%!TEX root = manypairs_main.tex
\section{Experiment}
\subsection{Experimental setup}
  \begin{figure}
    \includegraphics[width=.75\columnwidth]{exp_setup.eps}
    \caption{Schematic of the experimental set-up.
    Polarization correlations of entangled-photon
    pairs are measured by the polarization analyzers   M$_A$ and M$_B$,   each
    consisting of a half wave plate ($\lambda/2$) followed by a polarization beam
    splitter (PBS).
    All photons are detected by Avalanche photodetectors D$_H$ and
    D$_V$, and registered in a coincidence unit (CU).}
    \label{fig:setup}
  \end{figure}

  In our experiment (see figure~\ref{fig:setup}), the output of a
  grating-stabilized laser diode (LD, central wavelength 405\,nm) passes through a
  single mode optical fiber (SMF) for spatial mode filtering, and is focused to a
  beam waist of 80\,${\mu}$m into a 2\,mm thick BBO crystal cut for type-II phase-
  matching. There, photon pairs are generated via spontaneous parametric down-
  conversion (SPDC) in a slightly non-collinear configuration.  A half-wave plate
  ($\lambda$/2) and a pair of compensation crystals (CC) take care of the temporal
  and transversal walk-off~\cite{kwiat:95}. Two spatial modes ($A$, $B$) of down-
  converted light, defined by the SMFs for 810\,nm, are matched to the pump mode
  to optimize the collection~\cite{kurtsiefer:01}. In type-II SPDC, each down-
  converted pair consists of an ordinary and extraordinarily polarized photon,
  corresponding to horizontal ($H$) and vertical ($V$) in our setup.

  We use
  a pair of
  polarization controllers (PC)
  to minimize the polarization rotation caused by the SMFs to the collected modes.

  To arrive at an approximate singlet Bell
  state, the phase $\phi$ between the two decay possibilities in the polarization
  state \mbox{$\ket{\psi}=1/\sqrt{2}\left(\ket{H}_A\ket{V}_B+e^{i\phi}\ket{V}_A\ket{H}_B
  \right)$} is adjusted to $\phi=\pi$ by tilting the CC.

  In the polarization analyzers (inset of figure~\ref{fig:setup}), photons from SPDC are
  projected onto arbitrary linear polarization by $\lambda$/2 plates, set to half
  of the analyzing angles $\theta_{A(B)}$, and polarization beam splitter
  (extinction ratio 1/2000  and 1/200 respectively for transmitted and reflected
  arm) in each analyzer. Photons are detected by avalanche photo diodes (APD), and
  corresponding detection events from the same pair identified by a coincidence
  unit if they arrive within $\approx\pm$3\,ns of each other.

  The quality of polarization entanglement is tested by probing the polarization
  correlations in a basis complementary to the intrinsic HV basis of the crystal.
  With interference filters (IF) of 5\,nm bandwidth (FWHM) centered at 810\,nm, in
  the $45^\circ$ linear polarization basis we observe a visibility $V_{45}$ =
  98.68$\pm$0.20\%. The visibility in the natural H/V basis of the type-II down-
  conversion process reaches $V_{\rm HV}$ = 99.67$\pm$0.12\%.

  The non-ideal visibility is due to the non-perfect neutralization of the polarization rotation caused by the SM fibers.

\subsection{Measurement and Post-processing}
  In the realization of this proof of principle experiment, we did not intend to
  provide a loophole-free demonstration.
  Due to the limited efficiency of the APD
  detectors, we assume
  that the fraction of the photon we detected is a fair representation of the
  entire ensemble (fair sampling assumption).
  % a fair sampling.
  Similarly, even if Alice and Bob are not
  space-like separated, we assume that no communication happens between the two
  measurements. Moreover, the basis choice is not random, as expected in an ideal
  Bell-like experiment. We instead set the basis and record the number of events
  in a fixed time. We are assuming that the state generated by the source, and all
  the other parameters of the experiment, do not change between experimental runs.

  The basic measurement lasts $60\,$s, during which we record an average of
  $16\times10^3$ \mbox{two-fold} coincidences between detectors at Alice and Bob.
  A
  detection event at the transmitted output of each PBS is associated with 0,
  reflected one with 1. Three- and four-fold coincidences, as well as two-fold
  coincidences between detectors belonging to the same party, correspond to
  multiple pairs of photos generated within the coincidence time window. The rate
  of these events is negligible, therefore we discarded them.

  In order to avoid biases due to the asymmetries in detector efficiencies, to
  measure one basis $(A_j,B_k)$ we also measure three complementary basis:
  $(A_j+45^\circ,B_k)$, $(A_j, B_k+45^\circ)$, and $(A_j+45^\circ,B_k+45^\circ)$.
  A rotation by $45^\circ$ effectively swaps the roles of the transmitted and
  reflected detectors. Each party, when measuring on the rotated basis, needs to
  apply a $NOT$ operation to the measurement outcome.
  We repeat the measurement for a range of $\beta$, and the corresponding four bases defined by Eq.~(\ref{eq:basis}).

  To replicate the many-box scenario, for every set of measurement angles we organize the sequence of results into clusters of size $n$.
  For each cluster we calculate the majority (parity) binning using Eq.~(\ref{eq:majority_binning}) (Eq.~(\ref{eq:parity_binning})).
  Following the procedure described by Eqs.~(\ref{eq:bipartite_corr}) and~(\ref{eq:CHSH_ineq}), we obtain a value of $S_n$ for every $n$ of interest.
  To evaluate the error associated to every $S_n$, the same procedure is repeated~1000 times, shuffling the order of the results every time before the clustering.
