 Solution

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

Taking the outer product for A and averaging the two orthogonal projections for B gives

ρ_{A} = \frac{1}{2} (1111) = \frac{I + σ _{x}}{2}, ρ_{B} = \frac{1}{2} (1001) = \frac{I}{2} .

The off-diagonal entries in

ρ_{A}

retain quantum coherence between the

z

-basis states. They are absent from B's classical quantum state ensemble. In particular,

Tr ρ_{A}^{2} = 1

and

Tr ρ_{B}^{2} = 1/2

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