@@ -38,7 +38,7 @@ Bell derived a discriminator through \textit{Bell's inequality}, which was only

Bell concludes his proof saying that "there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant" \cite{bell1964}.

A more general form called the \textit{CHSH} inequality named after J. Clauser, M. Horne, A. Shimony and R. Holt is more general and can be used to perform a Bell test in this simulation \cite{clauser1969}. However, this goes beyond the scope of this report and has indeed been performed in \cite{michielsen2013}. De Raedt et al found in their simulations that for short enough \textit{time windows}$W$ the CHSH inequality is indeed violated. Thus, in our model in section \ref{sec:model} we decide to explain and model this time window $W$\cite{deraedt2012}.

A more general form called the \textit{CHSH} inequality named after J. Clauser, M. Horne, A. Shimony and R. Holt and can be used to perform a Bell test in this simulation \cite{clauser1969}. However, this goes beyond the scope of this report and has indeed been performed in \cite{michielsen2013}. De Raedt et al. found in their simulations that for short enough \textit{time windows}$W$ the CHSH inequality is indeed violated. Thus, in our model in section \ref{sec:model} we decide to explain and model this time window $W$\cite{deraedt2012}.

\section{Model and method}

\label{sec:model}

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@@ -96,17 +96,17 @@ We separate the simulation conceptually into three parts: (1) photon pair genera

The following explanation follows the path of an exemplary photon through the entire experimental process. For each physical state, evolution and measurement the associated equivalent in the computation model is described.

Each photon's state is sufficiently described by only the orientation of its linear polarization $\vartheta$, which we represent as a real number.\footnote{One real number is sufficient here, since we are only concerned about linear polarization.} Consequently, two maximally entangled photons will have a difference of $\Delta\vartheta=\frac{\pi}{2}$ in their orientation, corresponding to two orthogonal polarization states. For example, if one photon is initialized with horizontal polarization ($\vartheta=0$), then the other will have vertical polarization ($\vartheta=\frac{\pi}{2}$). Since, apart from their relative entanglement, the polarization angle of each photon is unknown, these values are initialized using the random variable $x_0$, which is generated once for each photon pair.

Each photon's state is sufficiently described by the orientation of its linear polarization $\vartheta$, which we represent as a real number.\footnote{One real number is sufficient here, since we are only concerned about linear polarization.} Consequently, two maximally entangled photons will have a difference of $\Delta\vartheta=\frac{\pi}{2}$ in their orientation, corresponding to two orthogonal polarization states. For example, if one photon is initialized with horizontal polarization ($\vartheta=0$), then the other will have vertical polarization ($\vartheta=\frac{\pi}{2}$). Since, apart from their relative entanglement, the polarization angle of each photon is unknown, these values are initialized using the random variable $x_0$, which is generated once for each photon pair.

In the same way the orientation of the half-wave plates inside the observations stations is modelled with real numbers. In each given simulation run, they will be configured with a given relative angle of $\Delta\varphi$. In practice, these settings are generally chosen to be $0, 45^\circ, 22.5^\circ$ and $67.5^\circ$ respectively — the "Bell test angles" — these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality. However, to get a smoother curve for the $E_{12}(\Delta\varphi)$ dependence that we are interested in, we choose 32 settings of different relative angles between the half-wave plates in the range [0, $2\pi$).

A projective measurement is conducted in each OS by determining the eigenvalue of the respective photon using

A projective measurement is conducted in each OS by determining the measurement outcome of the respective photon using

which yields the result $x_i =+1$ for $r_x =0$ whenever the angular offset $< 45^\circ$ as is coined by the \textit{Malus rule}. The pseudo-random number $r_x$ is required to capture the statistical nature of quantum mechanical measurements. The number is generated for each step of the simulation individually. While the photon's state itself is initialized randomly, the measurement outcome still depends on chance. That is, unless the measurement takes place in an eigenbasis where polarizer and polarization are perfectly aligned or anti-aligned and consequently, $x_i$ unambiguously yields +1 or -1 respectively \cite{deraedt2012, michielsen2013}.

which yields the result $x_i =+1$ for $r_x =0$ whenever the angular offset $< 45^\circ$ as is coined by the \textit{Malus rule}. The pseudo-random number $r_x$ is required to capture the statistical nature of quantum mechanical measurements. It is generated for each step of the simulation individually. While the photon's state itself is initialized randomly, the measurement outcome still depends on chance. That is, unless the measurement takes place in an eigenbasis where polarizer and polarization are perfectly aligned or anti-aligned and consequently, $x_i$ unambiguously yields +1 or -1 respectively \cite{deraedt2012, michielsen2013}.

In addition to the outcome $x_i$, each OS determines a relative\footnote{relative to the generation of the photon} timestamp of the measurement, which depends on the random variable $r_t$, the polarization angle of the photon $\vartheta_i$ and is fixed to the range [0, $T_0$). In particular, it is given by equation (\ref{eq:timestamp}). The dependence on the photon's polarization is caused by \textit{retardation properties} due to the different propagation velocities of electromagnetic waves along different axes of the half-wave plate lattice \cite{michielsen2013}.

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@@ -115,7 +115,7 @@ In addition to the outcome $x_i$, each OS determines a relative\footnote{relativ

t_i = T_0 r_t \sin^4(2 (\varphi_i - \vartheta_i))

\end{equation}

The generated timestamps are used for the simulation of coincident measurements. In reality, the experimenter has no direct information about which photon belongs to which photon pair. Therefore, the coincidence condition is used to match two photons of the same pair during data analysis and discard events not involved in a clear coincidence. Since this coincidence measurement is not immune to errors in reality -- e.g. two actually correlated photon might not trigger the coincidence -- it is sensible to incorporate this behavior in our simulation. Specifically, if the absolute difference of $t_1$ and $t_2$ is larger than the pre-defined coincidence window of $W =1$, then the current measurement is discarded. To analyze the effects of considering coincidence, we perform the simulation once with the coincidence requirement and once without it.

The generated timestamps are used for the simulation of coincident measurements. In reality, the experimenter has no direct information about which photon belongs to which photon pair. Therefore, the coincidence condition is used to match two photons of the same pair during data analysis and discard events not associated with a clear coincidence. Since this coincidence measurement is not immune to errors in reality -- e.g. two actually correlated photons might not trigger the coincidence -- it is sensible to incorporate this behavior in our simulation, as was already motivated in section \ref{sec:background}. Specifically, if the absolute difference of $t_1$ and $t_2$ is larger than the pre-defined coincidence window of $W =1$, then the current measurement is discarded. To analyze the effects of considering coincidence, we perform the simulation once with the coincidence requirement and once without it.

\subsection{Evaluation}

\label{sec:evaluation}

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@@ -168,7 +168,7 @@ The second implementation is voluntary work leveraging various Numpy features to

\label{fig:correlation}

\end{figure}

The simulated measurement results as described in section \ref{sec:evaluation} are plotted in figures \ref{fig:correlation} and \ref{fig:expectation_value} for both finite $W$ and $W \to\infty$ along with their respective theoretical expectation for a quantum mechanical system. It is apparent that the simulation results agree with the quantum mechanical predictions as long as the model includes a finite time window $W$. The individual expectation values in Fig. \ref{fig:expectation_value} are equal to zero, since the initial polarization of each photon is completely random and therefore equally likely to result in a measurement outcome of +1 or -1.

The simulated measurement results as described in section \ref{sec:evaluation} are plotted in figures \ref{fig:correlation} and \ref{fig:expectation_value} for both finite $W$ and $W \to\infty$ along with their respective theoretical expectation for a quantum mechanical system. It is apparent that the simulation results at least qualitatively agree with the quantum mechanical predictions as long as the model includes a finite time window $W$. The individual expectation values in Fig. \ref{fig:expectation_value} are equal to zero, since the initial polarization of each photon is completely random and therefore equally likely to result in a measurement outcome of +1 or -1.

\begin{figure}[h!]

\centering

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@@ -181,10 +181,10 @@ 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 $45^\circ$ offset to the detectors of the observation stations. As a result, all measurement outcomes are equally likely for both detectors 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 coincidence 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.

The difference between the orange datasets which contains all events and the blue datasets 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 detectors 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 coincidence 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}.

\section{Discussion}

We presented the correlation function $E_{12}(\varphi)$ in Fig. \ref{fig:correlation} and the product of the single-particle expectation values $E_1(\varphi)E_2(\varphi)$ in Fig. \ref{fig:expectation_value}. For sufficiently small time windows $W$, the simulation results violate the Bell-CHSH-inequality, which means that non-locality and equivalently entanglement is an inherent requirement of the EPRB theory. This in itself does not verify the validity of quantum theory. However, in this case, the results are indeed compatible with the predictions made by quantum mechanics. Our results indicate that there is no fundamental reason as to why precision experiments do not yield the precise correspondence to the predictions that quantum theories are so famous for.

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We presented the correlation function $E_{12}(\varphi)$ in Fig. \ref{fig:correlation} and the product of the single-particle expectation values $E_1(\varphi)E_2(\varphi)$ in Fig. \ref{fig:expectation_value}. For sufficiently small time windows $W$, the simulation results violate the Bell-CHSH-inequality, which means that non-locality and equivalently entanglement is an inherent requirement of the EPRB theory. This in itself does not verify the validity of quantum theory. However, in this case, the results are indeed compatible with the predictions made by quantum mechanics. Our results indicate that there is no fundamental reason as to why precision experiments do not yield the precise correspondence to the predictions that quantum theories are so famous for.