Maximal coin-position entanglement generation in a quantum walk for the third step and beyond regardless of the initial state

We study maximal coin-position entanglement generation via a discrete-time quantum walk, in which the coin operation is randomly selected from one of two coin operators set at each step. We solve maximal entanglement generation as an optimization problem with quantum process fidelity as the cost function. Then we determine the maximal entanglement that can be rigorously generated for any step beyond the second regardless of initial conditions with appropriate coin sequences. The simplest coin sequence comprising Hadamard and identity operations is equivalent to the generalized elephant quantum walk, which exhibits an increasingly faster spreading in terms of probability distribution. Experimentally, we demonstrate a ten-step quantum walk driven by such coin sequences with linear optics and thereby show the desired high-dimensional bipartite entanglement as well as the transport behavior of faster spreading.

Figure 1

(a) The geometric representation of the optimal coin sequence ${\mathit{C}}_T$ that can generate maximal entanglement irrelevant of initial state ${|\theta ,\varphi \rangle }_{\text{c}}^{\text{in}}$. (b) Results of optimization of $1-{\mathcal{F}}_{{\mathit{C}}_T}$ with ${\widehat{C}}_t\in \{\widehat{C}\left({\gamma }_0\right),\widehat{C}\left({\gamma }_1\right)\}$ at $T=5, T=10$, and $T=20$. The values of ${\gamma }_{(0,1)}$ are taken from $0^{\circ }$ to ${90}^{\circ }$ with an interval of $1^{\circ }$. (c) The maximal ${\mathcal{F}}_{{\mathit{C}}_T}$ in a QW from $T=1$ to $T=20$ with coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$ (blue up-pointing triangle), $\{\widehat{H},{\widehat{\sigma }}_z\}$ (green down-pointing triangle), $\{\widehat{H},{\widehat{\sigma }}_x\}$ (purple circle), $\{\widehat{H},\widehat{F}\}$ (red square), and $\left\{\widehat{H}\right\}$ (yellow diamond). (d) Fifty $\mathit{b}$'s that can achieve ${\mathcal{F}}_{{\mathit{C}}_T}=1$ at $T=20$ with coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$, where 0 represents $\widehat{H}$ and 1 represents $\widehat{\mathbb{1}}\}$.

Figure 2

(a) Detailed sketch of the setup to realize the ten-step DTQW. (b) Coin operations realized in experiment. A half-wave plate (HWP) set at $22.5^{\circ }$ corresponds to operation $\widehat{H}$ and a quarter-wave plate (QWP) set at ${45}^{\circ }$ corresponds to operation $\widehat{F}$. No wave plate needs to be arranged if the operation is $\widehat{\mathbb{1}}$. (c) Symbols used in panels (a) and (b): periodically poled potassium titanyl phosphate, PPKTP; polarization beam splitter, PBS; half-wave plate, HWP; quarter-wave plate, QWP; beam displacer, BD; and single photon detector, SPD.

Figure 3

(a) Experimental results of ${\mathcal{F}}_{{\mathit{C}}_T}$ with coin sets $\{\widehat{H},\widehat{\mathbb{1}}\}$ (blue up-pointing triangles), $\{\widehat{H},\widehat{F}\}$ (red squares), and $\left\{\widehat{H}\right\}$ (yellow diamonds) at $T=2$ to $T=10$. (b) Average entanglement $\langle {\mathcal{S}}_{\text{E}}\rangle$ over 296 initial coin states with the reconstructed ${\chi }_{{\mathit{C}}_T}^{\text{exp}}$. (c) Geometric representation of reconstructed ${\chi }_{{\mathit{C}}_T}^{\text{exp}}$ with the coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$ at $T=4$, 6, 8, and 10. (d) Geometric representation of reconstructed ${\chi }_{{\mathit{C}}_T}^{\text{exp}}$ with the coin set $\{\widehat{H},\widehat{F}\}$ at $T=4$, 6, 8, and 10.

Figure 4

Measured $\mathcal{P}(x,T)$ with initial coin states (a) $|H\rangle$ and (c) $|L\rangle$ driven by the coin sequences with coin sets $\{\widehat{H},\widehat{\mathbb{1}}\}, \{\widehat{H},\widehat{F}\}$, and $\left\{\widehat{H}\right\}$, respectively. The calculated ${\mathcal{S}}_{\text{S}}\left(T\right)$ with measured $\mathcal{P}(x,T)$ of input states (b) $|H\rangle$ and (d) $|L\rangle$, where the results of coin sets $\{\widehat{H},\widehat{\mathbb{1}}\}, \{\widehat{H},\widehat{F}\}$, and $\left\{\widehat{H}\right\}$ are shown with blue up-pointing triangles, red squares, and yellow diamonds, respectively.

Figure 5

(a) The experimental results of the second moment $m\left(t\right)$ in a ten-step QW. The blue up-pointing triangles represent the results with the coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$. The red squares and yellow diamonds represent the results with the coin sets $\{\widehat{H},\widehat{F}\}$ and $\left\{\widehat{H}\right\}$, respectively. The black line is $m\left(t\right)=t$. (b) The simulated 500-step QW with the coin sets $\{\widehat{H},\widehat{\mathbb{1}}\}$ and $\left\{\widehat{H}\right\}$. For QW with the coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$, the coin operation is randomly selected from the coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$ at each step. The average of $m\left(t\right)$ (blue line) is calculated on a sample of 1000 different ${\mathit{C}}_T$ values, and the blue shade corresponds to the standard deviation of $m\left(t\right)$. The insert is $m\left(t\right)$ from $t=1$ to $t=20$.

Figure 6

(a) Geometric representation of the reconstructed ${\chi }_{{\mathit{C}}_T}^{\text{exp}}$ with the coin set $\{\widehat{H},\widehat{\mathbb{1}}\}$ at $T=3$, 5, 7, and 9. (b) Geometric representation of the reconstructed ${\chi }_{{\mathit{C}}_T}^{\text{exp}}$ with the coin set $\{\widehat{H},\widehat{F}\}$ at $T=3$, 5, 7, and 9.