%!TEX root = manypairs_main.tex
\section{Theory}

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We consider the many-box scenario, in which $n$ bipartite boxes are distributed to two parties, and each party only observes the sum of all his boxes' outputs. We write this sum $a_{xy}=\sum_i a_{xy}^i$ for Alice and $b_{xy}=\sum_i b_{xy}^i$ for Bob, when inputs $x$ and $y$ are used by the respective parties. For boxes with two binary measurements, this scenario involves 2 settings for each party, each with $n+1$ possible outcomes.

The correlations observed in this scenario are nonlocal iff they violate a [(n+1, n+1) (n+1, n+1)] Bell inequality. Some of these Bell inequalities are liftings from simpler scenarios, such as the [(2, 2) (2, 2)] scenario. Liftings can be seen as inequalities that are tested on the statistics obtained after performing a binning of the $n+1$ outputs into fewer values. We consider here two particular binnings.

\subsection{Majority voting}
Here we consider the binning obtained by comparing the observed output to a fixed threshold $t=n/2$. If the outcome is larger than $t$, we produce `+1', otherwise we produce `-1', i.e.
\begin{equation}\label{eq:majority_binning}
    a_{xy}' = \text{sign}(a_{xy}-t).
\end{equation}
This particular binning was already considered in several studies~\cite{...}. In particular, numerical evidences showed that it lead to a violation of the CHSH inequality for large $n$.

In this case, the CHSH expression can be written as ...

\subsection{Parity binning}
Let us now consider the parity binning which consists in only remembering the parity of the observed outcome. This brings us to the 2-outcomes scenario, for which the only Bell inequality of interest for our matter is CHSH.

Denoting by $a_{xy}^i$ and $b_{xy}^i$ the outputs of box $i$ upon measurement $x$ and $y$, we can write the binned outcomes of Alice as
\begin{equation}\label{eq:parity_binning}
a'_{xy}=(-1)^{\sum_i a_{xy}^i}.
\end{equation}
The bipartite correlator after binning, and in presence of $n$ boxes, can therefore be written as
\begin{equation}\label{eq:bipartite_corr}
\begin{split}
E_{xy}^{(n)} &= \langle (-1)^{\sum_i a_{xy}^i} \times (-1)^{\sum_i b_{xy}^i} \rangle \\
&= \langle \prod_i (-1)^{a_{xy}^i + b_{xy}^i} \rangle\\
&=(E_{xy})^n,
\end{split}
\end{equation}
where $E_{xy}$ is the 1-box correlator (which is identical for all boxes). The CHSH inequality for $n$ boxes can thus be conveniently written as
\begin{equation}\label{eq:CHSH_ineq}
S_n = (E_{11})^n + (E_{12})^n + (E_{21})^n - (E_{22})^n \leq 2.
\end{equation}

Obviously, the usual CHSH statistics don't violate this inequality for large n. Let us however consider measurement of a singlet according to
\begin{align}\label{eq:basis}
A_1 &= \sigma_z, & A_2 &= \cos(2\beta)\sigma_z + \sin(2\beta)\sigma_x\nonumber\\
B_1 &= \cos\beta\ \sigma_z + \sin\beta\ \sigma_x, & B_2 &= \cos\beta\ \sigma_z - \sin\beta\ \sigma_x.
\end{align}
This gives
\begin{equation}
S_n=3\cos^n\beta - \cos^n(3\beta).
\end{equation}
Choosing $\beta=\frac{\beta_0}{\sqrt{n}}$, we find
\begin{equation}
S_n \xrightarrow{n\to\infty} 3e^{-\beta_0^2/2} -e^{-9\beta_0^2/2} = 8\cdot 3^{-9/8} \simeq 2.32
\end{equation}
for $\beta_0 = \sqrt{\log(3)}/2 \simeq 0.524$.

Hence, the CHSH inequality with parity binning can be violated substantially even in the macroscopic limit of large number of pairs $n$. This contrasts with the majority voting case, for which evidence suggests that the maximal violation decreases as a function $n$.


\subsection{Werner states}
Let us now analyse the situation we can expect if the state of each box is slightly noisy. Here we assume that it is of the form
\begin{equation}
\rho=V\ketbra{\psi^-}{\psi^-}+(1-V)\id/4,
\end{equation}
with almost perfect visibility $V=1-\epsilon$.

It is straightforward to see that in this case, the CHSH after parity binning is simply rescaled as
\begin{equation}
S_n(\epsilon)=V^n\cdot S_n(\epsilon=0).
\end{equation}
Therefore, whenever $V<1$, the violation disappears in the macroscopic limit : $S_n(\epsilon)\xrightarrow{n\to\infty} 0 < 2$.

Assuming a fixed small $\epsilon$, the value of CHSH for large $n$ and the above settings is
\begin{equation}
\begin{split}
S_n&\simeq 8\cdot3^{-9/8} (1 - \epsilon)^n\\
&\simeq 8\cdot3^{-9/8} (1-n\epsilon)
\end{split}
\end{equation}
This value is two for the critical number of pairs
\begin{equation}
n_c \simeq \frac{1-3^{9/8}/4}{\epsilon} \simeq \frac{0.14}{\epsilon}
\end{equation}
In other words, the critical visibility required to see a violation with $n$ pairs is
\begin{equation}
V_c \simeq 1-\frac{1-3^{9/8}/4}{n} \simeq 1-\frac{0.14}{n}
\end{equation}

Also show the plot of the violation as a function of $n$!

\subsubsection{Comparison with the majority binning}
In comparison, for the majority voting strategy, the violation decreases much faster as a function of $n$. Therefore a large violation for large $n$ is not expected. The violation tail, however is quite long. In fact, it is conjectured that the critical visibility (for a Werner state again) is
\begin{equation}
V_c \simeq 1-\frac{0.64}{n}.
\end{equation}

Therefore, the critical number of pairs where a violation can still be observed (for the same noise $\epsilon$) is about 3$\sim$4 times the one for the parity binning.

