Probabilistic quantum phase-space simulation of Bell violations and their dynamical evolution

Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase-space methods, namely, the positive-$P$ representation. In this approach the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. Nevertheless, the representation is exactly equivalent to quantum mechanics. A number of states violating Bell inequalities are sampled, demonstrating that these quantum paradoxes can be treated with probabilistic methods. We treat quantum dynamics by simulating the time evolution of the Bell state formed via parametric down-conversion and discuss multimode generalizations.

Figure 1

Simulated moment-based CHD Bell violation $S_{\mathrm{CHD}}\left(\phi \right)$ as a function of the relative polarizer angle $\phi$ for one photon pair using the positive-$P$ distribution. Green dashed lines show the result of static sampling with $2^{18}$ samples. Blue solid lines show the results of the dynamical simulation with $2^{18}$ trajectories at dimensionless time $\tau =0.1$. For each of the sampled states, the filled region represents the range of the estimated error around the mean of $S_{\mathrm{CHD}}\left(\phi \right)$. The exact quantum mechanical prediction of this value is represented by the gray dotted line.

Figure 2

Simulated moment-based Bell violation $S_{\mathrm{CHD}}\left(\phi \right)$ as a function of the relative polarizer angle $\phi$ for two photon pairs using the positive-$P$ distribution. Green dashed lines show the result of static sampling with $2^{24}$ samples. Blue solid lines show the results of a dynamical simulation with $2^{18}$ trajectories at dimensionless time $\tau =0.1$. For each of the sampled states, the filled region represents the range of the estimated error around the mean of $S_{\mathrm{CHD}}\left(\phi \right)$. The exact quantum mechanical prediction of this value is represented by the gray dotted line.

Figure 3

Evolution of a single probability $P_{++}^{AB}({\theta }^{\prime },{\varphi }^{\prime })$ demonstrating that the sampling error increases after $\tau =0.5$.

Figure 4

Evolution of the violation of the Clauser-Horne Bell-type inequality for the state generated by the parametric down-conversion process (24): Plotted is the ratio $S_{\mathrm{CH}}$, defined in Eq. (62) for the angle choices $\theta =0$, $\varphi =\frac{\pi }{8}$, ${\theta }^{\prime }=\frac{\pi }{4}$, and ${\varphi }^{\prime }=3\frac{\pi }{8}$. Violation of the Bell inequality occurs when $S_{\mathrm{CH}}>1$. The filled region represents the range of the estimated error around the mean of $S_{\mathrm{CH}}$. The horizontal dotted line is the expected value with no high-order components. Here we consider $2^{18}$ samples.

Figure 5

Angular dependence of the simulated Clauser-Horne Bell-type inequality for the state generated by the parametric down-conversion process (24): Plotted is the ratio $S_{\mathrm{CH}}$, defined in Eq. (62) as a function of the relative polarizer angle $\phi =\varphi -\theta ={\varphi }^{\prime }-{\theta }^{\prime }={\theta }^{\prime }-\varphi$ at dimensionless time $\tau =0.1$. The filled region represents the range of the estimated error around the mean of $S_{\mathrm{CH}}$. Here we consider $2^{18}$ samples.

Figure 6

Evolution of the violation of the CHSH Bell inequality for the state generated by the parametric down-conversion process (24): Plotted is the ratio $S_{\mathrm{CHSH}}$, defined in Eq. (67) for the angle choices $\theta =0$, $\varphi =\frac{\pi }{8}$, ${\theta }^{\prime }=\frac{\pi }{4}$, and ${\varphi }^{\prime }=3\frac{\pi }{8}$. Violation of the Bell inequality occurs when $S_{\mathrm{CHSH}}>1$. The blue lines corresponds to the case with postselection or heralding, where we consider the projector operator defined in Eq. (63) and exclude the joint null events from the statistics, while the red lines corresponds to the simulation of the CHSH Bell-type inequality without postselection. In each case, the filled region represents the range of the sampled error. The horizontal axis is the expected value at $\tau =0$.

Figure 7

Angular dependence of the simulated CHSH Bell inequality for the state generated by the parametric down-conversion process (24): Plotted is the ratio $S_{\mathrm{CHSH}}$, defined in Eq. (67) as a function of the relative polarizer angle $\phi =\varphi -\theta ={\varphi }^{\prime }-{\theta }^{\prime }={\theta }^{\prime }-\varphi$ for dimensionless time $\tau =0.1$. In this case we consider the postselection or heralding process. The filled region represents the range of the sampled error.