Past exam of the quantum information course of the University of Cambridge / 2015 / Paper 1 / 1 / c

Write $H=g{I}_S\otimes Z_D\otimes X_E$, with $Z_D={\Lambda }_{\uparrow }-{\Lambda }_{\downarrow }$. The orthogonal projection operators obey ${\Lambda }_{\uparrow }{\Lambda }_{\downarrow }=0$ and ${\Lambda }_j^2={\Lambda }_j$. Hence $[Z_D,{\Lambda }_j]=0$, and the remaining tensor factors commute with detector operators. ThereforeThe environment interaction preserves these detector populations, making their basis a candidate pointer basis. It can change their relative coherence without flipping the stored labels.