= Solution
The original <Von Neumann entropy> depends on its <eigenvalues>, with $0\log0$ defined by continuity:
$$
\boxed{\frac{S_{\rm before}}{k_B}
=-\frac34\log\frac34-\frac14\log\frac14
=2\log2-\frac34\log3
\simeq0.562335.}
$$
After <rank-one dephasing>, the <eigenvalues> are the retained diagonal populations $1/2,1/4,1/4$, giving
$$
\boxed{\frac{S_{\rm after}}{k_B}=\frac32\log2\simeq1.039721,\qquad
\frac{\Delta S}{k_B}=\frac34\log3-\frac12\log2\simeq0.477386>0.}
$$
<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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