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\begin{poster}
\large \sf

%%%%%%%%%%%%%%%%%%%%%
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\begin{center}
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    %%% Title
    \begin{minipage}[c]{0.65\textwidth}
        \begin{center}
            {\sc \Huge Exploring the Limits of Non-Locality
            }\\[9mm]
            {\sc \Huge
            with Pairs of Photons}\\[9mm]
            % {\LARGE Alessandro~Cer\`{e}$^1$,
            %     Hou~Shun~Poh$^1$,
            %     Siddarth~K.~Josh$^{1}$,\\
            %     Marcin~Markiewicz$^{2}$,
            %     Pawe{\l}~Kurzy\'nski$^{1,3}$,\\
            %     Dagomir~Kaszlikowski$^{1,4}$,
            %     Ad\'{a}n~Cabello$^{5}$,
            %     and Christian~Kurtsiefer$^{1,4}$}\\[9mm]
            % { {$^1$Centre for Quantum Technologies, National University of Singapore, Singapore;\\
            %     $^2$Institute of Physics, Jagiellonian University, Krakow, Poland;\\
            %     $^3$Faculty~of~Physics,  Adam~Mickiewicz~University, Pozna\'{n}, Poland;\\
            %     $^4$Department of Physics, National University of Singapore, Singapore;\\
            %     $^5$Departamento de F\'{i}sica Aplicada II, Universidad de Sevilla, Sevilla, Spain.
            %     }}
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\begin{center}
            {\LARGE Alessandro~Cer\`{e}$^1$,
                Hou~Shun~Poh$^1$,
                Siddarth~K.~Josh$^{1}$,
                Marcin~Markiewicz$^{2}$,
                Pawe{\l}~Kurzy\'nski$^{1,3}$,\\
                Dagomir~Kaszlikowski$^{1,4}$,
                Ad\'{a}n~Cabello$^{5}$,
                and Christian~Kurtsiefer$^{1,4}$}\\[9mm]
            { {$^1$Centre for Quantum Technologies, National University of Singapore, Singapore;\\
                $^2$Institute of Physics, Jagiellonian University, Krakow, Poland;\\
                $^3$Faculty~of~Physics,  Adam~Mickiewicz~University, Pozna\'{n}, Poland;\\
                $^4$Department of Physics, National University of Singapore, Singapore;\\
                $^5$Departamento de F\'{i}sica Aplicada II, Universidad de Sevilla, Sevilla, Spain.
                }}

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%\vspace{0.1cm}

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\vspace{0cm}
\begin{multicols}{2}
\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
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{\large
Bell inequalities are a powerful tool in discriminating whether Nature is better described by classical theories or non-local ones.
We
% present our efforts in
experimentally test Bell inequalities and the predictions of standard quantum mechanics using a source of polarization entangled photon pairs.
% , in particular our most recent two works. In the first, we
An attempt to saturate the Tsirelson bound, the well known $2\sqrt{2}$ predicted by standard quantum mechanics,
% . This attempt
allowed us to exclude some of the alternative, non-local theories [1].
% In the second,
We also present an different approach to discriminate between classical and non-classical system based on the computational complexity of the output, as opposed to the statistical nature of the standard Bell tests [2].}

\vspace{1cm}
\begin{center}
  {\bf \huge \textsf {Experimental Setup}}
\end{center}
\vspace{.5cm}

\setcounter{figure}{0}
\begin{figurehere}
     \begin{center}
    %\centering
    %\def\svgwidth{\columnwidth}
    %\input{setup.pdf_tex}
        \includegraphics[width=1\columnwidth]{setup}
            \caption{
            Schematic of the experimental set-up.
            We built a typical polarization-entangled photon pair source in a crossed-ring configuration at 810 nm.
            Careful balancing of the $\ket{HV}$ and
            $\ket{VH}$ contribution and
            adjustment of their relative phase using the compensation crystals (CC)
            % adjusted to generated the polarization state
            yield a very clean singlet state
            \mbox{$\ket{\psi}=1/\sqrt{2}\left(\ket{H}_A\ket{V}_B-\ket{V}_A\ket{H}_B\right)$}.
            We minimize contributions from higher order parametric conversion processes
            by restricting the pump power to $7/,$mW. We observed both in the natural H/V and complementary $\pm$\,45$^\circ$ basis, high visibility of $V=99.9\pm0.1\%$, indicating a high quality of polarization entanglement.
            In the first experiment, polarization analysis is being perform using film polarizers (extinction ration 10$^{-5}$) in front of the collection optics. This eliminates any unwanted polarization state transformations due to fiber neutralization.
            In the second experiment,polarization is projected
            % onto arbitrary linear polarization
            by $\lambda/2$ plates and polarization beam splitter in each analyzer after the single mode fibre (inset of figure 1).
            % For both experiments,
            Photons are detected by APDs, and corresponding detection events from the same pair identified by a coincidence unit if they arrive within $\approx\pm$1.2\,ns of each other.
            % To arrive at a very clean singlet state, we carefully align the photon pair collection to balance the two photon pair contributions $\ket{HV}$ and
            % $\ket{VH}$, and adjust their relative phase with the CC.
            %  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).
            }
    \end{center}
\end{figurehere}

\vspace{1cm}
\begin{center}
  {\bf \huge \textsf {Approaching Tsirelson bound}}
\end{center}

\begin{figurehere}
    \begin{center}
        \includegraphics[width=0.75\columnwidth]{figure3.eps}
        \caption{Selected experimental tests of the CHSH Bell inequality with results close to the Tsirelson (T) and Grinbaum (G) bounds in photonic systems (circles),
        atoms and ions (diamonds),
        Josephson junctions (square),
        and nitrogen-vacancy centers in diamond (triangle).
        See [1] for details.}
    \end{center}
\end{figurehere}
Assuming local realism, Bell inequality (CHSH inequality~[3]) $|S|\le 2$ is not violated.
% Tsirelson [4] showed that,
According to quantum theory, $|S|$ has an upper bound of $2 \sqrt{2} \approx 2.82843$ (Tsirelson bound). Other non-local theories predict different upper bounds.
For example, a model proposed by Grinbaum~[5] predicts a bound of~2.82537(2) (Grinbaum bound), slightly smaller than the Tsirelson bound,
but also consistent with all the available experimental results obtained so far.
The underlying theory
tries to address the cut between the observer and the observed system in quantum mechanics,
supporting the hypothesis that quantum theory is only an effective description of a more fundamental theory.

% For the first experiment, the violation of the CHSH inequality,
In this experiment, we collected for 312 times a complete set of 16 measurements needed to evaluate the $S$ parameter, for a total of~33,184,329 pair events.
% We collect coincidence events for each of the 16 settings required to evaluate $S$ for 1 minute, and then repeat again the whole set. Within 312 such complete sets, we registered a total of 33,184,329 pair events.As a result, we obtain in this experiment,
We obtained a final
a value of $S=2.82759\,\pm\,0.00051$, or a separation of $2\sqrt{2}-S=0.00084\pm0.00051$ from the Tsirelson bound.
This value violets Grinbaum bound, thus refuting his theory.
% This violation of the Grinbaum bound suggests that quantum mechanics is not just an effective description of a more fundamental theory but a fundamental theory in itself.

\vspace{1cm}
\begin{center}
  {\bf \huge \textsf {Quantum test based on compression software}}
\end{center}\vspace{1cm}
Standard Bell tests are based on statistical inference of correlations between measurements.
We adopt an algorithmic approach and replace correlation with a distance based on Kolmogorov complexity~[6].
We estimate the Kolmogorov complexity of a string $x$ by the length of a string after compression with a commercial compression software, $C(x)$.
The Normalized Compression Distance (NCD)
% tells us how
is a measure of how
similar two strings $x$ and $y$ are:
\begin{equation}
  NCD(x,y)\,=\,\frac{C(x,y)\,-\,\min\{C(x), C(y)\}}{\max\{C(x), C(y)\}}\,.
\end{equation}

\vspace{0cm}
\vfill
\columnbreak

% where $C(x,y)$, the resulting file size after compression with a commercial compression software, gives the estimate of its Kolmogorov complexity which in general cannot be evaluated.
\begin{minipage}{0.5\linewidth}
     \centering
         \includegraphics[width=.9\columnwidth]{figure1}
\end{minipage}
\begin{minipage}{0.5\linewidth}
  \begin{figurehere}
         \caption{Model of
         the typical bipartite Bell-type experiment:
         two spatially separated universal Turing machines (UTMs): UTM$_A$ and UTM$_B$,
         fed a common input program $\Lambda$ and independent basis choice a$_j$and b$_k$,
         and corresponding outputs, $x$ and $y$.
         A model of a bipartite system to reproduce correlated strings $x$ and $y$ generated from measurements on a bipartite system with local UTMs and a common program $\Lambda$.
         }
  \end{figurehere}
\end{minipage}
A local hidden variable theory that describes the outputs two machines can be encoded in a common input program $\Lambda$, and additional programs for each UTM encoding the sequence of local measurement settings $a_j$ and $b_k$ ($j,k=0,1$). An experimental result that cannot be simulated by the UTMs would therefore falsify any local realistic description of that process.
%
% Standard Bell tests are based on statistical inference of correlations between measurements.
% We adopt an algorithmic approach and replace correlation with a distance measurement, the Normalized Compression distance (NCD), based on Kolmogorov complexity~[6].
%
% consider Kolmogorov complexity~[6] of the output of a long sequence of measurements. Under such an approach, the typical bipartite Bell-type experiment can be thought of as having two spatially separated universal Turing machines (UTMs): UTM$_A$ and UTM$_B$, with the corresponding outputs, $x$ and $y$.

We define an algorithmic Bell inequality given by:
\begin{eqnarray}
    S_A &=& NCD(x_{0}, y_{1}) - NCD(x_{0},y_{0}) -
    NCD(x_{1},y_{0})\,-\,NCD(x_{1},y_{1}) \leq 0.
\end{eqnarray}
If $S_A$ is positive the local realism hypothesis is violated.


% \vspace{0cm}
% \vfill
% \columnbreak

\begin{figurehere}
    \centering
    \includegraphics[width=0.75\columnwidth]{figure4.eps}
        \caption{Plots of $S_A$ versus angle of separation $\theta$. (a) Result obtained from Eq.~(2), (b) result obtained from using the LZMA compressor on numerically generated data, c) measurement of $S$ in the experiment shown in figure~1,and (d) longer measurement at the optimal angle $\theta=8.6^\circ$.}
\end{figurehere}

We experimentally tested inequality~(2) using the setup of figure~1,
choosing set of polarization projection
$\vec{a_0}\cdot\vec{b_1}=\cos 3\theta\,$ and
$\vec{a_0}\cdot\vec{b_0}=\vec{a_1}\cdot\vec{b_0}=\vec{a_1}\cdot\vec{b_1}=\cos
\theta$,
% , violation of the algorithmic Bell inequality, the inequality is experimentally tested by evaluating $S_A$ in Eq.~(1)
for a range of the angle $\theta$.
% between each of the four settings; the results [points (c), (d) in figure 4] are consistently lower than the trace (a) calculated via entropy, and than a simulation with the same compressor (b). This is because the compression software we used (LZMA Utility) is not working exactly at the Shannon limit, and also due to imperfect state generation and detection.

To estimate an uncertainty of the experimentally obtained values for $S_A$, we set $\theta=8.6^\circ$, for which we expect the maximum violation, and collected results from a larger number of photon pairs. We repeated the measurement of $S$ 8 times, and considered the average value and standard deviation of this set obtaining the final result of $S(\theta=8.6^\circ)\,=\,0.0494\,\pm\,$0.0076.
% This demonstrates that the algorithmic approach is complementary to the orthodox Bell inequality approach to quantum nonlocality that is statistical in its nature.
\begin{center}
  {\bf \huge \textsf {Choice of compressor}}
\end{center}\vspace{1cm}


\begin{figurehere}
  \begin{minipage}{0.5\linewidth}
    \includegraphics[width=\columnwidth]{comp3.eps}
  \end{minipage}
  \begin{minipage}{0.5\linewidth}
    \includegraphics[width=\columnwidth]{comp4.eps}
  \end{minipage}
  \caption{Compression overhead for different programs as a function of input length (left) and degree of correlation (right).}
\end{figurehere}
\begin{flushleft}
    \begin{center}{\bf \LARGE \textsf {References}}\end{center}
    {\small
    [1] H.~S.~Poh, S.~K.~Joshi, A.~Cer\`e, A.~Cabello, and C.~Kurtsiefer, Phys.~Rev.~Lett.~\textbf{115}, 180408 (2015).
    \newline
    [2] H.~S.~Poh, M.~Markiewicz, P.~Kurzynski, A.~Cer\`e, D.~Kaszlikowski, and C.~Kurtsiefer,New~J.~Phys.~\textbf{18}, 035011 (2016).
    \newline
    [3] J.~F.~Clauser, M.~A.~Horne, A.~Shimony, and R.~A.~Holt, Phys.~Rev.~Lett.~\textbf{23}, 880 (1969).
    \newline
    [4] B.~Cirel’son, Lett.~Math.~Phys.~\textbf{4}, 93 (1980).
    \newline
    [5] A.~Grinbaum, Found.~Phys.~10.1007/s10701-015-9937-y, (2015).
    }
\end{flushleft}

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\end{poster}

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