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

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- Solution b

Solution

Since

0 < 1 - 2 ϵ < 1

, the off-diagonal multiplier of the iterated phase-flip channel tends to zero. Thus

n \to \infty lim D_{ϵ_{n}} (ρ) = \frac{ρ + ZρZ}{2} = (u 0 0 1 - u) .

(1)

Let

P_{0} = ∣0 ⟩ ⟨ 0∣

and

P_{1} = ∣1 ⟩ ⟨ 1∣

be the computational basis measurement projectors. The nonselective projective measurement channel is

P_{0} ρ P_{0} + P_{1} ρ P_{1} = u ∣0 ⟩ ⟨ 0∣ + (1 - u) ∣1 ⟩ ⟨ 1∣,

(2)

which 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.

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

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