Figure 2

(a) We plot the state fidelity \(\pmb{\max _{s}\max_{k}{\mathcal {F}}(N)}\) evaluated over the target state \(\pmb{( \vert {-N+k} \rangle_{p} +s \vert {N-k} \rangle_{p})/\sqrt{2}}\) (with \(\pmb{s=\pm1}\), \(\pmb{k=0,1,2,3}\)) and the position state of the walker after \(\pmb{N=10}\) (magenta, top curve), \(\pmb{N=30}\) (green, second curve), \(\pmb{N=50}\) (blue, third curve) and \(\pmb{N=100}\) (red, bottom curve) steps, plotted against the coin parameter θ. The initial monotonically decreasing trait of each curve corresponds to the fidelity with the state with \(k=0\). The first ripple is for \(k=1\), and so on. (b) Same as panel (a) but for the conditional state of the walker achieved by projecting the coin onto its initial state. Despite the similarity between the curves shown in panel (a) and (b), they are not related by a scaling factor: the ratio between one of the curves in panel (b) and the corresponding one in panel (a) is a non-constant function of both N and θ.