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

For nonnegative integers $n$, assume the storage channel over successive intervals of length $\tau$ composes as $D_{ϵ}^n$. This is the memoryless interpretation of the specified storage law. In the computational basis, write the density operator asConjugation by the Pauli Z gate reverses the off-diagonal entries and leaves the diagonal entries unchanged, soAfter $n$ applications of the phase-flip channel, the multiplier is $(1-2ϵ{)}^n$. Comparing with the multiplier $1-2{ϵ}_n$ of a single phase-flip channel givesFor $n=0$ this gives the identity channel. Equivalently, composing two consecutive phase errors cancels them because $Z^2=I$, so their effective probability obeys ${ϵ}_{n+1}=ϵ+(1-2ϵ){ϵ}_n$. The same expression solves this recurrence. The iterated phase-flip channel formula presumes this composition law; specifying a channel at one time alone would not constrain a memory-bearing environment at later times.