Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 51 / 4 / b / Solution

Since $0<1-2ϵ<1$, the off-diagonal multiplier of the iterated phase-flip channel tends to zero. ThusLet $P_0=|0⟩⟨0|$ and $P_1=|1⟩⟨1|$ be the computational basis measurement projectors. The nonselective projective measurement channel iswhich is exactly the displayed limit. Its diagonal entries retain the original outcome probabilities; the measurement outcome is ignored and the coherences are erased. The limiting evolution is therefore the completely dephasing channel.