Quantum walker as a probe for its coin parameter

In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\theta$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space $H_w\left(\theta \right)$ increases with $\theta$ and with time, which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare $H_w\left(\theta \right)$ to the full QFI $H_f\left(\theta \right)$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\theta$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.

Figure 1

Probability distribution of unbounded DTQW in one dimension after 200 time steps for different values of $\theta$. The insets show the corresponding distributions for a DTQW bounded in the region $[-50,50]$. In both cases the initial state of the system has been set to $\frac{1}{\sqrt{2}}(|\uparrow \rangle +|\downarrow \rangle )\otimes |x=0\rangle \equiv |+\rangle \otimes |0\rangle$.

Figure 2

The full QFI $H_f\left(\theta \right)$ as a function of time for different values of $\theta$. The initial state of the system is $|+\rangle \otimes |0\rangle$. The full QFI $H_f\left(\theta \right)$ is the same for bounded and unbounded DTQWs.

Figure 3

The walker's position space QFI $H_w\left(\theta \right)$ as a function of time for different values of $\theta$. Due to interference effects, we see clear differences between the walker's position space QFIs of bounded and unbounded DTQWs. The initial state of the system in both cases is $|+\rangle \otimes |0\rangle$.

Figure 4

Density plot of the degree of interference $\mu$ for bounded and unbounded DTQWs as a function of the time steps and the position. Panels (a) and (b) describe the behavior of $\mu$ for $\theta =\pi /8$: left panel for unbounded and right panel for bounded DTQW, respectively. Similarly, (c) and (d) correspond to the dynamics for $\theta =\pi /4$ and (e) and (f) correspond to $\theta =3\pi /8$. The bounded walker is moving in the interval $[-50,50]$. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.

Figure 5

Ratio of quantum Fisher information in position Hilbert space ${\mathcal{H}}_p$ to quantum Fisher information in complete Hilbert space $\mathcal{H}={\mathcal{H}}_c\otimes {\mathcal{H}}_p$ for unbounded discrete-time quantum walk for 200 time steps and different values of $\theta$. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.

Figure 6

The walker's position space QFI $H_w\left(\theta \right)$ as a function of $\theta$ evaluated after a different number of time steps. The differences between the QFI $H_w\left(\theta \right)$ of unbounded and bounded DTQWs is again due to interference effects in bounded DTQW. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.

Figure 7

QFI in position space ($H_w$) as a function of time steps and $\theta$ for DTQW. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.

Figure 8

The Fisher information $F_x\left(\theta \right)$ (left panels) and $F_{xl}\left(\theta \right)$ (right panels) as a function of time for unbounded DTQW and different values of $\theta$. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$. Both sets of plots are for unbounded DTQW with $F_x\left(\theta \right)$ referring to the information extractable by a full position measurement, whereas $F_{xl}\left(\theta \right)$ quantifies the information that may be gained by measurements with limited access to the position of the walker (see text for details). The insets of the right panels are legends for the region $S$, accessible by the measurement.

Figure 9

The ratio $F_x\left(\theta \right)/H_f\left(\theta \right)$ between the position Fisher information and the full quantum Fisher information as a function of time and for different values of $\theta$. The insets show the ratio $F_x\left(\theta \right)/H_w\left(\theta \right)$ between the position Fisher information and the walker's position space quantum Fisher information. All the plots refer to unbounded DTQW. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.

Figure 10

The standard deviation for split-step quantum walk as function of ${\theta }_1$ for different values of ${\theta }_2$ after 100 steps of walk. Standard deviation is always bounded by the larger $\theta$ parameter. Therefore it only gives information of the evolution parameter with the higher value. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.

Figure 11

Quantum Fisher information in position space with respect to the evolution parameters ${\theta }_2$ (left) and ${\theta }_1$ (right) when other parameters are fixed. Having clearly distinct plots for both parameters after 100 time steps helps us to uniquely probe both parameters independently. The initial state of the walker is $|+\rangle \otimes |x=0\rangle$.