††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Noise-robust proofs of quantum network nonlocality

Sadra Boreiri  Department of Applied Physics, University of Geneva,
Switzerland  Bora Ulu  Department of Applied Physics, University of Geneva,
Switzerland  Nicolas Brunner  Department of Applied Physics, University of
Geneva, Switzerland  Pavel Sekatski  Department of Applied Physics, University
of Geneva, Switzerland

###### Abstract

Quantum networks allow for novel forms of quantum nonlocality. By exploiting
the combination of entangled states and entangled measurements, strong
nonlocal correlations can be generated across the entire network. So far, all
proofs of this effect are essentially restricted to the idealized case of pure
entangled states and projective local measurements. Here we present noise-
robust proofs of network quantum nonlocality, for a class of quantum
distributions on the triangle network that are based on entangled states and
entangled measurements. The key ingredient is a result of approximate rigidity
for local distributions that satisfy the so-called “parity token counting”
property with high probability. Considering quantum distributions obtained
with imperfect sources, we obtain noise robustness up to $\sim 80\%$ for
dephasing noise and up to $\sim 0.67\%$ for white noise. Additionally, we can
prove that all distributions in the vicinity of some ideal quantum
distributions are nonlocal, with a bound on the total-variation distance. Our
work opens interesting perspectives towards the practical implementation of
quantum network nonlocality.

##  I Introduction

A growing interest has recently been devoted to the question of quantum
nonlocality in networks, see e.g. [1] for a review. A general framework has
been developed to investigate nonlocal correlations in networks featuring
independent sources [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. A central motivation
comes from the fact that the network setting allows for novel forms of
nonlocality compared to the more standard Bell test, where a single source
distributes a physical resource (e.g. an entangled quantum state) to the
parties.

Figure 1: The triangle network features three distant parties, connected
pairwise by three independent sources distributing physical systems.

Notably, it is possible to observe quantum nonlocality with parties performing
a fixed measurement [4, 3]; see also [12] for a recent experiment. That is, in
each round of the experiment, each party provides an output, but they receive
no input. This in contrast to the standard Bell test where the use of
measurement inputs is essential.

Another key aspect of the network setting is the possibility to exploit joint
entangled measurements (where one or more eigenstates are non-separable) [13].
Similarly as in quantum teleportation, the combination of entangled states and
entangled joint measurements allows for strong correlations to be distributed
across the entire network. Notably, this leads to novel forms of quantum
nonlocal correlations that are genuine to the network structure [14], in the
sense that they crucially rely on the use of entanglement joint measurements
[15, 16]. This is possible for a simple triangle network, as in Fig. 1.

While these works represent significant progress in our understanding of
quantum nonlocality, they suffer from one major limitation. Namely, these
proofs of genuine quantum network nonlocality are derived in an ideal
noiseless setting, where the sources distribute ideal (pure) quantum states
and nodes perform ideal (projective) quantum measurements. However, these
proofs no longer apply when any realistic level of noise is considered. This
represents a major hurdle towards any practical implementation or application
of these ideas, as any experiment features a certain amount of noise
originating from unavoidable technical imperfections. Moreover, numerical
analysis based on neural networks indicates noise-robust nonlocality [17].

One can draw a parallel with the situation of nonlocality in the mid-sixties.
Indeed, John Bell’s initial proof of nonlocality applied only to the idealized
case of a pure entangled state and projective local measurements [18]. A key
step was then made by Clauser, Horne, Shimony and Holt who derived a noise-
robust proof of quantum nonlocality via their famous CHSH Bell inequality
[19], which enabled the first Bell experiments. This is still today the most
commonly used test of nonlocality, and the basis of many applications within
the device-independent approach (see e.g. [20]).

In this work we address the question of noise-robust proofs of quantum network
nonlocality. We derive methods for this problem and discuss a number of
examples. Our main contributions are the following. We present a general
family of quantum distributions on the triangle network (without inputs) and
prove their nonlocality in the ideal (noiseless) case. These distributions are
based on the concept of “parity token counting” (PTC) and include as a special
case the distributions of Renou et al. [14], from now on referred to as RGB4.
In turn, we derive noise-robust nonlocality proofs for these quantum
distributions. The key ingredient is a result of approximate rigidity for
local distributions that are PTC, which allows us to deal with noise. In
particular, we identify quantum distributions that remain nonlocal when up to
$\sim 80\%$ dephasing noise, or $\sim 0.67$% white noise, is added at each
source. Moreover, we characterize the distributions in the vicinity of a given
nonlocal distribution. For a specific quantum distribution, we can show that
any distribution in a $\sim 0.27$% total-variation distance ball around it is
nonlocal. This can be viewed as a Bell inequality, in the sense that it
guarantees that a full-measure region of the correlation set is nonlocal.

The paper is organized as follows. In Section II, we provide the general
context, introduce notations, and review the concept of parity token counting
(PTC) distributions. In Section III, we prove results on approximate PTC
rigidity for local distributions. This allows us to bound local correlations
that approximately verify the PTC condition. Additionally, we derive an
analytical outer relaxation of the set of binary output correlations
compatible with a triangle-local model. In Section IV, we introduce a general
family of quantum nonlocal distributions. We first prove their nonlocality in
the noiseless case. In turn, based on the results of Section III, we prove
results on the nonlocality of these distributions in the noisy case. Finally,
we conclude in Section VI and discuss future perspectives.

##  II Background

To start we recall some background notions and fix the notations. Consider a
triangle network as in Fig. 1, where three independent sources (labelled
$\alpha$, $\beta$ and $\gamma$) distribute physical systems to three distant
parties (nodes $A$, $B$ and $C$). Each party thus receives two physical
subsystems (from two different sources), and produces a measurement outcome,
denoted by the classical variables $a$, $b$ and $c$ (of finite cardinality $d$
111For simplicity we assume here that the cardinality is the same for all
outputs.). The experiment results in the joint probability distribution
$P(a,b,c)$.

We aim to understand which distributions are compatible with the triangle
causal network depicted in Fig. 1. The set of possible distributions depends
on the underlying physical theory. We are interested here in distributions
that can be realized in classical physics (where the sources produce
correlated classical variables) and in quantum theory (where the sources can
produce entangled states). Our focus will be on quantum distributions that
cannot be realized classically, hence demonstrating network nonlocality.

First, a distribution $P_{L}(a,b,c)$ is said to be triangle-local if it admits
a classical model, i.e. it can be decomposed as

$P_{L}(a,b,c)=\mathds{E}\Big{(}P_{A}(a|\beta,\gamma)P_{B}(b|\gamma,\alpha)P_{C}(c|\alpha,\beta)\Big{)},$
(1)

where $\mathds{E}$ is the expected value over the independently distributed
classical variables $\alpha,\beta,\gamma$. Furthermore, the conditional
probabilities $P_{X}(x|\xi,\xi^{\prime})$ can be considered deterministic
without loss of generality, and represented by the response functions
$x(\xi,\xi^{\prime})$. A distribution that does not admit a decomposition of
the form (1) is termed triangle-nonlocal, or simply nonlocal in the following.

Second, we say that a distribution $P_{Q}(a,b,c)$ is triangle-quantum if it
can be written as

$\begin{split}P_{Q}(a,b,c)=\tr\Big(&\rho^{\alpha}_{B_{L}C_{R}}\otimes\rho^{\beta}_{C_{L}A_{R}}\otimes\rho^{\gamma}_{A_{L}B_{R}}{\\\
}\times E_{A_{L}A_{R}}^{a}\otimes E_{B_{L}B_{R}}^{b}\otimes
E_{C_{L}C_{R}}^{c}\Big{missing})\end{split}$ (2)

where $\rho^{\xi}$ denote the quantum states distributed by the sources
(density operators), and $\\{E_{YY^{\prime}}^{x}\\}$ denote the measurements
performed at each node (POVMs). Note that one should pay attention to the
order of subsystems when computing (2).

For a given output cardinality $d$ we denote the set of all triangle-local
distributions $\mathcal{L}_{\triangle}^{d}$ and the set of all triangle-
quantum distributions $\mathcal{Q}_{\triangle}^{d}$. In general,
characterizing these sets is a challenging problem. Outer approximations can
be derived [21, 6, 9, 8], but typically only provide loose bounds.

For $d\geq 3$, it is known that
$\mathcal{L}_{\triangle}^{d}\subset\mathcal{Q}_{\triangle}^{d}$, i.e. there
exists quantum distributions that are nonlocal (see [22] and references
therein). For $d=2$, this is still an open question.

Known proofs of quantum nonlocality for the triangle network are of two types.
The first are distributions that can be viewed as a clever embedding of a
standard Bell test (e.g. CHSH) on the triangle network without inputs. The
initial construction was derived by Fritz [4]; see also [23, 24, 25, 22]. For
this class of distributions, noise-robust methods have been recently developed
[26], leading to a first experiment [12]. Due to their connection with
standard Bell tests, these distributions can be implemented without the need
of any entangled measurements. Hence the nonlocality of these distributions
originates solely from the use of an entangled state.

In contrast, there exist quantum distributions on the triangle network based
on the judicious combination of entangled states and measurements, as first
shown by Renou and colleagues [14]. It was recently shown that this
distribution requires in fact the use of entangled measurements [16], hence
demonstrating genuine network quantum nonlocality [15]. So far, these
nonlocality proofs are essentially restricted to an ideal noiseless scenario
see e.g. [14, 27, 28, 22, 29]). That is, they consider a setting where the
shared states $\rho^{\xi}$ are pure and the local measurements
$\\{E_{YY^{\prime}}^{x}\\}$ are projective. However, these proofs no longer
apply when any realistic amount of noise is included222Note the semi-
analytical methods of inflation [6] can detect a specific instance of the RGB4
distribution, but the resulting noise-robustness is around $10^{-7}$ for white
noise at the sources, so irrelevant from any practical perspective.. On the
other hand, numerical analysis based on machine learning suggests that the
nonlocality of these quantum distributions is in fact robust to noise [17], up
to $\approx 10\%$ white noise at the sources for some instances of the RGB4
distribution. Deriving noise-robust analytical proofs is therefore an
important open problem, which we address in this work.

###  II.1 PTC distributions and rigidity

A key class of distributions for our work are termed parity token counting
(PTC). These are local distributions on the triangle network, obtained via the
following model. Each source possesses a single token that it sends to either
one of its connected parties with some given probability. For instance, the
source $\alpha$ sends its token to Bob with probability $p_{\alpha}$ and to
Charlie with probability $1-p_{\alpha}$. Then each party outputs the parity of
the total number of received tokens. Therefore if we define the binary random
variable $t_{\alpha}\in\\{0,1\\}$ as the number of tokens $\alpha$ sends to
Bob, $t_{\beta}\in\\{0,1\\}$ as the number of tokens $\beta$ sends to Charlie,
and $t_{\gamma}\in\\{0,1\\}$ as the number of tokens $\gamma$ sends to Alice,
then for example Alice is receiving $t_{\gamma}$ from the source $\gamma$ and
$t_{\beta}\oplus 1$ from the source $\beta$ and its response function woud be
$a(\beta,\gamma)=t_{\gamma}\oplus t_{\beta}\oplus 1$, where $\oplus$ denotes
the sum modulo two.

The distribution $P(a,b,c)$ resulting from the above PTC model has binary
outputs ($d=2$). By construction, any such distribution satisfies

$a\oplus b\oplus c=1,$ (3)

which we refer to as the PTC condition.

Let us now explore the reverse link. Starting from a distribution that
satisfies the PTC condition, what can we infer on the underlying classical
model? Interestingly, PTC distributions demonstrate a form of rigidity [22];
note that this property of rigidity was initially discussed for token counting
models [27]. Rigidity means that a PTC distribution can only be obtained via a
PTC model (up to irrelevant relabellings). That is, for any local model
leading to a PTC distribution, there exists for each source a token function

$\begin{split}&T_{\xi}:\xi\to t_{\xi}\in\\{0,1\\}\qquad\text{such that}\\\
&x(\xi,\xi^{\prime})=t_{\xi}\oplus t_{\xi^{\prime}}\oplus 1.\end{split}$ (4)

In addition, if the probability distribution is such that
$E_{a},E_{b},E_{b}\neq 0$ for $E_{x}=\text{Pr}(x=0)-\text{Pr}(x=1)$, the token
functions can be chosen to fulfill

$p_{\alpha}(t):=\text{Pr}(t_{\alpha}=t)=\frac{1}{2}\left(1+(-1)^{t}\sqrt{\left|\frac{E_{b}E_{c}}{E_{a}}\right|}\right)$
(5)

and similar equalities of the other sources. This means that there is
essentially only a unique local model over the triangle which can simulate
such a distribution. In [22] one can find a simple characterization of the
local PTC set $\mathcal{L}_{\triangle|PTC}^{2}$, containing all triangle-local
distributions that satisfy the PTC condition of Eq. (3). We find that this set
covers about 12.5% of the volume333We have computed the volume of the set by
uniformly sampling $10^{6}$ points $P$ from the PTC slice
$\mathcal{P}_{PTC}^{2}$ and verifying for each point if it admits a local
model or not. For symmetric distributions $E_{A}=E_{B}=E_{C}$ we find that the
set $\mathcal{L}_{\triangle|PTC}^{2}$ is characterized by a simple inequality
$P(1,1,1)\geq\frac{1}{4}$. of $\mathcal{P}_{PTC}^{2}$ – the set of binary
probability distributions $P(a,b,c)$ fulfilling Eq. (3).

Remarkably the PTC rigidity, albeit formulated differently, is also true for
any quantum PTC distribution [16]. In particular, this implies that in the
slice of the probability set where the parity condition in Eq. (3) holds there
is no separation between the quantum and local sets, i.e.
$\mathcal{Q}_{\triangle|PTC}^{2}=\mathcal{L}_{\triangle|PTC}^{2}$.

##  III Approximate PTC rigidity for local distributions

Let us now consider a local distribution $P(a,b,c)$ for which the parity
condition only holds approximately, i.e.

$\text{Pr}(a\oplus b\oplus c=1)\geq 1-\varepsilon.$ (6)

Intuitively, one expects the underlying local model to verify some approximate
form of rigidity. This is what we now establish.

Since the response functions are deterministic444Note that any measurement
that is not described by a deterministic response function can be decomposed
as an additional local source of randomness followed by a deterministic
response function. In this case, the randomness source can be merged with one
of the sources present in the network, and all of the following arguments can
be applied at the level of the merged sources., there is a set

$\Lambda=\\{\bm{\xi}|a(\beta,\gamma)\oplus b(\alpha,\gamma)\oplus
c(\alpha,\beta)=1\\}$ (7)

containing all the values $\bm{\xi}=(\alpha,\beta,\gamma)$ for which the
parity condition in Eq. (3) is true. By assumption we have
$\text{Pr}(\bm{\xi}\in\Lambda)\geq 1-\varepsilon$.

Let us now take any value $\gamma_{*}$ for which
$\text{Pr}(\bm{\xi}\in\Lambda|\gamma=\gamma_{*})\geq 1-\varepsilon$. Such a
value must exist, since this probability can not be strictly lower than its
average on all $\gamma$. For the chosen value of $\gamma$ we define the token
function to be

$t_{\gamma_{*}}=1.$ (8)

Next, take $S=\\{(\alpha,\beta)|(\alpha,\beta,\gamma_{*})\in\Lambda\\}$ to be
the set of all pairs $(\alpha,\beta)$ for which the parity condition holds
when $\gamma=\gamma_{*}$. On this set we now define the token functions
$t_{\alpha}$ and $t_{\beta}$ as

$\begin{split}t_{\alpha}&=b(\alpha,\gamma_{*})\\\
t_{\beta}&=a(\beta,\gamma_{*}).\end{split}$ (9)

Let us check that for $\gamma=\gamma_{*}$ and any values $\alpha$ and $\beta$
in $S$, the PTC rigidity of Eq.(4) holds. By construction we have
$t_{\beta}\oplus t_{\gamma_{*}}\oplus 1=t_{\beta}=a(\beta,\gamma_{*})$ and
$t_{\alpha}\oplus t_{\gamma_{*}}\oplus 1=b(\alpha,\gamma_{*})$, while

$t_{\beta}\oplus t_{\alpha}\oplus 1=a(\beta,\gamma_{*})\oplus
b(\alpha,\gamma_{*})\oplus 1=c(\alpha,\beta)$ (10)

follows form the parity condition at $(\alpha,\beta,\gamma_{*})\in\Lambda$. In
addition, we have a lower bound on the probability that $(\alpha,\beta)$ fall
in $S$

$\text{Pr}\big{(}(\alpha,\beta)\in
S\big{)}=\text{Pr}(\bm{\xi}\in\Lambda|\gamma=\gamma_{*})\geq 1-\varepsilon.$
(11)

Finally, consider the set

$\Upsilon=\\{\bm{\xi}|(\alpha,\beta)\in S\text{ and }\bm{\xi}\in\Lambda\\}.$
(12)

For all the values $\gamma$ that appear in $\Upsilon$ we define

$t_{\gamma}=1\oplus a(\beta,\gamma)\oplus a(\beta,\gamma_{*}).$ (13)

Let us verify that with this definition, PTC rigidity holds on the set
$\Upsilon$. We have

$\begin{split}t_{\gamma}\oplus t_{\beta}\oplus 1&=a(\beta,\gamma)\oplus
a(\beta,\gamma_{*})\oplus a(\beta,\gamma_{*})=a(\beta,\gamma)\\\
t_{\gamma}\oplus t_{\alpha}\oplus 1&=a(\beta,\gamma)\oplus
a(\beta,\gamma_{*})\oplus b(\alpha,\gamma_{*})\\\ &=a(\beta,\gamma)\oplus
c(\alpha,\beta)\oplus 1=b(\alpha,\gamma)\end{split}$ (14)

and $t_{\alpha}\oplus t_{\beta}\oplus 1=c(\alpha,\beta)$ has already been
shown. It remains to bound the probability that the local variables belong to
$\Upsilon$

$\begin{split}\text{Pr}(\bm{\xi}\in\Upsilon)&=1-\text{Pr}(\bm{\xi}\notin\Lambda\text{
or }(\alpha,\beta)\notin S)\\\ &\geq
1-\text{Pr}(\bm{\xi}\notin\Lambda)-\text{Pr}((\alpha,\beta)\notin S)\\\ &\geq
1-2\varepsilon.\end{split}$ (15)

In summary, we have shown that the following result holds.

Result 1. (Approximate PTC rigidity) For any triangle local model leading to a
distribution $P(a,b,c)\in\mathcal{L}_{\triangle}^{2}$ that fulfills the
approximate parity condition $\text{Pr}(a\oplus b\oplus c=1)\geq
1-\varepsilon$, there exists a subset $\Upsilon$ of local variable values
$\bm{\xi}=(\alpha,\beta,\gamma)$ with $\text{Pr}(\bm{\xi}\in\Upsilon)\geq
1-2\varepsilon$, on which one can define the token functions
$T_{\alpha},T_{\beta},T_{\gamma}$ that satisfy the PTC rigidity
$a(\beta,\gamma)=t_{\beta}\oplus t_{\gamma}\oplus 1$ etc.

The set $\Upsilon$ gives rise to the following subnormalized distribution
$P(a,b,c,\bm{\xi}\in\Upsilon)$ which satisfies
$\sum_{a,b,c}P(a,b,c,\bm{\xi}\in\Upsilon)\geq 1-2\varepsilon$. For simplicity
let us assume that this bound is tight (otherwise one can repeat the following
discussion with a second parameter $\varepsilon^{\prime}\leq\varepsilon$ and
reach the same final conclusions), it follows that

$P(a,b,c,\bm{\xi}\in\Upsilon)+2\varepsilon P^{\prime}(a,b,c)=P(a,b,c),$ (16)

where $P^{\prime}(a,b,c)$ is some555Note that we know the values
$P^{\prime}(a,b,c)$ on the outcomes with wrong parity and will exploit this
fact later. (non necessarily local) distribution.

Now, let us prolong the token functions $T_{\xi}$ to all the values of the
local variables that do not appear in $\Upsilon$, this can be done by e.g.
assigning $t_{\xi}=0$ to all such values. With the token functions defined
everywhere we can now define a new probability distribution

$\widetilde{P}(a,b,c)=\mathds{E}\Big{(}P_{\scriptscriptstyle
ptc}(a|t_{\beta},t_{\gamma})P_{\scriptscriptstyle
ptc}(b|t_{\alpha},t_{\gamma})P_{\scriptscriptstyle
ptc}(c|t_{\alpha},t_{\beta})\Big{)}$ (17)

where the outputs are obtained with the PTC response functions

$P_{\scriptscriptstyle
ptc}(x|t_{\xi},t_{\xi^{\prime}})=\begin{cases}1&x=t_{\xi}\oplus
t_{\xi^{\prime}}\oplus 1\\\ 0&x\neq t_{\xi}\oplus t_{\xi^{\prime}}\oplus
1\end{cases}.$ (18)

By construction this distribution is triangle local and PTC. Furthermore, we
know that $P(a,b,c)$ and $\widetilde{P}(a,b,c)$ agree on the subset
$\bm{\xi}\in\Upsilon$, hence the identity

$P(a,b,c,\bm{\xi}\in\Upsilon)+2\varepsilon\widetilde{P}^{\prime}(a,b,c)=\widetilde{P}(a,b,c)$
(19)

holds for some PTC (non necessarily local) distributions
$\widetilde{P}^{\prime}$. Combining Eqs. (16,19) we conclude that the observed
local distribution must satisfy
$P(a,b,c)=\widetilde{P}(a,b,c)-2\varepsilon\widetilde{P}^{\prime}(a,b,c)+2\varepsilon
P^{\prime}(a,b,c)$, which can be nicely expressed in terms of total variation
distance

$\delta(P,\widetilde{P})=\frac{1}{2}\sum_{a,b,c}|P(a,b,c)-\widetilde{P}(a,b,c)|\leq
2\varepsilon.$ (20)

We have thus shown that any local distribution $P(a,b,c)$ satisfying the
approximate parity condition of Eq. (6), must be close to the local PTC set

$\delta(P,\mathcal{L}^{2}_{\triangle|PTC})=\min_{\widetilde{P}\in\mathcal{L}_{\triangle|PTC}^{2}}\delta(P,\widetilde{P})\leq
2\varepsilon$ (21)

In turn, the approximate parity conditions is equivalent to a bound on the
total variation distance between $P$ and the PTC slice of the probability set
$\delta(P,\mathcal{P}_{PTC}^{2})=\min_{Q\in\mathcal{P}_{PTC}^{2}}\delta(P,Q)\leq\varepsilon.$
Therefore, we have shown the following result.

Result 2.(Relaxation of the local set $\mathcal{L}_{\triangle}^{2}$) Any
triangle local distribution $P(a,b,c)\in\mathcal{L}_{\triangle}^{2}$ that is
close to the PTC slice of the probability set
$\delta(P,\mathcal{P}_{PTC}^{2})\leq\varepsilon$ is also close to the local
PTC set $\delta(P,\mathcal{L}^{2}_{\triangle|PTC})\leq 2\varepsilon$

Figure 2: A graphical representation of the construction used in the proof of
approximate PTC rigidity. The coordinates along the axes represent different
values of the local variables $\beta$ and $\gamma$, here discredited to 15
possible values each. The color of each square gives the value of Alice’s
outcome $a(\beta,\gamma)=0$ (yellow) and $a(\beta,\gamma)=1$ (blue). On the
left, the lighter region depicts the set $\Upsilon$ on which the binary token
functions $t_{\beta}$ and $t_{\gamma}$ are defined (given below the axes) and
satisfy $a(\beta,\gamma)=t_{\beta}\oplus t_{\gamma}\oplus 1$. On the right,
the token functions are prolonged on the whole set of local variable by
setting them to 0 outside of $\Upsilon$. The values obtained by applying the
PTC response function $a(\beta,\gamma)=t_{\beta}\oplus t_{\gamma}\oplus 1$
(right) are not guaranteed to match the observed values $a(\beta,\gamma)$
(left) outside of $\Upsilon$.

In words the approximate rigidity allows us to define a cone of distributions
$\widehat{\mathcal{L}}^{2}_{\triangle}$ that are accessible when going away
from the PTC slice. Loosely speaking – if we only move a distance
$\varepsilon$ from $\mathcal{L}_{\triangle|PTC}^{2}$ perpendicular to the PTC
slice we can not move more than a distance $\varepsilon$ inside the slice. Any
distribution outside of the cone is guaranteed to be nonlocal, i.e. we have
established an outer approximation of the local set
$\mathcal{L}_{\triangle}^{2}\subset\widehat{\mathcal{L}}^{2}_{\triangle}$. It
is an open question whether all quantum distributions are also contained in
this cone.

To illustrate Result 2 we discuss the nonlocality of the noisy W distributions
(see Appendix A).

##  IV Noise-robust nonlocality in $\mathcal{Q}^{4}_{\triangle}$

We now consider a particular family of four-outcome quantum distributions on
the triangle. Those are obtainded from the following quantum models. Each
source distributes a two-qubit state

$\ket{\psi}=\lambda_{0}\ket{01}+\lambda_{1}\ket{10},$ (22)

with real and positive $\lambda_{0}^{2}+\lambda_{1}^{2}=1$, to the
neighbouring parties. While each party performs the same projective
measurement
$\\{\Pi_{x_{1},x_{2}}=\outerproduct{\phi_{x_{1},x_{2}}}{\phi_{x_{1},x_{2}}}\\}$
on the two qubits it receives from neighbouring sources, with the POVM
elements given by projectors on the states

$\begin{split}\ket{\phi_{1,0}}&=u\ket{0,1}+v\ket{1,0}\\\
\ket{\phi_{1,1}}&=v\ket{0,1}-u\ket{1,0}\\\
\ket{\phi_{0,0}}&=w\ket{0,0}+z\ket{1,1}\\\
\ket{\phi_{0,1}}&=z\ket{0,0}-w\ket{1,1}\end{split}$ (23)

with real positive $u,v,w,z$ satisfying $u^{2}+v^{2}=w^{2}+z^{2}=1$. The
resulting family of quantum distribution is termed “tilted Bell state
measurements” (TBSM) distribution, since the local measurements can be viewed
as tilted Bell state measurements.

$\displaystyle P(1,i,1,j,1,k)$
$\displaystyle=(\lambda_{0}^{3}u_{i}u_{j}u_{k}+\lambda_{1}^{3}v_{i}v_{j}v_{k})^{2}$
(24) $\displaystyle P(1,i,0,j,0,k)$
$\displaystyle=(\lambda_{0}\lambda_{1}^{2}u_{i}w_{j}z_{k}+\lambda_{1}\lambda_{0}^{2}v_{i}w_{k}z_{j})^{2}\,\circlearrowleft$
(25)

where $u_{0}=u,\,v_{0}=v\,,w_{0}=w\,,z_{0}=z,\,u_{1}=v,\,v_{1}=-u,\,w_{1}=z$
and $z_{1}=-w$

By setting $w=1-z=0$ one recovers the RGB4 family of distributions [14], while
merging the outcomes $(0,0)$ and $(0,1)$ leads to the three-output family of
quantum distributions introduced in [22].

As a warm up, we now present a proof of nonlocality of the distributions in
Eq. (24) for a subsets of parameter values $(\lambda_{0},u,m)$. It has the
advantage of streamlining the proof techniques used in [14] and subsequently
[22], while relying on the same basic idea, This will be helpful when we move
to the noisy case.

###  IV.1  Nonlocality of the noiseless TBSM distributions

The nonlocality proof proceeds by contradiction, so let us assume that the
distribution is local. It is easy to see that when the second output is
discarded, the resulting binary distributions

$P(a_{1},b_{1},c_{1})=\sum_{a_{2},b_{2},c_{2}}P(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2})$
(26)

satisfy the parity condition $a_{1}\oplus b_{1}\oplus c_{1}=1$. By rigidity,
it follows that for any underlying local model there exist the binary token
functions $T_{\xi}$ that explain the distribution

$P(\bm{x}_{1})=\mathds{E}\Big{(}P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})\Big{)},$ (27)

where we introduced the compact notation $\bm{x}_{i}=(a_{i},b_{i},c_{i})$,
$\bm{t}=(t_{\alpha},t_{\beta},t_{\gamma})$ and

$P_{\scriptscriptstyle PTC}(\bm{x}_{1}|\bm{t})=P_{\scriptscriptstyle
ptc}(a_{1}|t_{\beta},t_{\gamma})P_{\scriptscriptstyle
ptc}(b_{1}|t_{\alpha},t_{\gamma})P_{\scriptscriptstyle
ptc}(c_{1}|t_{\alpha},t_{\beta}),$ (28)

with the response functions of Eq. (18). Furthermore, if
$E_{a_{1}},E_{b_{1}},E_{c_{1}}\neq 0$ the token distribution
$p(\bm{t})=p_{\alpha}(t_{\alpha})p_{\beta}(t_{\beta})p_{\gamma}(t_{\gamma})$
is known exactly via Eq. (5).

This conclusion is of course independent of whether we discard the outcomes
$\bm{x}_{2}$ or not. Hence, for any local model resulting in the TBSM
distribution, we can define the random variable $\bm{t}$, indicating the
behavior of the tokens, using the Chain rule we have
$P(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{t}}p(\bm{t})P(\bm{x}_{2}|\bm{t})P(\bm{x}_{1}|\bm{t},\bm{x}_{2})$.
From the rigidity, we can infer that $\bm{x}_{1}$ is solely a function of
$\bm{t}$, leading to

$\begin{split}P(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{t}}p(\bm{t})P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2}|\bm{t}).\end{split}$ (29)

Here, all the distributions
$P(\bm{x}_{2}|\bm{t})\in\mathcal{L}_{\triangle}^{2}$ must be triangle local.
Furthermore, these distributions are not independent, since the output of a
party (e.g. $a_{2}$) can not be influenced by the source that is not connected
to it (that is by $\alpha$ and the token value $t_{\alpha}$). This observation
implies the following constraints on the marginal distributions
$P(a_{2}|\bm{t})=\sum_{b_{2},c_{2}}P(\bm{x}_{2}|\bm{t})$

$P(a_{2}|t_{\alpha}=0,t_{\beta},t_{\gamma})=P(a_{2}|t_{\alpha}=1,t_{\beta},t_{\gamma})\quad\circlearrowleft$
(30)

and similar constraints for all permutations of parties, as indicated by the
symbol $\circlearrowleft$. The above discussion can be summarized as follows.

Result 3. (Nonlocality of TBSM distributions) For any triangle local
distribution $P(\bm{x}_{1},\bm{x}_{2})\in\mathcal{L}_{\triangle}^{4}$
satisfying $a_{1}\oplus b_{1}\oplus c_{1}=1$ , there exist eight probability
distributions $P(\bm{x}_{2}|\bm{t})$ such that

$\begin{split}&(3.i)\,P(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{t}}p(\bm{t})P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2}|\bm{t})\\\
&(3.ii)\,P(a_{2}|t_{\alpha}=0,t_{\beta},t_{\gamma})=P(a_{2}|t_{\alpha}=1,t_{\beta},t_{\gamma})\quad\circlearrowleft\\\
&(3.iii)\,P(\bm{x}_{2}|\bm{t})\in\mathcal{L}_{\triangle}^{2}\qquad\forall\bm{t}\end{split}$
(31)

where $P_{\scriptscriptstyle PTC}$ is the parity token counting response
function and the token distribution
$p(\bm{t})=p_{\alpha}(t_{\alpha})p_{\beta}(t_{\beta})p_{\gamma}(t_{\gamma})$
is uniquely determined from $P(\bm{x}_{1})$ via Eq. (5) (except degenerate
cases were it is not unique).

If we were able to show that these conditions can not be satisfied together,
we could conclude that the distribution $P(\bm{x}_{1},\bm{x}_{2})$ in
nonlocal. Yet, the locality condition $(3.iii)$ here is very challenging to
formalize since we do not have a good characterization of the local set
$\mathcal{L}_{\triangle}^{2}$. A simple solution is to relax it and keep the
other two conditions. This relaxation gives rise to a simple linear program
(LP) that must be feasible for any triangle local $P(\bm{x}_{1},\bm{x}_{2})$
with $a_{1}\oplus b_{1}\oplus c_{1}=1$, that we give in appendix B.

Figure 3: Nonlocality of noiseless TBSM distributions. Nonlocal region
depicted in yellow. Here distributions are parametrized via angles:
$u=\cos(\phi_{u})$, $w=\cos(\phi_{w})$. Parameter: $\lambda_{0}^{2}=0.225$.

In Fig. 3, we present the results of the LP for $\lambda_{0}^{2}=0.225$. One
notes that the Result 3 can be used to prove the nonlocality of the noisy
distribution, if the noise does not perturb the parity of the $\bm{x}_{1}$
outcomes. For example, the local dephasing noise given by the channel
$\mathcal{E}_{d}:\outerproduct{\psi}{\psi}\mapsto\varrho$ with

$\varrho=(1-d)\outerproduct{\psi}{\psi}+d\left(\lambda_{0}^{2}\outerproduct{01}{01}+\lambda_{1}^{2}\outerproduct{10}{10}\right)$
(32)

is of this type. In the appendix B we give the LP which finds the minimal
amount of dephasing noise such that the distribution becomes triangle local.
Its results are presented in Fig. 4 for $\lambda_{0}^{2}=0.225$.

Figure 4: Dephasing Noise Robustness. The amount of local dephasing noise $d$
in Eq. (32) below which the TBSM distribution is guaranteed to be nonlocal.
Parameters: $u=\cos(\phi_{u})$, $w=\cos(\phi_{w})$ and
$\lambda_{0}^{2}=0.225$.

Nevertheless, this still only applies to a slice of the quantum set
$\mathcal{Q}_{\triangle}^{4}$. In the next section we derive an approach which
singles out the nonlocality of a full-measure subset of triangle quantum
distributions. Interestingly, we observe that the largest robustness to white
noise, as well as for total-variation distance, is obtained for the same
parameters $\lambda_{0}^{2}=0.225,\phi_{u}=0.424$.

###  IV.2 Nonlocality of the noisy TBSM distributions

Our main idea now is pretty simple – adjust the LP based nonlocality proof of
the previous section to the setting of approximate PTC rigidity discussed in
Section III.

Indeed, let us consider any triangle local distribution

$P(\bm{x}_{1},\bm{x}_{2})=\mathds{E}\big{(}P_{A}(a_{1},a_{2}|\beta,\gamma)P_{B}(b_{1},b_{2}|\alpha,\gamma)P_{C}(\dots)\big{)}$
(33)

such that $\text{Pr}(a_{1}\oplus b_{1}\oplus c_{1}=1)\geq 1-\varepsilon$. By
approximate PTC rigidity we know that there exists a local PTC distribution
$\widetilde{P}(\bm{x}_{1})$ such that
$\delta(P(\bm{x}_{1}),\widetilde{P}(\bm{x}_{1}))\leq 2\varepsilon.$ This
distribution was constructed by applying the token functions $T_{\xi}$ to the
sources, and replacing the detectors with the parity token counting response
functions (See Fig. 2). This construction does not require us to discard the
outcomes $\bm{x}_{2}$, and we can define a full distribution

$\widetilde{P}(\bm{x}_{1},\bm{x}_{2})=\mathds{E}\big{(}P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P_{A}(a_{2}|\beta,\gamma)P_{B}(b_{2}|\alpha,\gamma)\dots).$
(34)

The distribution $\widetilde{P}$ is triangle local, by construction it agrees
with the initial distribution on outcomes $\bm{x}_{2}$
($\widetilde{P}(\bm{x}_{2})=P(\bm{x}_{2})$), and is PTC local for the outcomes
$\bm{x}_{1}$. Furthermore, we know that the two distributions match
$P(\bm{x}_{1},\bm{x}_{2},\bm{\xi}\in\Upsilon)=\widetilde{P}(\bm{x}_{1},\bm{x}_{2},\bm{\xi}\in\Upsilon)$
on the subset $\Upsilon$ of local variable values with
$\text{Pr}(\bm{\xi}\in\Upsilon)\geq 1-2\varepsilon$, which is equivalent to a
bound on their total variation distance

$\delta(P(\bm{x}_{1},\bm{x}_{2}),\widetilde{P}(\bm{x}_{1},\bm{x}_{2}))\leq
2\varepsilon.$ (35)

We have thus reached the following conclusion.

Result 4. (Nonlocality of noisy TBSM distribution) For any triangle local
distribution $P(\bm{x}_{1},\bm{x}_{2})\in\mathcal{L}_{\triangle}^{4}$
satisfying $\text{Pr}(a_{1}\oplus b_{1}\oplus c_{1}=1)\geq 1-\varepsilon$,
there exists a triangle local distribution
$\widetilde{P}(\bm{x}_{1},\bm{x}_{2})$ such that

$\begin{split}&(4.i)\,\widetilde{P}(\bm{x}_{1},\bm{x}_{2})=0\quad\text{for}\quad
a_{1}\oplus b_{1}\oplus c_{1}\neq 1\\\
&(4.ii)\,\delta(P(\bm{x}_{1},\bm{x}_{2}),\widetilde{P}(\bm{x}_{1},\bm{x}_{2}))\leq
2\varepsilon.\\\
&(4.iii)\,\sum_{\bm{x}_{1}}\widetilde{P}(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{x}_{1}}P(\bm{x}_{1},\bm{x}_{2})\end{split}$
(36)

In turn, the distribution $\widetilde{P}$ satisfies the premises of the Result
3. We could thus verify the nonlocality of $P$ by showing that no distribution
$\widetilde{P}$ compatible with both Result 3 and 4 exists. A complication
here is that the relation (5) between the distribution
$\widetilde{P}(\bm{x}_{1})$ and the underlying token distribution $p(\bm{t})$
is non-linear. It is thus easier to introduce the token distributions
$p_{\alpha},p_{\beta},p_{\gamma}$ as additional variables rather then
computing them from $\widetilde{P}$. That way we can combine results 3 and 4
into the following corollary.

Corollary of 4. For any triangle local distribution
$P(\bm{x}_{1},\bm{x}_{2})\in\mathcal{L}_{\triangle}^{4}$ satisfying
$\text{Pr}(a_{1}\oplus b_{1}\oplus c_{1}=1)\geq 1-\varepsilon$, there exist
probability distributions $P(\bm{x}_{2}|\bm{t})$, $p_{\xi}(t_{\xi})$ and
$\widetilde{P}(\bm{x}_{1},\bm{x}_{2})$ such that
$(4.ii),(4.iii),(3.ii),(3.iii)$ and

$(*)\,\widetilde{P}(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{t}}p(\bm{t})P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2}|\bm{t})$ (37)

for
$p(\bm{t})=p_{\alpha}(t_{\alpha})p_{\beta}(t_{\beta})p_{\gamma}(t_{\gamma})$.

Figure 5: White Noise Robustness. Maximal amount of white noise below which we
can prove the nonlocality of TBSM distributions. The largest amount of noise
that can be tolerated is here $\omega=0.6\%$; note that this value can be
improved slightly (see main text). Parameters: $u=\cos(\phi_{u})$,
$w=\cos(\phi_{w})$ and $\lambda_{0}^{2}=0.225$.

As in the ideal case in order to verify this feasibility problem we start by
forgetting the condition $(3.iii)$ requiring the distributions
$P(\bm{x}_{2}|\bm{t})$ to be local. Note also that the condition $(4.ii)$ on
the total variation distance $\delta(P,\widetilde{P})\leq 2\varepsilon$ is
equivalent to the existence of two probability distributions
$P^{\prime}(\bm{x}_{1},\bm{x}_{2})$ and
$\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})$ fulfilling $P+2\varepsilon
P^{\prime}=\widetilde{P}+2\varepsilon\widetilde{P}$. By introducing these
probabilities as variables we can rewire the conditions $(4.iii)$ and $(*)$ as

$\displaystyle(4.iii^{\prime})\,$
$\displaystyle\sum_{\bm{x}_{1}}\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{x}_{1}}P^{\prime}(\bm{x}_{1},\bm{x}_{2})$
(38) $\displaystyle(*^{\prime})\,$ $\displaystyle
P(\bm{x}_{1},\bm{x}_{2})+2\varepsilon\big{(}P^{\prime}(\bm{x}_{1},\bm{x}_{2})-\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})\big{)}$
$\displaystyle=\sum_{\bm{t}}p(\bm{t})P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2}|\bm{t})$ (39)

and get rid of $(4.ii)$ and the variable
$\widetilde{P}(\bm{x}_{1},\bm{x}_{2})$. Finally, it is also possible to
directly bound the values of the token probabilities $p_{\xi}(t_{\xi})$ from
$(4.ii)$. One easily sees that the total variation distance implies
$|\widetilde{E}_{x_{1}}-E_{x_{1}}|\leq 4\varepsilon$. In fact, a slightly
better bound can be obtained by using the fact that $\widetilde{P}$ is in the
PTC slice

$|\widetilde{E}_{x_{1}}-E_{x_{1}}^{\Lambda}|\leq 3\varepsilon,$ (40)

where $E_{x_{1}}^{\Lambda}=\text{Pr}(x_{1}=0,a_{1}\oplus b_{1}\oplus
c_{1}=1)-\text{Pr}(x_{1}=1,a_{1}\oplus b_{1}\oplus c_{1}=1)$. Hence, if all
the correlators for the observed distribution satisfy
$|E^{\Lambda}_{x_{1}}|\geq 3\varepsilon$ one can use Eq. (5) to obtain

$\begin{split}\frac{1}{2}&\left(1+\sqrt{\frac{(|E_{b_{1}}^{\Lambda}|-3\varepsilon)(|E_{c_{1}}^{\Lambda}|-3\varepsilon)}{|E_{a_{1}}^{\Lambda}|+3\varepsilon}}\right)\leq
p_{\alpha}(0)\\\
&\leq\frac{1}{2}\left(1+\sqrt{\frac{(|E_{b_{1}}^{\Lambda}|+3\varepsilon)(|E_{c_{1}}^{\Lambda}|+3\varepsilon)}{|E_{a_{1}}^{\Lambda}|-3\varepsilon}}\right)\quad\circlearrowleft.\end{split}$
(41)

In the Appendix C we show how using this bound the feasibility problem in Eq.
(37) can be relaxed to an LP, or a tighter collection of LPs for a grid of
values $p_{\xi}(0)$. To illustrate this approach we now consider the family of
noisy TBSM distributions, where local white noise is added to the pure state
in Eq. (22) leading to the density operator

$\rho=(1-\omega)\outerproduct{\psi}{\psi}+\omega\frac{\mathds{1}}{2}\otimes\frac{\mathds{1}}{2}.$
(42)

The resulting distribution satisfies $\text{Pr}(a_{1}\oplus b_{1}\oplus
c_{1}=1)=\frac{1}{2}\left(1+(1-\omega)^{3}\right)$. In Fig. 5 we plot the
maximal amount of white noise $\omega$ for which we can prove the nonlocality
of the distribution for $\lambda_{0}^{2}=0.225$ and different measurement
parameters. The maximal value that we find is about $\omega=0.67\%$ at
$\phi_{u}=0.424$.

In Appendix D we also discuss the nonlocality of the distributions for white
noise at the measurements, measurements with a no-click probability, as well
as photon-number-resolving detectors with limited efficiency in the case of
the single-photon entanglement implementation [28].

Figure 6: Total Variation. The size of the total variation distance ball
$\mathcal{B}_{\varepsilon}(\bar{P})$ in Eq. (43) around the ideal TBSM
distributions, for which we prove nonlocality. The maximal value we find is
$\varepsilon=0.24\%$, which can be improved to $\varepsilon=0.27\%$ (see main
text). Parameters: $u=\cos(\phi_{u})$, $w=\cos(\phi_{w})$ and
$\lambda_{0}^{2}=0.225$.

###  IV.3 Nonlocality of regions of the quantum set.

So far we argued how the nonlocality of a given distribution
$P(\bm{x}_{1},\bm{x}_{2})$ can be demonstrated. However, to better understand
the separation between the quantum and the local sets it is more insightful to
derive expressions which single out whole regions of the quantum sets that are
guaranteed to be nonlocal, similarly to the usual Bell tests. In the case of
nonlocality without inputs, considered here, such expression must be
nonlinear. We now derive such ”Bell tests” in terms of total variation
distance balls around a target quantum distribution.

Given a target quantum distribution
$\bar{P}(\bm{x}_{1},\bm{x}_{2})\in\mathcal{Q}_{\triangle}^{4}$ define the
total variation distance ball as the following subset of the correlation set

$\mathcal{B}_{\varepsilon}(\bar{P})=\\{P\in\mathcal{P}_{\triangle}^{4}|\delta(P,\bar{P})\leq\varepsilon\\}.$
(43)

We will now show that for well chosen target distributions $\bar{P}$ and small
enough $\varepsilon$ the whole set $\mathcal{B}_{\varepsilon}(\bar{P})$ is
nonlocal.

Concretely, consider $\bar{P}$ such that $a_{1}\oplus b_{1}\oplus c_{1}=1$.
Any distribution $P\in\mathcal{B}_{\varepsilon}(\bar{P})$ must satisfy

$\text{Pr}(a_{1}\oplus b_{1}\oplus c_{1}=1)\geq 1-\varepsilon.$ (44)

In addition, if the ball contains some triangle local distribution $P$ by
Result 4 there must exist a local PTC distribution $\widetilde{P}$ with
$\delta(\widetilde{P}(\bm{x}_{1},\bm{x}_{2}),P(\bm{x}_{1},\bm{x}_{2}))\leq
2\varepsilon$ by (4.ii) and
$\delta(\widetilde{P}(\bm{x}_{2}),P(\bm{x}_{2}))=0$ by (4.iii). By the
triangle inequality $\widetilde{P}$ must also satisfy

$\displaystyle\delta(\widetilde{P}(\bm{x}_{1},\bm{x}_{2}),\bar{P}(\bm{x}_{1},\bm{x}_{2}))$
$\displaystyle\leq\delta(\widetilde{P},P)+\delta(P,\bar{P})\leq 3\varepsilon$
(45) $\displaystyle\delta(\widetilde{P}(\bm{x}_{2}),\bar{P}(\bm{x}_{2}))$
$\displaystyle\leq 0+\delta(P,\bar{P})\leq\varepsilon.$ (46)

Leading to the following conclusion.

Result 5. (Nonlocality of a ball $B_{\varepsilon}(\bar{P})$) If the ball
$B_{\varepsilon}(\bar{P})\subset\mathcal{P}_{\triangle}^{4}$ contains triangle
local distributions, there exists a triangle local distribution
$\widetilde{P}(\bm{x}_{1},\bm{x}_{2})$ such that

$\begin{split}&(5.i)\,\widetilde{P}(\bm{x}_{1},\bm{x}_{2})=0\quad\text{for}\quad
a_{1}\oplus b_{1}\oplus c_{1}\neq 1\\\
&(5.ii)\,\delta(\bar{P}(\bm{x}_{1},\bm{x}_{2}),\widetilde{P}(\bm{x}_{1},\bm{x}_{2}))\leq
3\varepsilon.\\\
&(5.iii)\,\delta(\bar{P}(\bm{x}_{2}),\widetilde{P}(\bm{x}_{2}))\leq\varepsilon.\end{split}$
(47)

Naturally, this can be combined with the Result 3 in order to show that no
local distribution $\widetilde{P}$ fulfilling $(5.i)-(5.iii)$ can exist. The
argumentation is fairly similar to the corollary of the result 4, and it is
pointless to repeat in the main text. In the Appendix E we present an LP
relaxation of this feasibility problem. Note that the above formulation of
Result 5 is elegant but not exactly tight with the argument leading to it.
This is because the conditions (5.ii) and (5.iii) ignore the fact that the
triangle inequality in Eqs. (45,46) is applied for the same distribution
$P(\bm{x}_{1},\bm{x}_{2})$, which must satisfy
$\delta(\widetilde{P}(\bm{x}_{2}),P(\bm{x}_{2}))=0$. The LP derived in the
appendix directly relies on the Eqs. (45,46). In Fig. 6 we plot the size of
the ball $\varepsilon$ around the TBSM distributions with
$\lambda_{0}^{2}=0.225$ which we prove to be nonlocal. The largest value we
find is $\varepsilon=0.27\%$ and it is achieved for an RGB4 distribution
($w=0$).

##  V Discussion

We have derived noise-robust proofs of quantum nonlocality in the triangle
network without inputs. Our key ingredient is a result of approximate rigidity
for parity token (PTC) local distributions. This allowed us to obtain bounds
on the admissible noise for a general family of quantum distributions that we
introduced, generalizing the RGB4 distributions [14]. We characterized noise-
robustness in terms of limited visibility for the source states, considering
dephasing and white noise. Alternatively, we can also characterize
distributions in the vicinity of a target distribution, via a bound on the
total-variation distance.

Our work opens different perspectives. First, it would be interesting to see
if our methods can be applied to larger networks and distributions satisfying
PTC. This may lead to proofs of network nonlocality that are more robust to
noise compared to the present results.

Another direction is to see how to strengthen the noise robustness for the
nonlocality proof in our Result 4. A promising option is to revisit Result 3,
and strengthen the last condition (3.iii). We used the standard implementation
of the inflation technique, but stronger versions could be obtained by
exploiting nonlinear constraints. While we could prove a robustness to white
noise up to $0.67\%$, there is still potential for significant improvements;
indeed numerical methods [17] suggest a robustness of up to $\sim 10\%$. Any
progress in this direction would be relevant, in particular from the
perspective of experimental demonstrations. Concerning dephasing noise, we
demonstrated strong robustness, up to $\sim 80\%$. We conjecture that there
exist quantum distributions that can tolerate an amount of dephasing noise
arbitrarily close to one.

It would also be interesting to see if the property of approximate PTC
rigidity can be extended from local models to quantum ones. If not, this could
indicate the existence of quantum nonlocal correlations in the triangle
network with binary outputs (and no inputs).

_Acknowledgements.—_ We thank Wolgang Dür for discussions. We acknowledge
financial support from the Swiss National Science Foundation (project 2000021
and NCCR SwissMAP) and by the Swiss Secretariat for Education, Research and
Innovation (SERI) under contract number UeM019-3.

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##  Appendix A The noisy W-distribution

In this section we illustrate the Result 2 and discuss the nonlocality of
noisy W-distributions. Such distributions $P_{W}^{(\varepsilon,\delta)}$ are
given by

$P_{W}^{(\varepsilon,\delta)}(100)=\frac{1-\varepsilon-\delta}{3}\circlearrowleft,\qquad
P_{W}^{(\varepsilon,\delta)}(111)=\delta$ (48)

and the other terms left unspecified. It satisfies the PTC condition with
probability $1-\varepsilon$. Hence, by our Result 2 we know that there must
exist a local PTC distribution $P(a,b,c)$ such that
$\delta(P_{W}^{(\varepsilon,\delta)},P)\leq 2\varepsilon$. With
straightforward algebra, this condition can rewritten as

$\Delta^{(\varepsilon,\delta)}(P)=\sum_{(a,b,c)=(100),(010),(001)}\left|\frac{1-\varepsilon-\delta}{3}-P(a,b,c)\right|+|\delta-P(111)|-3\varepsilon\leq
0.$ (49)

Now, the set of the PTC distributions can be parameterized with the three
“token probabilities” $p_{\xi}(0)=q_{\xi}\in[0,1]$ leading to

$\displaystyle P(111)$
$\displaystyle=q_{\alpha}q_{\beta}q_{\gamma}+(1-q_{\alpha})(1-q_{\beta})(1-q_{\gamma})$
(50) $\displaystyle P(100)$
$\displaystyle=(1-q_{\alpha})q_{\beta}q_{\gamma}+q_{\alpha}(1-q_{\beta})(1-q_{\gamma})\qquad\circlearrowleft.$
(51)

Therefore the implication of the result two is equivalent to

$\Delta_{min}^{(\varepsilon,\delta)}=\min_{q_{\alpha},q_{\beta},q_{\gamma}}\Delta^{(\varepsilon,\delta)}(q_{\alpha}q_{\beta}q_{\gamma})\leq
0,$ (52)

since there must exist a combination of $q_{\alpha},q_{\beta},q_{\gamma}$ for
which $\Delta^{(\varepsilon,\delta)}(P)\leq 0$ holds. To prove that a
distribution $P_{W}^{(\varepsilon,\delta)}$ is not triangle local it is thus
sufficient to show that $\Delta_{min}^{(\varepsilon,\delta)}>0$.

Figure 7: In yellow we show the parameter region $(\varepsilon,\delta)$ where
the noisy W-distributions $P_{W}^{(\varepsilon,\delta)}(a,b,c)$ in Eq. (48)
appear to be nonlocal.

To perform the minimization in Eq. (52), we used the Nelder-Mead or simplex
search method [30] via the Python library SciPy [31]. The results of the
minimization are reported in Fig. 7. Note that this is an empirical
minimization algorithm, which does not provide a convergence guarantee.
Therefore, we have only proven the nonlocality of a given distribution under
the assumption that the global minimum was found. Nevertheless, the goal
function $\Delta^{(\varepsilon,\delta)}(q_{\alpha},q_{\beta},q_{\gamma})$ is a
smooth and well-behaved function of three variables on a compact interval, and
with more computational resources it is possible to do the minimization in a
guaranteed manner, e.g. using a branch-and-bound algorithm.

##  Appendix B Linear Program for TBSM when PTC is satisfied exactly

######  Definition 1.

For a binary tuple of outputs $\bm{x}=(a,b,c)$, and a binary tuple of tokens
$\bm{t}=(t_{\alpha},t_{\beta},t_{\gamma})$, we define the PTC response
function as follows:

$\displaystyle P_{\scriptscriptstyle
PTC}(\bm{x}|\bm{t})=\begin{cases}1&a=t_{\beta}\oplus t_{\gamma}\oplus 1,\
b=t_{\gamma}\oplus t_{\alpha}\oplus 1\ ,c=t_{\alpha}\oplus t_{\beta}\oplus
1\\\ 0&o.w.\end{cases}.$

This function is indeed, an indicator function of whether the outputs $\bm{x}$
and tokens $\bm{t}$ are consistent. Note that clearly for any $\bm{t}$ there
is only one consistent $\bm{x}$, and more interestingly for any PTC tuple
$\bm{x}=(a,b,c)$, i.e. $a\oplus b\oplus c=1$, there are exactly two consistent
tokens $\bm{t}$, which are flipped of each other. For example for
$\bm{x}=(1,1,1)$ we find $\bm{t}=(1,1,1)$, or $\bm{t}=(0,0,0)$.

Noiseless case: nonlocality of $P(\bm{x}_{1}\bm{x}_{2})$ can be checked by the
infeasibility of the following linear program with 8 probability distributions
(64 variables) $P(\bm{x}_{2}|\bm{t})$ , where we use the compact notation
$\bm{x}_{i}=(a_{i},b_{i},c_{i})$, $\bm{t}=(t_{\alpha},t_{\beta},t_{\gamma})$

$\begin{split}&(i)\,P(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{t}}p(\bm{t})P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2}|\bm{t})\\\ &(ii)\
\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2}|0,t_{\beta},t_{\gamma})}-\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2}|1,t_{\beta},t_{\gamma})=0\quad\circlearrowleft\end{split}$
(53)

Note that as mentioned before, for any PTC tuple of $\bm{x}_{1}$, there are
exactly two consistent $\bm{t}$, therefore the sum in $(i)$, is only over two
terms.

Dephasing noise: the local dephasing noise given by the channel
$\mathcal{E}_{d}:\outerproduct{\psi}{\psi}\mapsto\varrho$ with

$\varrho=(1-d)\outerproduct{\psi}{\psi}+d\left(\lambda_{0}^{2}\outerproduct{01}{01}+\lambda_{1}^{2}\outerproduct{10}{10}\right)$
(54)

where $\ket{\psi}=\lambda_{0}\ket{01}+\lambda_{1}\ket{10}$. Note that the
resulting distribution $P$ in the presence of the dephasing noise is still a
PTC distribution and can be written as

$P=(1-d)^{3}P_{Q}+(1-(1-d)^{3})P_{C}$

Where $P_{Q}$ is the quantum distribution obtained without the presence of
dephasing noise($d=0$), and $P_{C}$ is the classical distribution($d=1$),
therefore we have, $P=P_{Q}+(P_{C}-P_{Q})e$ for $e:=(1-(1-d)^{3})$ and using
(53) finding the minimum amount of dephasing noise $d$ for which the
distribution becomes local can be writen as

$\displaystyle\text{Minimize \ }e$ (55) s.t. $\displaystyle(i)\
P_{Q}(\bm{x}_{1}\bm{x}_{2})=\sum_{\bm{t}}{p(\bm{t})P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2}|\bm{t})}+\big{(}P_{Q}(\bm{x}_{1}\bm{x}_{2})-P_{C}(\bm{x}_{1}\bm{x}_{2})\big{)}e$
(56) $\displaystyle(ii)\
\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2}|0,t_{\beta},t_{\gamma})}-\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2}|1,t_{\beta},t_{\gamma})=0\quad\circlearrowleft$
(57)

Solving this LP, we find the optimum value $e^{*}$, which translates to the
optimum dephasing noise $d^{*}=1+\sqrt[3]{e^{*}-1}$. We used MOSEK Optimizer
API for Python to solve the Linear Program [32].

##  Appendix C Linear Program for noisy TBSM distributions

######  Lemma 1.

For probability distributions $p(x)$ and $q(x)$ the following statements are
equivalent,

$\displaystyle(i)\ \text{There exist probability distributions $p^{\prime}$
and $q^{\prime}$ such that }p(x)=q(x)+\varepsilon p^{\prime}(x)-\varepsilon
q^{\prime}(x)$ $\displaystyle(ii)\ \delta_{TV}(p,q)\leq\varepsilon$

###### Proof.

$(i)\Rightarrow(ii):$

$\displaystyle\delta_{TV}(p,q)=\frac{1}{2}\sum_{x}|p(x)-q(x)|=\frac{\varepsilon}{2}\sum_{x}|p^{\prime}(x)-q^{\prime}(x)|=\varepsilon\
\delta_{TV}(p^{\prime},q^{\prime})\leq\varepsilon$

$(ii)\Rightarrow(i):$ let’s define $r(x)=p(x)-q(x)$, then we have
$\sum_{x}r(x)=0$ and $\sum_{x}|r(x)|\leq\varepsilon$ we define
$p^{\prime}(x)=\frac{|r(x)|}{\sum_{x}|r(x)|}$ and
$q^{\prime}(x)=p^{\prime}(x)-\frac{r(x)}{\varepsilon}$.

∎

LP for an observed distribution with $\text{Pr}\bm{(a_{1}\oplus b_{1}\oplus
c_{1}=1)\geq 1-\varepsilon}$: Nonlocality of an observed distribution
$P(\bm{x}_{1}\bm{x}_{2})$ can be checked by the infeasibility of the following
quadratic constrained program with 6 probability distributions
$P(\bm{x}_{2},\bm{t}),P^{\prime}(\bm{x}_{1},\bm{x}_{2}),\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2}),p_{\alpha}(t),p_{\beta}(t),p_{\gamma}(t)$.

$\displaystyle(*^{\prime})\,P(\bm{x}_{1},\bm{x}_{2})=\underbrace{\sum_{\bm{t}}P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2},\bm{t})}_{\widetilde{P}(\bm{x}_{1}\bm{x}_{2})}-2\varepsilon
P^{\prime}(\bm{x}_{1},\bm{x}_{2})+2\varepsilon\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})$
$\displaystyle(4.iii^{\prime})\,\sum_{\bm{x}_{1}}\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})=\sum_{\bm{x}_{1}}P^{\prime}(\bm{x}_{1},\bm{x}_{2})$
(58) $\displaystyle(ii)\
p_{\alpha}(1)\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2},0,t_{\beta},t_{\gamma})}-p_{\alpha}(0)\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2},1,t_{\beta},t_{\gamma})=0\quad\circlearrowleft$
(59)

Note that
$P(t_{\alpha},t_{\beta},t_{\gamma})=p_{\alpha}(t_{\alpha})p_{\beta}(t_{\beta})p_{\gamma}(t_{\gamma})$
is implied by the last constraint. Furthermore, this constraint is coming from
the network structure which implies
$P(a_{2}|0,t_{\beta},t_{\gamma})=P(a_{2}|1,t_{\beta},t_{\gamma})$. Moreover,
note that here $p_{\alpha}(t),p_{\beta}(t),p_{\gamma}(t)$ correspond to the
PTC distribution $\widetilde{P}$ and not the observed distribution $P$,
therefore we do not know their values and we consider them as additional
variables. Note that the quadratic constrains in $(iii)$ are not convex, i.e.
if we write these constrains in the form of $x^{T}Qx$, the matrix $Q$ is not
positive semidefinite.

Note that although we do not know the distributions
$p_{\alpha}(t),p_{\beta}(t),p_{\gamma}(t)$ but we can bound them:

First notice that
$p_{\alpha}(t_{\alpha}=0)=\frac{1}{2}\big{(}1+\sqrt{|\frac{\widetilde{E}_{b_{1}}\widetilde{E}_{c_{1}}}{\widetilde{E}_{a_{1}}}|}\big{)}$.
Where, $\widetilde{E}_{x_{1}}=\widetilde{P}(x_{1}=0)-\widetilde{P}(x_{1}=1)$.
It’s straightforward to see that the total variation distance leads to the
conclusion that $|\widetilde{E}_{x_{1}}-E_{x_{1}}|\leq 4\varepsilon$. In fact,
a slightly better bound can be obtained by using the fact that $\widetilde{P}$
is in the PTC slice, see section C.1 below,

$|\widetilde{E}_{x_{1}}-E_{x_{1}}^{\Lambda}|\leq 3\varepsilon,$ (60)

where $E_{x_{1}}^{\Lambda}=P(x_{1}=0,a_{1}\oplus b_{1}\oplus
c_{1}=1)-P(x_{1}=1,a_{1}\oplus b_{1}\oplus c_{1}=1)$ is computed from the PTC
part of the observed distribution $P$. Hence, if all the correlators for the
observed distribution satisfy $|E^{\Lambda}_{x_{1}}|\geq 3\varepsilon$ one can
use Eq. (5) to obtain

$\begin{split}l_{\alpha}:=\frac{1}{2}\left(1+\sqrt{\frac{(|E_{b_{1}}^{\Lambda}|-3\varepsilon)(|E_{c_{1}}^{\Lambda}|-3\varepsilon)}{|E_{a_{1}}^{\Lambda}|+3\varepsilon}}\right)\leq\
p_{\alpha}(0)\
\leq\frac{1}{2}\left(1+\sqrt{\frac{(|E_{b_{1}}^{\Lambda}|+3\varepsilon)(|E_{c_{1}}^{\Lambda}|+3\varepsilon)}{|E_{a_{1}}^{\Lambda}|-3\varepsilon}}\right):=u_{\alpha}\quad\circlearrowleft.\end{split}$
(61)

Therefore $(ii)$ can be written as:

$\displaystyle\
l_{\alpha}\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2},0,t_{\beta},t_{\gamma})}-(1-l_{\alpha})\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2},1,t_{\beta},t_{\gamma})\leq
0$ (62) $\displaystyle\
u_{\alpha}\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2},0,t_{\beta},t_{\gamma})}-(1-u_{\alpha})\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2},1,t_{\beta},t_{\gamma})\geq
0.$ (63)

Finally, note that the LP has to be satisfied for some values
$p_{\xi}(t_{\xi})$ of the token probabilities in the intervals
$[l_{\alpha},u_{\alpha}]$. In practice, it is convenient to divide each
interval into smaller intervals
$[l_{\alpha}+k\frac{u_{\alpha}-l_{\alpha}}{M},l_{\alpha}+(k+1)\frac{u_{\alpha}-l_{\alpha}}{M}]$,
and verify that no solution can be found within any of the $M^{3}$ possible
combinations of intervals. This gives a better result as compared to the
harsher relaxation in Eqs. (62,63).

Furthermore, to see how good is this relaxation of the $p_{\xi}(t_{\xi})$
variable to a finite set of intervals, we also check the feasibility of the LP
for the honest values of $p_{\xi}(0)=q_{\xi}=\lambda_{0}^{2}$, i.e. the values
which appear in the quantum strategy and are thus feasible by construction.
This gives an upper bound on the maximum amount of noise for which the
original feasibility problem could guarantee that the distribution is
nonlocal.

As an illustration, we can obtain bounds on the white noise robustness of the
RGB4 distributions, see Fig. 8.

Figure 8: Noise robustness of the RGB4 distributions. Maximal amount of white
noise below which we can prove the nonlocality of RGB4 distributions (TBSM
distributions with $w=1$).

###  C.1 Development of the $3\varepsilon$ bound.

Note that we have:

$\displaystyle
P(\bm{x}_{1},\bm{\xi}\in\Lambda)=P(\bm{x}_{1},\bm{\xi}\in\Upsilon)+\varepsilon
P^{\prime\prime}(\bm{x}_{1})$
$\displaystyle\widetilde{P}(\bm{x}_{1})=P(\bm{x}_{1},\bm{\xi}\in\Upsilon)+2\varepsilon\widetilde{P^{\prime}}(\bm{x}_{1})$

Which can be rewritten as :

$\displaystyle\widetilde{P}(\bm{x}_{1})=P(\bm{x}_{1},\bm{\xi}\in\Lambda)+2\varepsilon\widetilde{P}^{\prime}(\bm{x}_{1})-\varepsilon
P^{\prime\prime}(\bm{x}_{1})$

Therefore with $E_{x_{1}}^{\Lambda}=P(x_{1}=0,a_{1}\oplus b_{1}\oplus
c_{1}=1)-P(x_{1}=1,a_{1}\oplus b_{1}\oplus c_{1}=1)$ we have

$\displaystyle\widetilde{E}_{x_{1}}=E_{x_{1}}^{\Lambda}+2\varepsilon\widetilde{E}^{\prime}_{x_{1}}-\varepsilon
E^{\prime\prime}_{x_{1}}$

Therefore,

$\displaystyle\widetilde{E}_{x_{1}}\in[E_{x_{1}}^{\Lambda}-3\varepsilon,E_{x_{1}}^{\Lambda}+3\varepsilon].$

##  Appendix D Other noise models

#### White noise at the measurement.

Alternatively to the white noise at the source, discussed in the main text,
one can consider white noise affecting the measurements. In this case the POVM
elements describing the measurement are replaced by

$E_{x_{1},x_{2}}=(1-\omega^{\prime})\outerproduct{\phi_{x_{1},x_{2}}}{\phi_{x_{1},x_{2}}}+\omega^{\prime}\frac{\mathds{1}}{2}\otimes\frac{\mathds{1}}{2}.$
(64)

Note that here for the resulting distribution we have $\text{Pr}(a_{1}\oplus
b_{1}\oplus c_{1}=1)=\frac{1}{2}\left(1+(1-\omega^{\prime})^{3}\right)$. In
Fig. 9(a), we depict the robustness to measurement white noise of the TBSM
distributions with $\lambda_{0}^{2}=0.225$. The maximal amount of noise for
which we can prove that nonlocality of the distribution is given by
$\omega^{\prime}=0.61\%$ for the case $\phi_{u}=0.425$ and $w=1$.

Considering the combined effect of white noise at the sources and the parties
we get, $\text{Pr}(a_{1}\oplus b_{1}\oplus
c_{1}=1)=\frac{1}{2}\left(1+(1-\omega^{\prime})^{3}(1-\omega)^{3}\right)$. In
Fig. 9(b), we depict the nonlocality of the TBSM distribution with
$\phi_{u}=0.425$ and $w=1$ as a function of both the noise parameters.

(a) White noise at the masurement (b) Noisy states and measurements

Figure 9:  (a) Maximal amount of white noise at the level of measurements per
party below which we can prove the nonlocality of the TBSM distribution.
Parameters: $u=\cos(\phi_{u})$, $w=\cos(\phi_{w})$ and
$\lambda_{0}^{2}=0.225$, grid-size=10. (b) Noise robustness of the TBSM
distribution when noise is added to both states ($\omega$) and measurements
($\omega^{\prime}$). The distribution is nonlocal in the yellow region.
Parameters: $\lambda_{0}^{2}=0.225,\phi_{u}=0.424,\phi_{w}=0$, grid-size=50.

#### No-click noise.

Another natural imperfection to consider is the possibility that the
measurement does not produce an outcome with probability $p$. This is
described by five-outcome POVM given by

$\displaystyle E_{x_{1},x_{2}}$
$\displaystyle=(1-p)\outerproduct{\phi_{x_{1},x_{2}}}{\phi_{x_{1},x_{2}}}\qquad\forall
x_{1},x_{2}=0,1$ (65) $\displaystyle E_{\textup{\,\o\,}}$
$\displaystyle=p\,\mathds{1}$ (66)

where $E_{\textup{\,\o\,}}$ is the element associated to the no-click event.
The probability distribution obtained with such measurements has five outcomes
for each party, and is not readily analyzed with the results presented in this
paper. However, a trick commonly used for device independent quantum key
distribution is to relabel the no-click outcome to one of the four ”good”
outcomes restoring output cardinality.

We find that the optimal strategy (for noise robustness) is for each party to
relabel the no-click outcome to $\textup{\,\o\,}\to(1,0)$, which corresponds
to the POVM

$\displaystyle E_{1,0}$
$\displaystyle=(1-p)\outerproduct{\phi_{1,0}}{\phi_{1,0}}+p\,\mathds{1}$ (67)
$\displaystyle E_{x_{1},x_{2}}$
$\displaystyle=(1-p)\outerproduct{\phi_{x_{1},x_{2}}}{\phi_{x_{1},x_{2}}}\qquad\forall(x_{1},x_{2})\neq(1,0).$
(68)

For this strategy we prove the nonlocality of the resulting distribution for
up to a no-click probability of $p=0.85\%$, see Fig. 10(a).

Note that if e.g. the sources produce two photons entangled in polarization
$\ket{0}=\ket{1_{h}}$ and $\ket{1}=\ket{1_{v}}$, a limited photon transmission
(or detector efficiency) $\eta$ would correspond to the no-click noise with
$(1-p)=\eta^{2}$.

#### Single photon entanglement implementation with loss.

For $w=1$ the TBSM distributions (RGB4) can be realised with single-photon
entanglement (vacuum state $\ket{0}$ and single photon $\ket{1}$), linear
optics and photon number resolving detectors [28]. In this case the dominant
noise is the limited efficiency of the detectors, which can be modeled as a
photon loss channel with transmission $\eta$. Formally, we have the channel

$\displaystyle\mathcal{E}(\rho)$ $\displaystyle=K_{0}\rho
K_{0}^{\dagger}+K_{1}\rho K_{1}^{\dagger}$ (69) $\displaystyle K_{0}$
$\displaystyle=\outerproduct{0}{0}+\sqrt{\eta}\outerproduct{1}{1}$ (70)
$\displaystyle K_{1}$ $\displaystyle=\sqrt{1-\eta}\outerproduct{0}{1}$ (71)

which is applied on all of the six modes. Alternatively, the noise can be
absorbed in the measurements by considering the POVM elements

$E_{x_{1},x_{2}}=\mathcal{E}^{*}\otimes\mathcal{E}^{*}(\Pi_{x_{1},x_{2}})=\sum_{i,j=0}^{1}K_{i}^{\dagger}\otimes
K_{j}^{\dagger}\,\Pi_{x_{1},x_{2}}\,K_{i}\otimes K_{j}.$ (72)

Recall that for $w=1$ the original projectors for $x_{1}=1$ read
$\Pi_{00}=\outerproduct{00}{00}$ and $\Pi_{01}=\outerproduct{11}{11}$, while
the lossy POVM elements read

$\displaystyle E_{0,0}$ $\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1-\eta&0&0\\\ 0&0&1-\eta&0\\\ 0&0&0&(1-\eta)^{2}\\\
\end{array}\right)\qquad\qquad\quad
E_{0,1}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\eta^{2}\\\ \end{array}\right)$ (81) $\displaystyle E_{1,0}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&\eta u^{2}&\eta uv&0\\\
0&\eta uv&\eta v^{2}&0\\\ 0&0&0&(1-\eta)\eta\\\ \end{array}\right)\qquad
E_{1,1}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&\eta v^{2}&-\eta uv&0\\\
0&-\eta uv&\eta u^{2}&0\\\ 0&0&0&(1-\eta)\eta\\\ \end{array}\right)$ (90)

in the computational basis $\\{\ket{00},\ket{01},\ket{10},\ket{11}\\}$.

The nonlocality of the resulting distributions can be detected up to a
transmission loss of $1-\eta=0.357\%$, see Fig. 10 (b).

(a) No-click scenario (b) Single-photon entanglement scenario with loss

Figure 10:  (a) Maximal no-click probability at each party below which we can
prove the nonlocality of the distribution. Parameters: $u=\cos(\phi_{u})$,
$w=\cos(\phi_{w})$ and $\lambda_{0}^{2}=0.225$, grid-size=10. (b) Maximal
amount of photon loss at measurements below which we can prove the nonlocality
of the RGB4 distribution (TBSM with $w=1$) for the single-photon entanglement
implementation. Here $\eta$ denotes the detection efficiency, i.e. the
probability that an emitted photon is detected. Parameters:
$u=\cos(\phi_{u})$, $w=1$ and $\lambda_{0}^{2}=0.225$

##  Appendix E Linear Program for the total variation distance

Given a target quantum distribution
$\bar{P}(\bm{x}_{1},\bm{x}_{2})\in\mathcal{Q}_{\triangle}^{4}$ define the
total variation distance ball as the following subset of the correlation set

$\mathcal{B}_{\varepsilon}(\bar{P})=\\{P\in\mathcal{P}_{\triangle}^{4}|\delta(P,\bar{P})\leq\varepsilon\\}.$
(91)

We will now demonstrate that for well chosen target distributions $\bar{P}$
and small enough $\varepsilon$ the entire set
$\mathcal{B}_{\varepsilon}(\bar{P})$ is nonlocal. Concretely, consider
$\bar{P}$ such that $a_{1}\oplus b_{1}\oplus c_{1}=1$. Any distribution
$P\in\mathcal{B}_{\varepsilon}(\bar{P})$ must satisfy

$\text{Pr}(a_{1}\oplus b_{1}\oplus c_{1}=1)\geq 1-\varepsilon.$ (92)

We will prove that all distributions in $\mathcal{B}_{\varepsilon}(\bar{P})$
are nonlocal by contradiction. Let us assume that the ball contains some
triangle local distribution $P$, then from Result 3 there must exist a local
PTC distribution $\widetilde{P}$ with
$\delta(\widetilde{P}(\bm{x}_{1},\bm{x}_{2}),P(\bm{x}_{1},\bm{x}_{2}))\leq
2\varepsilon$ and $\delta(\widetilde{P}(\bm{x}_{2}),P(\bm{x}_{2}))=0$. By
triangle inequality $\widetilde{P}$ must also satisfy

$\displaystyle\delta(\widetilde{P}(\bm{x}_{1},\bm{x}_{2}),\bar{P}(\bm{x}_{1},\bm{x}_{2}))$
$\displaystyle\leq\delta(\widetilde{P},P)+\delta(P,\bar{P})\leq 3\varepsilon$
(93) $\displaystyle\delta(\widetilde{P}(\bm{x}_{2}),\bar{P}(\bm{x}_{2}))$
$\displaystyle\leq 0+\delta(P,\bar{P})\leq\varepsilon.$ (94)

Therefore, from Eq. (93) and Lemma 1

$\bar{P}(\bm{x}_{1},\bm{x}_{2})=\underbrace{\sum_{\bm{t}}P_{\scriptscriptstyle
PTC}(\bm{x}_{1}|\bm{t})P(\bm{x}_{2},\bm{t})}_{\widetilde{P}(\bm{x}_{1}\bm{x}_{2})}-3\varepsilon\bar{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})+3\varepsilon\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})$
(95)

$\bar{P}(\bm{x}_{2})=\widetilde{P}(\bm{x}_{2})-\varepsilon\bar{P}^{\prime}(\bm{x}_{2})+\varepsilon\widetilde{P}^{\prime}(\bm{x}_{2})$
(96)

Moreover from this equation and Eq. (94) we get
$\delta(\widetilde{P}^{\prime}(\bm{x}_{2}),\bar{P}^{\prime}(\bm{x}_{2}))\leq
1/3$, therefore again by Lemma 1, we have

$\displaystyle(4.iii^{\prime})\,\sum_{\bm{x}_{1}}\widetilde{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})-\sum_{\bm{x}_{1}}\bar{P}^{\prime}(\bm{x}_{1},\bm{x}_{2})+\frac{1}{3}\widetilde{P}^{\prime\prime}(\bm{x}_{2})-\frac{1}{3}\bar{P}^{\prime\prime}(\bm{x}_{2})=0$
(97)

Moreover, like before, we use the network constraint that
$P(a_{2}|0,t_{\beta},t_{\gamma})=P(a_{2}|1,t_{\beta},t_{\gamma})$ written as
the following

$\displaystyle(ii)\
p_{\alpha}(0)\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2},0,t_{\beta},t_{\gamma})}-p_{\alpha}(1)\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2},1,t_{\beta},t_{\gamma})=0\quad\circlearrowleft$
(98)

Note that here $p_{\alpha}(t),p_{\beta}(t),p_{\gamma}(t)$ correspond to the
PTC distribution $\widetilde{P}$ which is not observed, but we can bound
$p_{\alpha}(t),p_{\beta}(t),p_{\gamma}(t)$:

Notice that
$p_{\alpha}(t_{\alpha}=0)=\frac{1}{2}\big{(}1+\sqrt{|\frac{\widetilde{E}_{b_{1}}\widetilde{E}_{c_{1}}}{\widetilde{E}_{a_{1}}}|}\big{)}$.
Where, $\widetilde{E}_{x_{1}}=\widetilde{P}(x_{1}=0)-\widetilde{P}(x_{1}=1)$.
It’s straightforward to see that the total variation distance
$\delta(\widetilde{P},\bar{P})\leq 3\varepsilon$, leads to the conclusion that
$|\widetilde{E}_{x_{1}}-\bar{E}_{x_{1}}|\leq 6\varepsilon$. Where
$\bar{E}_{x_{1}}=\bar{P}(x_{1}=0)-\bar{P}(x_{1}=1)$ is computed from the
target PTC distribution $\bar{P}$. Hence, if all the correlators for the
target distribution satisfy $|\bar{E}_{x_{1}}|\geq 6\varepsilon$ one can use
Eq. (5) to obtain

$\begin{split}l_{\alpha}:=\frac{1}{2}\left(1+\sqrt{\frac{(|\bar{E}_{b_{1}}|-6\varepsilon)(|\bar{E}_{c_{1}}|-6\varepsilon)}{|\bar{E}_{a_{1}}|+6\varepsilon}}\right)\leq\
p_{\alpha}(0)\
\leq\frac{1}{2}\left(1+\sqrt{\frac{(|\bar{E}_{b_{1}}|+6\varepsilon)(|\bar{E}_{c_{1}}|+6\varepsilon)}{|\bar{E}_{a_{1}}|-6\varepsilon}}\right):=u_{\alpha}\quad\circlearrowleft.\end{split}$
(99)

Therefore $(ii)$ can be written as:

$\begin{split}\
l_{\alpha}\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2},0,t_{\beta},t_{\gamma})}-(1-l_{\alpha})\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2},1,t_{\beta},t_{\gamma})\leq
0\\\ \
u_{\alpha}\sum_{b_{2},c_{2}}{P(a_{2},b_{2},c_{2},0,t_{\beta},t_{\gamma})}-(1-u_{\alpha})\sum_{b_{2},c_{2}}P(a_{2},b_{2},c_{2},1,t_{\beta},t_{\gamma})\geq
0\end{split}$ (100)

Combining Equations 95, 97 100, we get a set of linear constraints which
should be satisfied if there exist any local distribution inside the ball
$\mathcal{B}_{\varepsilon}(\bar{P})$. Therefore, the infeasibility of this LP,
proves non-locality of all of the points in
$\mathcal{B}_{\varepsilon}(\bar{P})$. (See Fig. 6). Again to find tighter
bounds we use the grid technique.