 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/2/e/solution

The original Von Neumann entropy depends on its eigenvalues, with

0 lo g 0

defined by continuity:

\frac{S _{before}}{k _{B}} = - \frac{3}{4} lo g \frac{3}{4} - \frac{1}{4} lo g \frac{1}{4} = 2 lo g 2 - \frac{3}{4} lo g 3 ≃ 0.562335.

1/2, 1/4, 1/4

\frac{S _{after}}{k _{B}} = \frac{3}{2} lo g 2 ≃ 1.039721, \frac{Δ S}{k _{B}} = \frac{3}{4} lo g 3 - \frac{1}{2} lo g 2 ≃ 0.477386 > 0.

dephasing removes coherences in the specified basis, increasing the system's mixedness while preserving those measurement probabilities. It cannot be a closed-system unitary transformation, since the eigenvalues change. The subsystem entropy increase does not by itself assert an increase in the entropy of a closed system plus its environment.

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