@@ -183,7 +183,7 @@ The correlation, however, depends on the relative angle between the half-wave pl

\cite{deraedt2012} goes further than this and demonstrates how for sufficiently small time windows $W$ Bell's inequality is violated and Einstein's principle of locality cannot hold as was thoroughly explained in section \ref{sec:background}.

The difference between the Fig. \ref{fig:expectation_value}, which contains all events and Fig. \ref{fig:correlation}, which contains only coincident events can be explained as follows: Without any coincidence restrictions, all combinations of $\Delta\vartheta$ and $\Delta\varphi$ are equally likely to occur. This includes cases, where the photon polarization has a 45\,° offset to the detectors of the observation stations. As a result, all measurement outcomes are equally likely for both detector individually, independent of the entanglement of the photons. Therefore, no correlation is observed for all such cases. Considering all possible cases (some where the correlation is maximal, some where there is no correlation and everything in between), this averages out to a total correlation of $\pm0.5\cdot |\braket{x_1 x_2}_{max}|$, where $|\braket{x_1 x_2}_{max}|$ is the maximal absolute observable correlation between $x_1$ and $x_2$. On the other hand, when we only consider coincident events, most of the cases where photon polarization and detection axes are aligned inconveniently (as described before) are already filtered out by the coincident check. This is because for $|\varphi_i -\vartheta_i| \approx90\,°$, the sine in (\ref{eq:timestamp}) gets maximal, which leads to a higher probability, of $|t_1- t_2|$ being larger than the coincidence window $W$. This effect is quite strong, since $T_0\gg W$. As a result, most of the events, where no correlation can be inferred from, are discarded and most of the residual events show near maximal correlation. Hence the maximal correlation of $\braket{x_1 x_2}_{max}=\pm1$ is nearly reached for coincidence measurements.

The difference between the Fig. \ref{fig:expectation_value}, which contains all events and Fig. \ref{fig:correlation}, which contains only coincident events can be explained as follows: Without any coincidence restrictions, all combinations of $\Delta\vartheta$ and $\Delta\varphi$ are equally likely to occur. This includes cases, where the photon polarization has a $45^\circ$ offset to the detectors of the observation stations. As a result, all measurement outcomes are equally likely for both detector individually, independent of the entanglement of the photons. Therefore, no correlation is observed for all such cases. Considering all possible cases (some where the correlation is maximal, some where there is no correlation and everything in between), this averages out to a total correlation of $\pm0.5\cdot |\braket{x_1 x_2}_{max}|$, where $|\braket{x_1 x_2}_{max}|$ is the maximal absolute observable correlation between $x_1$ and $x_2$. On the other hand, when we only consider coincident events, most of the cases where photon polarization and detection axes are aligned inconveniently (as described before) are already filtered out by the coincident check. This is because for $|\varphi_i -\vartheta_i| \approx90^\circ$, the sine in (\ref{eq:timestamp}) gets maximal, which leads to a higher probability, of $|t_1- t_2|$ being larger than the coincidence window $W$. This effect is quite strong, since $T_0\gg W$. As a result, most of the events, where no correlation can be inferred from, are discarded and most of the residual events show near maximal correlation. Hence the maximal correlation of $\braket{x_1 x_2}_{max}=\pm1$ is nearly reached for coincidence measurements.

Based on these findings, we conclude that given the underlying assumptions are correct, this simulation of the EPRB experiment clearly shows that the behavior of photons in the given scenario agrees with quantum theory. This was achieved through an event-based approach, which as a discrete model does not rely on any concept of quantum theory \cite{deraedt2012}.