Nonlocality, quantum correlations, and violations of classical realism in the dynamics of two noninteracting quantum walkers

That quantum correlations can be generated over time between the spin and the position of a quantum walker is indisputable. The creation of bipartite entanglement has also been reported for two-walker systems. In this scenario, however, since the global state lies in a fourpartite Hilbert space, the question arises as to whether genuine multipartite entanglement may develop in time. Also, since the spatial degrees of freedom can be viewed as a noisy channel for the two-spin part, one may wonder how other nonclassical aspects, such as Bell nonlocality, Einstein-Podolsky-Rosen steering, quantum discord, and symmetrical quantum discord, evolve in time during the walk. The lack of analytical and numerical evidences which would allow one to address these questions is possibly due to the usual computational difficulties associated with the recursive nature of quantum walks. Here, we work around this issue by introducing a simplified Gaussian model which proves to be very accurate within a given domain and powerful for analytical studies. Then, for an instance involving two noninteracting quantum walkers, whose spins start in the singlet state, we quantify the aforementioned nonclassical features as a function of time, and evaluate violations of both realism and related aspects of locality. In addition, we analyze situations in which the initial two-spin state is affected by white noise. The typical scenario found is such that while genuine fourpartite entanglement increases over time, all the investigated nonclassical features vanish (suddenly or asymptotically) except realism-based nonlocality. Moreover, realism is prevented for all finite times. Our findings open perspectives for the understanding of the dynamics of quantum resources in quantum walks.

Figure 1

Numerical probability distributions $|\langle x|{\psi }_t{\rangle |}^2$ of quantum walks with $\alpha =3\pi /4$ [see Eq. (1)] for local (${\sigma }_0=0.2$) and broad (${\sigma }_0=5$) Gaussian states, at $t=100$, as a function of the dimensionless position $x$. The greater the initial dispersion, the more effective the maintenance of the Gaussian shape over time.

Figure 2

Probability distribution $p_t(x_1,x_2)$ at $t=20$ of two quantum walkers with initial states (a) $|{\mathrm{\Psi }}_0\rangle$ [see Eq. (14)] and (b) $|{\psi }_0^{\uparrow }\rangle |{\psi }_0^{\downarrow }\rangle$ [see Eqs. (10)], both with ${\sigma }_0=5$. A correlated two-spin state is more effective in producing strong spatial anticorrelations.

Figure 3

Probability distributions $p_t\left(x_1\right)$ (black line), $p_t^{\uparrow }\left(x_1\right)$ (dotted red line), and $p_t^{\downarrow }\left(x_1\right)$ (dashed blue line) of finding particle 1 at position $x_1$, at position $x_1$ with spin up, and at position $x_1$ with spin down, respectively, at instants (a) $t=4$ and (b) $t=25$. The initial dispersion is ${\sigma }_0=5$.

Figure 4

(a) Contour plot for the asymptotic irreality ${\mathfrak{I}}_A\left({\rho }_{\infty }^{\epsilon =1}\right)$ of an observable $A={\widehat{v}}_1·\vec{\sigma }$ on ${\mathcal{H}}_{S_1}$, in spherical coordinates, for the no-noise regime $(\epsilon =1)$. The color scale goes from zero (blue) to $ln2$ (red). (b) Scaled irreality ${\tilde{\mathfrak{I}}}_A\left({\rho }_t^{\epsilon =1}\right)$ (cyan curves) as a function of the scaled time $\tau =t/{\sigma }_0$ for 200 randomly chosen measurement directions ${\widehat{v}}_1$. The scaled discord $\tilde{\mathcal{D}}\left({\rho }_t^{\epsilon =1}\right)$ (dashed black curve) defines a tight upper bound [see Eq. (50)]. In the inset, the difference $\mathrm{\Delta }:=\tilde{\mathcal{D}}\left({\rho }_t^{\epsilon =1}\right)-{\tilde{\mathfrak{I}}}_A\left({\rho }_t^{\epsilon =1}\right)$ is plotted for each of the 200 measurement directions, showing results never greater than 0.03.

Figure 5

Behavior of the normalized contextual realism-based nonlocality ${\eta }_{AB}\left({\rho }_t^{\epsilon =1}\right)/ln2$, in the no-noise regime ($\epsilon =1$), as a function of the scaled time $\tau =t/{\sigma }_0$, for observables $A$ and $B$ (contexts) obeying the code that follows. In continuous lines, from top to bottom: $A=B={\sigma }_y$ (black line), $A=B={\sigma }_z$ (red line), and $A=B=({\sigma }_x+{\sigma }_z)/\sqrt{2}$ (blue line); in dashed lines, from top to bottom: $A=({\sigma }_x-{\sigma }_z)/\sqrt{2}$ and $B={\sigma }_y$ (black line), $A={\sigma }_x$ and $B={\sigma }_y$ (red line), $A={\sigma }_z$ and $B={\sigma }_x$ (blue line), and $A=({\sigma }_x+{\sigma }_z)/\sqrt{2}$ and $B={\sigma }_y$ (green line).

Figure 6

All the (observable independent) quantumness quantifiers $\mathcal{Q}$ computed in this work for the two-spin state ${\rho }_t^{\epsilon }$ as a function of the scaled times $\tau =t/{\sigma }_0$ for (a) $\epsilon =1.0$ (left panels) and (b) $\epsilon =0.8$ (right panels). $\mathcal{Q}$ assumes, in upper panels, Bell nonolocality $\mathcal{B}$ (blue lower line), EPR steering $\mathcal{S}$ (red middle line), and entanglement $E$ (green upper line), and, in bottom panels, normalized (symmetrical) quantum discord $\mathcal{D}/ln2$ (black lower line) and normalized realism-based nonlocality $\mathcal{N}/ln2$ (purple upper line). The vertical dashed lines in the upper right panel refer to the sudden-death times given by Eqs. (34), (37), and (40). Quantumness typically decreases with both time and the amount $1-\epsilon$ of noise, but realism-based nonlocality survives.