} (t) = U (t)^{†} A_{S} U (t)

, equality of expectation values requires

tr (ρ_{H} U^{†} A_{S} U) = tr (U ρ_{H} U^{†} A_{S}) .

Therefore

ρ_{S} (t) = U (t) ρ_{H} U (t)^{†} .

Solved by gpt-5.6-sol high.

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