%\documentclass[aps,pra,twocolumn,showpacs,showkeys,groupedaddress]{revtex4}
%\documentclass[aps,prl,preprint,superscriptaddress]{revtex4}
\documentclass[aps,prl,twocolumn,superscriptaddress]{revtex4-1}

\usepackage{graphicx}
\usepackage{natbib}
\usepackage[breaklinks]{hyperref}

\newcommand{\ket}[1]{|{#1}\rangle}
\newcommand{\bra}[1]{\langle{#1}|}

\begin{document}

%\preprint{}
\title{Closely approaching Tsirelson's bound in a photon pair experiment}
\author{Hou~Shun~Poh}
\affiliation{Center~for~Quantum~Technologies, National~University~of~Singapore,
3 Science Drive 2, Singapore  117543}
\author{Siddarth K. Joshi}
\affiliation{Center~for~Quantum~Technologies, National~University~of~Singapore,
3 Science Drive 2, Singapore  117543}
\author{Alessandro~Cer\`{e}}
\affiliation{Center~for~Quantum~Technologies, National~University~of~Singapore,
3 Science Drive 2, Singapore  117543}
\author{Adan Cabello}
\affiliation{Departamento de F\'{i}sica Aplicada II, Universidad de Sevilla, E-41012, Sevilla, Spain}
\author{Christian~Kurtsiefer}
\affiliation{Center~for~Quantum~Technologies, National~University~of~Singapore,
3 Science Drive 2, Singapore  117543}
\affiliation{Department~of~Physics, National~University~of~Singapore,
2 Science Drive 3, Singapore  117542}
\date{\today}

\begin{abstract}

The Clauser-Horne-Shimony-Holt (CHSH) Bell inequality is a common measure to
probe non-classical correlations, as it is violated in quantum systems with
suitable measurement settings. The maximal 
violation allowed by quantum theory is given by the Tsirelson bound.
Finding experimentally a close proximity to this bound necessitates
complementary principles to recent alternative explanations for nonlocal
correlations. We present a CHSH test on a  maximally entangled polarization
state of photon pairs, and find a violation of 
$S=2.8276\,\pm\,0.00082$, or a distance of $S-2\sqrt{2}=0.0008\pm0.00082$ to
the Tsirelson bound.
%$S=2.826\pm0.0015$
%. {\bf (should we state that this is close to $2\sqrt2$?)}

\end{abstract}


% insert suggested PACS numbers in braces on next line
% \pacs{37.10.Gh, % Atom traps and guides
% 42.50.Ct,       % Quantum description of interaction of light and matter;
%                 % related experiments
% 32.90.+a        % Other topics in atomic properties and interactions of atoms;
%                 % with photons (restricted to new topics in section 32)
% }


\maketitle

% \section{Why is important to experimentally reach the Tsireslson bound?}

% {\bf (we need an intro)}

%\subsection{What is the Tsirelson bound?}
\section{Introduction}
Giving a quantitative form to concerns about ``elements of reality'' in early
quantum physics, the Bell inequality~\cite{Bell64} allowed for an experimental
comparison with more traditional descriptions of the physical world. Bell
inequalities, in particular in the form of Clauser-Horne-Shimony-Holt
(CHSH)~\cite{CHSH69}, have been violated in many experiments, substantially strengthening
the confidence in  quantum theory. With the concept of entanglement finding
its way in new computing and secure communication schemes, interest in
explaining this aspect of quantum physics via information-related principles
has developed.

Already in 1969, CHSH~\cite{CHSH69} generalized the Bell theorem and
noticed that quantum physics predicts a maximal violation of a Bell inequality
in a bipartite system, each with two measurement settings, and two
measurement outcomes. In 1980 Tsirelson proved that this maximal violation is
independent of the dimensionality of the quantum system \cite{Tsirelson80}.


% The Tsirelson bound is the maximum violation allowed by quantum theory of the
% Clauser-Horne-Shimony-Holt (CHSH) Bell inequality \cite{Bell64,CHSH69}. 
% Assuming that quantum theory is correct, this is the maximum non-locality
% for the simplest scenario: two parties, each with two measurement settings,
% each with two measurement outcomes. Already in 1969, Clauser et
% al.~\cite{CHSH69} generalized Bell theorem and noticed that, using two-qubit
% systems, there is a maximum value predicted by quantum mechanics. In 1980
% Tsirelson proved that this value is the maximum independently of the
% dimensionality of the quantum system \cite{Tsirelson80}.


% \subsection{Fundamental reasons}

The Tsirelson bound is a fundamental limit not only in quantum theory, but in
any theory that satisfies any one of the following principles: information
causality (IC) \cite{PPKSWZ09}, macroscopic locality (ML) \cite{NW09}, and
exclusivity (E) \cite{Cabello14}. In contrast, other physical principles allow
for values beyond the Tsirelson bound: non-signaling \cite{PR94},
non-triviality of communication complexity \cite{VanDam00,BBLMTU06}, and local
orthogonality \cite{FSABCLA13}. If the Tsirelson bound can be surpassed,
quantum theory would be wrong and neither IC, nor ML, nor E would hold in nature. If the Tsirelson bound can be reached but not surpassed, a fundamental prediction of quantum theory would be confirmed and those principles which exactly single out the Tsirelson bound could be the basis for a better understanding of quantum theory from fundamental physical principles. If the Tsirelson bound would be unreachable, fundamentally different principles and theories should be considered.

For any theory satisfying the E principle, the maximum violation of the CHSH
Bell inequality implies a similar upper bound for the violation of an, a
priori, completely unrelated inequality \cite{Yan13} involving different sets
of three sequential sharp measurements \cite{NBDASBC13}. If the maximum
violation of the CHSH Bell inequality is actually the Tsirelson bound, then
the bound implied by the E principle on this other inequality is exactly the
maximum violation predicted by quantum theory (and only reached with quantum
systems of dimension five or higher \cite{NBDASBC13}). 
%Therefore, experimentally reaching the Tsirelson bound can be seen as a
%confirmation of quantum theory in correlation scenarios more general than
%Bell inequality scenarios. 
Reciprocally, experimentally reaching the quantum maximum for this other inequality could be taken as an evidence of the impossibility of violations of the CHSH Bell inequality beyond the Tsirelson bound \cite{NBDASBC13}.

Continuous experimental progress made it possible to probe the
Tsirelson bound with decreasing uncertainty. Here, we report on a
an experiment with entangled photon pairs that pushes this uncertainty by
another order of magnitude compared to previous measurements.

% \subsection{Applications}

% Cryptographic security. Violations of the CHSH Bell inequality may imply cryptographic security, even if the measurements that lead to the violation are unknown to the legitimate parties. This is the so-called ``device-independent security'' for key distribution protocols \cite{ABGMPS07}. It can be proven that, under the assumption that the probability distributions
% are known without error, a necessary and sufficient condition for having probability distributions which are necessarily independent of the measurement results of any eavesdropper is that the observed probabilities are exactly the quantum probability distributions reaching the Tsirelson bound \cite{FFW11}. This result leads to a protocol for certifying device-independent security based on reaching the Tsirelson bound \cite{FFW11}.



% Physical characterization from non-local correlations. Measurements leading to non-local correlations can be treated as black boxes which, given an input (the measurement setting), return an output (the measurement result). Interestingly, the observation of violations of the CHSH Bell inequality and the assumption of non-signaling, allow to certify a variety of physical properties {\em without making any assumption or having any knowledge of the initial state or the inner working of the measurement devices}. For example, the value of the violation of the CHSH Bell inequality can be used to estimate the amount of quantum randomness produced by the black boxes \cite{PAMBMMOHLMM10}. The higher the violation, the larger the amount of certified randomness. Reaching the Tsirelson bound can also be used to certify that the state being measured is equivalent to a maximally entangled state and that the measurements performed by each party are of the type represented in quantum theory by anti-commutating operators \cite{MY04}. This may be adapted to practical methods to estimate the fidelity of the maximally entangled estates; e.g., a CHSH violation larger than 2.81, certifies a singlet fidelity greater than 96\% \cite{BNSVY13}. Moreover, saturating the Tsirelson bound can be used to certify that a general quantum computation was actually performed \cite{RUV13}.

\section{CHSH Inequality}
\label{sec:derivation_of_the_chsh_inequality}
To introduce our notation, we briefly recall the results from the CHSH
paper~\cite{CHSH69}. We consider the usual configuration formed by a source of
entangled photons and two spatially separated polarization analyzers $A$ and $B$ with possible outcomes $+$ and $-$.
%
The CHSH Bell inequality can be written as $|S|\le2$, where the parameter $S$
is a combination of polarization correlations $E$,
\begin{equation}
    S=E(a_0, b_0)-E(a_0, b_1)+E(a_1, b_0)+E(a_1, b_1)\,,
    \label{eq:bell_quantity}
\end{equation}
where $a$ and $b$ are the experimental settings, i.e., the angle of the
polarizer axes with respect to a reference. %vertical in our case.
The correlations $E$ can be calculated 
from probabilities $P_{++}$, $P_{+-}$ etc.\  of obtaining a coincidence event
with setting $a$, $b$, $a+90^\circ$, and $b+90^\circ$, respectively:
\begin{equation}
    E=P_{++}-P_{+-}-P_{-+}+P_{--}\,.
    \label{eq:correlation_function}
\end{equation}
% The probability distributions predicted by quantum mechanics determine the 
% expected value of $S$ as a function of the settings
% $a_0$, $b_0$, $a_1$, and $b_1$. For a maximally entangled polarization state
% of two photons the maximum value of $S\,=\,2\sqrt{2}$ occurs at
% $\theta=22.5^\circ$~\cite{CHSH69}.
For settings $a$, $b$ separated by $\theta=22.5^\circ$, the maximal value of
$S\,=\,2\sqrt{2}$ should be reached according to quantum
physics~\cite{CHSH69}, corresponding to the Tsirelson bound.

Experimentally, we estimate $E$ from the statistical frequency of coincidence
counts $N$ between $A$ and $B$,
% In a real experiment, assuming fair sampling~\cite{fairsampling}, $E$ can be approximated by
\begin{equation}
    E=\frac{N_{++}-N_{+-}-N_{-+}+N_{--}}{N_{++}+N_{+-}+N_{-+}+N_{--}}\,.
    \label{eq:fair_sampling}
\end{equation}
% where $N_{++}$ is the number of coincidence events registered for a certain integration time.


\section{Prior art}
The violation of Bell's inequality has been observed in many experiments with
exceedingly high statistical significance,
many of which based on the generation of correlated photon pairs,
using cascade decays in atoms~\cite{freedman:72,Aspect:1981ga},
or exploiting non linear optical processes as
in~\cite{kwiat95,weihs98,Tittel98, %yarnall07,gadway09,lima10,
NBDASBC13}.
Other successful demonstrations were based on
internal degrees of freedom of ions~\cite{rowe01,monroe04,matsukevich08} and
neutral atoms~\cite{hofmann:12}, Josephson Junctions~\cite{ansmann09}, and NV
centers in diamond~\cite{hanson12}.
Figure~\ref{fig:vio} summarizes the obtained Bell parameter and corresponding
uncertainty of several experimental tests.

\begin{figure}
  \begin{center}
    \includegraphics[width=\columnwidth]{vio2.eps}
  \end{center}
  \caption{\label{fig:vio}
    Selected experimental tests of a CHSH inequality approaching the Tsirelson
    bound in photonic systems (circles), atoms and ions (diamonds), Josephson
    junctions (square), and NV centers in diamond (triangle). Numbers
    represent the references; * corresponds to this work.
    % {\bf(for sorting out references later:)} (a) this work, (b)~\cite{NBDASBC13},
    % (c)~\cite{kwiat13} , %(d)~\cite{gadway09},
    % (e)~\cite{Tittel98},
    % (f)~\cite{aspect82}, (g)~\cite{weihs98}, 
    % % (h)~\cite{yarnall07},
    % % (i)~\cite{lima10},
    % (j)~\cite{kwiat95}, (k)~\cite{matsukevich08},
    % (l)~\cite{monroe04}, (m)~\cite{rowe01}, (n)~\cite{hanson12},
    % (o)~\cite{ansmann09}, (p)~\cite{hofmann:12}, (q)~\cite{freedman:72}
  }
\end{figure}



\section{This experiment}
Our experiment follows the concept in~\cite{kwiat95}
% with mode-matching techniques in~\cite{kurtsiefer:01},
and is shown in
Fig.~\ref{fig:experimental_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.

In the crystal, cut for type-II phase-matching,
spontaneous parametric down-conversion (SPDC) in a slightly non-collinear
configuration generates photon pairs.
Each down-converted pair consists of an ordinary and extraordinarily polarized photon, corresponding to
horizontal (H) and vertical (V) in our setup.
Two SMFs for 810\,nm define two spatial modes matched to the pump mode to optimize the collection~\cite{kurtsiefer:01}.
A half-wave plate ($\lambda$/2) and a pair of compensation
crystals (CC) take care of the temporal and transversal
walk-off~\cite{kwiat95}, and allow to adjust the phase between the two decay
components to obtain a singlet state
$\ket{\Psi^-}=1/\sqrt{2}\left(\ket{H}_A\ket{V}_B-\ket{V}_A\ket{H}_B\right)$.
%
\begin{figure}
    \begin{center}
        \includegraphics[width=0.8\columnwidth]{experimental_setup.eps}
        \caption{Schematic of the experimental set-up. Polarization
          correlations of entangled-photon pairs are measured by film
          polarizers (POL) placed in front of the collection optics. All
          photons are detected by silicon avalanche photodetectors D$_A$ and D$_B$, and registered in a coincidence unit (CU).}
        \label{fig:experimental_setup}
    \end{center}
\end{figure}
%

% To arrive at an approximate singlet Bell state, the phase $\phi$ between the two decay possibilities in the polarization state
% \begin{equation}
%     \ket{\psi}={1\over\sqrt{2}}\left(\ket{H}_A\ket{V}_B+e^{i\phi}\ket{V}\ket{H}_B\right)\,,
%     \label{eq:gen_ouput_state}
% \end{equation}
% is adjusted to $\phi=\pi$ by tilting the CC.
%
% Two film polarizers placed in front of the collection serve to project the down-converted photons onto any arbitrary linear polarization.
Film polarizers (specified extinction ratio $10^4$) perform the basis choice
and polarization projection. Photons are detected by avalanche photo diodes
(APDs, quantum efficiency $\approx$40\%), and corresponding 
detection events from the same pair identified by a coincidence unit (CU) 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. Furthermore, we
minimize contributions from higher order parametric conversion processes
\cite{Wasilewski08} by restricting the pump power below 7\,mW, leading to
average detection rates 
of 5016\,s$^{-1}$ and 4051\,s$^{-1}$ at the two the detectors, (both
uncorrected for dark counts), resulting in a detected photon pair rate
of about 567\,s$^{-1}$. The detectors exhibit dark count rates of
91.7\,s$^{-1}$ and 106.2\,s$^{-1}$, respectively.
This results in an accidental coincidence rate of 0.0067$\pm
0.0025$\,s$^{-1}$, determined by looking at coincidences in two time windows
shifted by 10\,ns and 25\,ns from where the ``true'' coincidence were
appearing.

We test the quality of polarization entanglement by measuring the polarization correlations
% in a basis complementary to the intrinsic HV basis of the crystal; for Bell states $\ket{\psi^\pm}$, strong polarization correlations are e.g. expected in a
in the
$\pm$\,45$^\circ$ linear polarization basis. With interference filters (IF) of 5\,nm bandwidth (FWHM) centered at 810\,nm, we observe a visibility $V_{45}=99.9\pm0.1\%$. The visibility in the natural H/V basis of the type-II down-conversion process also reaches $V_{\rm HV}=99.9\pm0.1\%$. This indicates
% a relatively
a high quality of polarization entanglement; the uncertainties in the
visibilities are obtained from propagated Poissonian counting statistics.

% Due to slight imperfection in the state preparation and wedge error in the polarizers, the setting list in Section~\ref{section:chsh_inequality} may not produce the maximum violation. Thus a characterization procedure to find the optimized angle for maximum violation is carried out. A detailed description of the characterization is provided in the appendix.

% \subsection{Optimization of Polarizer Settings for Maximum Violation}\label{section:optimization}

Due to imperfections in the state generation and errors in the setting of the
polarizers, the setting $\theta=22.5^\circ$ 
%s in Eq.~\ref{eq:optimal_setting} {\bf (broken
%  reference; does this refer to the $\theta$=22.5 degree?)}
% for the ideal state $\ket{\psi^-}$
may not yield the maximum possible violation.
% For this reason there is a need to experimental evaluate the optimized polarizer settings for our case.
In order to observe the largest possible violation, and get as close as possible to the Tsirelson bound,
we optimized the angular settings of the polarizers.

The optimization starts by setting $a\,=\,0^\circ$. This provides the initial reference axis and corresponds to $a_0$.
Rotating $b$ and recording the rate of coincidences, we identify the angles $b_0'$ and $b_1'$ that better match the expected correlation values. Setting $b=b_0'$, we repeat a similar procedure for $a$, obtaining $a_0'$ and $a_1'$.
This procedure converged to the resolution of the rotation motors (verified
repeatability/resolution 0.1$^\circ$).
% procedures involve firstly  and rotate $b$ while recording the number of
% coincidence events for a certain integration time. The data is fitted to a
% sinusoidal function and the maximum normalized to 0.5. Thus from the
% expected probability distribution the optimized setting for $b_0$ and $b_1$
% can be obtained. Similarly by fixing $b_0$, the optimized value for $a_1$
% can also be evaluated.
For our experimental demonstration the optimal angles are $a_0'\,=\,1.9^\circ$, $b_0'\,=\,22.9^\circ$, $a_1'\,=\,46.8^\circ$, and $b_1'\,=\,67.7^\circ$.
% The orthogonal settings occur very close the settings $a\,+\,90^\circ$ and $b\,+\,90^\circ$ indicating that the polarizer suffers from very little wedge error. The main contribution to the error in the setting is probably due to a slight systematic offset in the angular position of the two polarizers.

For evaluating how close we can come with the Bell test to the Tsirelson
bound with a known uncertainty, we need to integrate for a sufficiently long
time to acquire the necessary counting statistics, assuming we have the
usual Poissonian statistics implied by the time invariance of our
experiment. 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 via
Eqs.~(\ref{eq:bell_quantity}),(\ref{eq:fair_sampling}) a value of  
$S=2.8276\,\pm\,0.00082$, or a separation of $S-2\sqrt{2}=0.0008\pm0.00082$
from the Tsirelson bound. The uncertainty is obtained only by propagating
Poissonian counting statistics on the individual pair detections into the
expression for $S$, as any systematical errors (attacks on detectors
excluded~\cite{Gerhardt11}, i.e., under the fair sampling assumption) would
only lower the degree of violation.

\section{Conclusion}
Our search for the maximal violation of a CHSH-type Bell inequality
indicates that the Tsirelson bound can be reached, but not surpassed -- with
very low uncertainty, consistent with the conventional quantum physics
description. At least for the physical system we investigated, this suggests
that alternative descriptions based on non-signaling, non-triviality of
communication complexity, and local orthogonality principles
\cite{PR94,VanDam00,BBLMTU06} that allow to exceed the Tsirelson bound need to
be complemented with components that explain why the bound can not be overcome.

\section{Acknowledgments}
We acknowledge the support of this work by the National Research Foundation \&
Ministry of Education in Singapore, partly through the Academic Research Fund MOE2012-T3-1-009.


\bibliographystyle{apsrev4-1}
\bibliography{tsirelson.bib}{}

\end{document}
