# Measurement dependence can enhance security in a quantum network

Amit Kundu<EMAIL_ADDRESS>Department of Applied Mathematics, University
of Calcutta, Kolkata- 700009, India  Debasis Sarkar<EMAIL_ADDRESS>Department of Applied Mathematics, University of Calcutta, Kolkata- 700009,
India

###### Abstract

Network Nonlocality is an advanced study of quantum nonlocality that comprises
network structure beyond Bell’s theorem. The development of quantum networks
has the potential to bring a lot of technological applications in sevaral
quantum information processing tasks. Here, we are focusing on how the role of
the independence of the measurement choices of the end parties in a network
works and can be used to enhance the security in a quantum network. In both
three-parties two-sources bilocal network and four-parties three-sources star
network scenarios, we are able to show, a practical way to understand the
relaxation of the assumptions to enhance a real security protocol if someone
wants to breach in a network communications. Theoratically, we have proved
that by relaxing the independence of the measurement choices of only one end
party we can create a Standard Network Nonlocality(SNN) and more stronger Full
Network Nonlocality(FNN) and we can get maximum quantum violation by the
classical no-signalling local model. We are able to distinguish between two
types of network nonlocality in the sense that the FNN is stronger than SNN,
i.e., FNN states all the sources in a network need to distribute nonlocal
resources.

##  I Introduction

A network consists of several number of parties connected in many possible
ways carrying a correlation between them. This network structure is very
important in quantum information processing. Nowadays, the urge to understand
quantum correlations in network scenarios has reached much attention and
developmentnetwork1 ; network2 ; network3 . There is a lot of interest in the
case where several quantum sources share quantum correlations between many
connected parties in various ways and reveal quantum nonlocalityswap ; review
; triA ; noinbi ; silva23 ; neural . The crucial assumption for a quantum
network is the independence of the several quantum sources, along with the
assumptions of realism, locality and measurement independence, used to create
Bell-type inequalitiesbellpaper . In a typical Bell scenario, if a correlation
from a quantum source between two parties, say, Alice and Bob, violates the
celebrated Bell-CHSH inequality, then the correlation is non-local in nature.
By this non-local correlation, we mean that the correlation from the quantum
source cannot be simulated by a local deterministic model if those three
assumptions are intact. There are waysbanik3 ; banik4 ; banik5  to classically
reproduce nonlocality by relaxing some of the assumptions like measurement
independence. Inbanik12 ; banik4 ; Hall1 ; Hall2 , they simulated a classical
model that violates Bell-CHSH inequality by relaxing measurement independence.
They made a scenario where the choices of the input of the parties, Alice and
Bob, are not fully random. The hidden variables control the input choices and
outcomes. They also showed that 30% and 59% experimental measurement
dependence is needed to simulate quantum correlation for both-sided and one-
sided dependency, respectively.

Now, what about the measurement independence assumption in a network
structure. Till date we have no significant results. Here, in this letter, we
are focusing on how this assumption uncovers novel forms of quantum nonlocal
correlations inherent to the network structure. How one can characterizes the
forms of non-locality and explore their potential for applications in quantum
information processing by applying measurement dependence. Further, what if we
loose some of the assumptions and introduce measurement dependence for the end
parties in a network structure and what will be the status of the correlation
remain in the given network structure? The results that we are going to
explore are very important in the sense that they reveal new physics in a
network which will help us to understand the structure of the correlation
better and to apply it in practical implications, we name it Boss Breaches,
where the middle party has a lot of power to control the whole network
correlation just by influencing the one end party, asking to relax the
randomness of the choice of measurement. By controlling the correlation we
mean that Bob can simulate classically the whole quantum maximal correlation
distributed in the networks with his classical variable selector and no one
can know. This is true for both Bilocal network and star network.
In short, we will first describe the Network Nonlocality, specially Standard
Network Nonlocality and Full network Nonlocality in section II. In section
III, we will show how we are imposing mathematically measurement dependence to
the end party in a network. In the next two section we will show that we can
reach the maximum quantum violation of in Bilocal and Star network by
classical model and its practical aspects respectively. We will end this
literature with the conclusion.

##  II Network Nonlocality

Consider a network composed of $m$ number of sources with $k$ number of
parties having input choices $x_{k}$ and output $a_{k}$. The correlations
maintained in this network will satisfy a network local model if it obeys;

$p(a_{1}...a_{k}|x_{1}...x_{k})=\idotsint
d\lambda_{1}\rho_{1}(\lambda_{1})...d\lambda_{m}\rho_{m}(\lambda_{m})\prod_{i}p(a_{i}|x_{i},{\bar{\lambda}}_{i})$
(1)

where $\lambda_{i}$’s are the local variables associated with each source, and
$\rho_{i}$’s are the probability densities, and $\bar{\lambda_{i}}$’s are the
set of local hidden variables of the sources connected with the
$i^{\prime}{th}$ party, otherwise the correlation will be called non n-local.
This scenario will reduce to the typical Bell scenario if $m=1$ and $k=2$.
In this letter, we will consider first the trivial network scenario called the
bilocal scenario when $m=2$, $k=3$ and then a star network scenario with
$m=3$, $k=4$nlocal . For the bilocal scenario, probability distribution
decomposed as,

$p(a_{1},a_{2},a_{3}|x_{1},x_{2},x_{3})=\int\int
d\lambda_{1}\rho_{1}(\lambda_{1})d\lambda_{2}\rho_{2}(\lambda_{2})p(a_{1}|x_{1},\lambda_{1})p(a_{2}|x_{2},\lambda_{1},\lambda_{2})p(a_{3}|x_{3},\lambda_{3})$

can be called bilocal in nature and by violating

$\sqrt{I}+\sqrt{J}\leq 1,$ (2)

we define a correlation to be non-bilocal(equ:4, 5 for $I$ and $J$). Ingisin17
, they showed if only one source is Bell nonlocal, then also the whole
correlation over the bilocal network will violate bilocal inequality. Later
inKundu20 ; andreoli , they generalised it in the n-local scenario. In recent
times, A. Pozas- Krestjens et. al. introduced a new idea called Full Network
Nonlocality(FNN) where by creating the maximum quantum violation of the
bilocal inequality with a local and PR box correlation shared between three
parties in a bilocal scenario, they concluded that the inequality(equ:2) only
confirms Standard Network Nonlocality(SNN). But in the trivial star network
scenario with three sources and four parties they showed that the correlation
will provide $S=\sqrt{2}$, where, generally for n-local scenario,

$S_{n}=\sum_{j=1}^{2^{n-1}}|I_{j}|^{\frac{1}{n}}\leq 2^{n-2}$ (3)

with
$I_{j}=\frac{1}{2^{n}}\sum_{x_{1}...x_{n}}(-1)^{g_{j}(x_{1}...x_{n})}\langle
A_{x_{1}}^{1}...A_{x_{n}}^{n}B^{j}\rangle$nlocal ; Kundu20 , coming from three
maximally entangled states shared from three different sources(fig:2) is Full
Network Nonlocal correlation, i.e., it is not reproducible by making any of
the three sources local.

##  III Imposing Measurement Dependence

Now, relaxing the assumption of measurement independence, we will show that
under certain amount of measurement dependence, only one party measurement
dependence can classically simulate not only maximal quantum correlation in
bilocal scenario(SNN) but also maximal quantum correlation in star shaped
3-local scenario(FNN). Both are defined by the maximal violation of the equ:2
and equ:3. This also allows us to use this maximal correlation in various
information protocol tasks by controlling the randomness of the measurement
choices. For the more complex quantum network structure the dependence of
measurement choices is increasing in very interesting way with the complexity
of the structure which we will discuss later.

Consider the scenario with three parties sharing two sources of independent
hidden states; this is the simplest case where our assumption of independent
sources with hidden states make sense. In such tripartite scenario, Bell’s
locality assumption reads

$P(a,b,c|x,y,z)=\int
d\lambda\rho(\lambda)P(a|x,\lambda)P(b|y,\lambda)P(c|z,\lambda).$

Here given the hidden state $\lambda$, the outputs $a$, $b$ and $c$ of the
three parties, for inputs $x$, $y$ and $z$ are determined, respectively, by
the local distributions $P(a|x,\lambda)$, $P(b|y,\lambda)$ and
$P(c|z,\lambda)$. The hidden states $\lambda$ follow the distribution
$\rho(\lambda)$, normalized such that $\int\rho(\lambda)d\lambda$=1. If we
assume that the response of the three parties depends only on the states
$\lambda_{1}$ or $\lambda_{2}$ characterizing the systems that they receive
from the sources $S_{1}$ or $S_{2}$, respectively, we write

$P(a,b,c|x,y,z)=\int\int
d\lambda_{1}d\lambda_{2}\rho(\lambda_{1},\lambda_{2})P(a|x,\lambda_{1})P(b|y,\lambda_{1},\lambda_{2})P(c|z,\lambda_{2}).$

This is a Bilocal correlation and it will jump to the tripartite Bell
correlation if $\lambda_{1}$ = $\lambda_{2}$ = $\lambda$. The crucial
assumption for this correlation is that the two sources are independent to
each other which is covered by the condition

$\rho(\lambda_{1},\lambda_{2})=\rho(\lambda_{1})\rho(\lambda_{2})$

with following distributions $\int\rho(\lambda_{1})d\lambda_{1}$ =
$\int\rho(\lambda_{2})d\lambda_{2}$ = 1.

Figure 1: Bilocal Network scenario

In cava11 ; cava12 , the first non-linear inequalities that allow one to
efficiently capture 2-local correlations were derived. Here we focus on an
inequality presented in cava12 , which we will refer to as the Bilocal
inequality(equ:2). Consider that Alice and Charlie receive binary inputs, $x$,
$x^{\prime}$ and $z$, $z^{\prime}$ and must provide binary outputs, denoted by
$a$ = $\pm$1 and $c$ = $\pm$1, respectively. The middle party Bob always
performs the same measurements with four possible outcomes, the usual BSM(Bell
State Measurements). Denote Bob’s outcome by two bits $B_{0}$ = $\pm$1 and
$B_{1}$ = $\pm$1\. The Bilocal Inequality reads
$S_{biloc}\equiv\sqrt{I}+\sqrt{J}\leq 1$ where

$I\equiv\frac{1}{4}\langle(A_{x}+A_{x^{\prime}})B_{0}(C_{z}+C_{z^{\prime}})\rangle$
(4)

$J\equiv\frac{1}{4}\langle(A_{x}-A_{x^{\prime}})B_{1}(C_{z}-C_{z^{\prime}})\rangle$
(5)

A non-bilocal correlation will violate this inequality. But after that, no
signalling correlations also form a polytope, bounded by $\max(|I|,|J|)\leq
1$; In particular, a no-signalling correlation such that $a+b+c=xy+yz$(mod 2),
can easily be realised with two PR boxes, reaches the value $I$ = $J$ =1. We
now introduced here the degree of measurement dependence for this scenario.
The dependence means that the hidden variable distribution controls the
measurement choices and it is quantified by the variational distance between
the distribution of the shared parameter for any pair of measurement settings.
Here we impose one-sided measurement dependence, i.e., on Alice’s side or
Charlie’s side, as the bilocal scenario carries a structural symmetry. We
define the degree of measurement dependence,
$M_{1}=\int|\rho(\lambda_{1}|x)-\rho(\lambda_{1}|x^{\prime})$ and
$M_{2}=\int|\rho(\lambda_{2}|z)-\rho(\lambda_{2}|z^{\prime})$ for Alice’s side
and Charlie’s side respectively. Clearly the values of $M_{1}$ and $M_{2}$ can
vary from 0 to 2. The experimental measurement independence, i.e., the
fraction of measurement independence corresponding to a given model is defined
as; $F_{1}/(F_{2})=(1-\frac{M_{1}}{2})/(1-\frac{M_{2}}{2})$. Applying those
conditions to the non linear Bilocal inequality, we have the modified
inequality, $S_{biloc}\equiv\sqrt{I}+\sqrt{J}\leq 1+\sqrt{\frac{M_{1}}{2}}$ or
$S_{biloc}\equiv\sqrt{I}+\sqrt{J}\leq 1+\sqrt{\frac{M_{2}}{2}}$ for Alice’s
part or Charlie’s part respectively. See apendix A for the proof.

##  IV Classical model for SNN and FNN

The largest known quantum violation by the bilocal inequality is
$S_{biloc}=\sqrt{2}$ and it is obtained by each source emitting singlet states
$|\psi^{-}\rangle$ = $(|01\rangle-|10\rangle)/\sqrt{2}$. In a no-signaling
deterministic model, the one-sided degree of measurement dependence that is
required to reproduce the correlation corresponding to the value of the
maximum quantum violation is $M_{1}=2(\sqrt{2}-1)^{2}$, and the experimental
measurement independence $F_{1}=2\sqrt{2}-2=82\%$. Here measurement dependence
$M_{1}=2$ and $F_{1}=0$ provides the value of the inequality from the P-R
correlation where no experimental independence is there. Here we provide a no
signalling deterministic model that reproduces the above correlation and also
compatible with the results.

B | $|B|$ | $\lambda_{1}$ | X | X’ | Z | Z’ | $\rho(\lambda_{1}|\lambda_{2},X,Z)$ | $\rho(\lambda_{1}|\lambda_{2},X,Z^{\prime})$ | $\rho(\lambda_{1}|\lambda_{2},X^{\prime},Z)$ | $\rho(\lambda_{1}|\lambda_{2},X^{\prime},Z^{\prime})$
---|---|---|---|---|---|---|---|---|---|---
$B_{0}$ | 00 | $\lambda_{1,1}$ | -a | -a | -a | -a | 0 | 0 | p | p
01 | $\lambda_{1,2}$ | b | b | b | b | 1 | 1 | 1-p | 1-p
$B_{1}$ | 10 | $\lambda_{1,1}$ | a | a | -a | a | 0 | 0 | p | p
11 | $\lambda_{1,2}$ | b | b | b | -b | 1 | 1 | 1-p | 1-p

In the table we describe the singlet correlation in a bilocal network with
measurement dependency only on the Alice’s side with the outcome containing
the variables $\lambda_{1,1}$ and $\lambda_{1,2}$ for the measurement setting
$X$ and $X^{\prime}$. We define $B_{0}$, $B_{1}$ as a selector of hidden
variable($\lambda^{\prime}s$) distribution. $a$ and $b$ can pick any value
$\pm 1$. The probability distribution is defined by a single parameter $0\leq
p\leq 1$ here. With some simple calculation, we can get the value of $I=4$ and
$J=4p$; this violates the Bilocal inequality $S_{biloc}$ by $\sqrt{p}$. So the
degree of measurement dependence is $M_{1}=2p$ which clearly shows that $0\leq
M_{1}\leq 2$ and $0\leq F_{1}\leq 1$. Therefore, for
$S_{biloc}=1-\sqrt{\frac{M_{1}}{2}}$, the amount of one-sided measurement
independence is $F=1+\sqrt{p}$ in agreement with the analytical measurement
independence of $82\%$. Here one can also see the maximum Bilocal violation(PR
correlation) can be simulated by only sacrificing the full measurement
independence of one party($p=1$).

Figure 2: 3-local scenario

We are focusing now on the trivial star network scenario with three sources
and four parties connected illustrated in the fig:2. Here the end parties,
$Alices$, are linked to a single part $Bob$ who performs a fixed measurement
with an outcome represented by a string $b=b_{1}b_{2}b_{3}\in(0,1)^{3}$. For
this scenario we can get a inequality from equ:3, which is

$\sqrt[3]{|I_{1}|}+\sqrt[3]{|I_{2}|}+\sqrt[3]{|I_{3}|}+\sqrt[3]{|I_{4}|}\leq
2$

. Now maximally entangled quantum states from three sources violates this
inequality with the value of $2\sqrt{2}$. From fullNN , we can say that
quantum correlation reaching $2\sqrt{2}$ are Full Network Nonlocal. We are
going to show that this Full Network Nonlocal correlation can be simulated by
no-signalling classical local model just by imposing $92\%$ measurement
dependence on only one party. With similar approach of bilocal inequality, we
have the modified inequality for four parties star network by imposing any of
that $Alices$ and it would provide us(Appendix B),

$\sqrt[3]{|I_{1}|}+\sqrt[3]{|I_{2}|}+\sqrt[3]{|I_{3}|}+\sqrt[3]{|I_{4}|}\leq
2+\sqrt[3]{4M_{1}}$ (6)

In a no-signaling deterministic model, the degree of measurement dependence
for one party that is required to reproduce the correlation corresponding to
the value of the maximum quantum violation is $M_{1}=2(\sqrt{2}-1)^{3}$, and
the experimental measurement independence $F_{1}=1-\frac{M_{1}}{2}=92\%$. Here
we provide a no signalling deterministic model that reproduces the above
correlation and also compatible with the results.

B | $|B|$ | $\lambda_{1}$ | X | X’ | Y | Y’ | Z | Z’ | $\rho_{\lambda 1|\lambda 2,1}$ | $\rho_{\lambda 1|\lambda 2,2}$ | $\rho_{\lambda 1|\lambda 2,3}$ | $\rho_{\lambda 1|\lambda 2,4}$ | $\rho_{\lambda 1|\lambda 2,1^{\prime}}$ | $\rho_{\lambda 1|\lambda 2,2^{\prime}}$ | $\rho_{\lambda 1|\lambda 2,3^{\prime}}$ | $\rho_{\lambda 1|\lambda 2,4^{\prime}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$B^{1}$ | 000 | $\lambda_{1,1}$ | a | a | a | a | a | a | 0 | 0 | 0 | 0 | p | p | p | p
001 | $\lambda_{1,2}$ | b | b | b | b | b | b | 1 | 1 | 1 | 1 | 1-p | 1-p | 1-p | 1-p
$B^{2}$ | 010 | $\lambda_{1,1}$ | a | a | -a | a | a | a | 0 | 0 | 0 | 0 | p | p | p | p
011 | $\lambda_{1,2}$ | b | -b | b | -b | b | b | 1 | 1 | 1 | 1 | 1-p | 1-p | 1-p | 1-p
$B^{3}$ | 100 | $\lambda_{1,1}$ | a | -a | -a | -a | -a | a | 0 | 0 | 0 | 0 | p | p | p | p
101 | $\lambda_{1,2}$ | b | b | b | b | b | -b | 1 | 1 | 1 | 1 | 1-p | 1-p | 1-p | 1-p
$B^{4}$ | 110 | $\lambda_{1,1}$ | a | a | a | -a | a | -a | 0 | 0 | 0 | 0 | p | p | p | p
111 | $\lambda_{1,2}$ | -b | b | b | -b | b | -b | 1 | 1 | 1 | 1 | 1-p | 1-p | 1-p | 1-p

Here, $\rho_{\lambda 1|\lambda 2,1}$, $\rho_{\lambda 1|\lambda 2,2}$,
$\rho_{\lambda 1|\lambda 2,3}$ and $\rho_{\lambda 1|\lambda 2,4}$ are written
in short form of $\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X,Y,Z}$,
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X,Y,Z^{\prime}}$,
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X,Y^{\prime},Z}$ and
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X,Y^{\prime},Z^{\prime}}$
respectively and similarly for $\rho_{\lambda 1|\lambda 2,1^{\prime}}$,
$\rho_{\lambda 1|\lambda 2,2^{\prime}}$, $\rho_{\lambda 1|\lambda
2,3^{\prime}}$ and $\rho_{\lambda 1|\lambda 2,4^{\prime}}$, they are
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X^{\prime},Y,Z}$,
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X^{\prime},Y,Z^{\prime}}$,
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X^{\prime},Y^{\prime},Z}$ and
$\rho_{\lambda_{1}|\lambda_{2},\lambda_{3},X^{\prime},Y^{\prime},Z^{\prime}}$
respectively.

In this table we describe the maximal quantum correlation in a four parties
star network with measurement dependency only on one of the Alices side with
the outcome containing the variables $\lambda_{1,1}$ and $\lambda_{1,2}$ for
the measurement setting $X$ and $X^{\prime}$. Here similarly we define
$B^{1}$, $B^{2}$, $B^{3}$ and $B^{4}$ as a selector of hidden
variable($\lambda^{\prime}s$) distribution and $a$, $b$ pick the value $1$.
The same simple calculation provides the value of
$\sqrt[3]{|I_{1}|}+\sqrt[3]{|I_{2}|}+\sqrt[3]{|I_{3}|}+\sqrt[3]{|I_{4}|}\leq
2+2\sqrt[3]{p}$; this violates the quantity $S_{3}$(equ:3) by $2\sqrt[3]{p}$.
So the degree of measurement dependence is $M_{1}=2p$ which clearly shows that
$0\leq M_{1}\leq 2$ and $0\leq F_{1}\leq 1$. Therefore, for
$S_{3}=2+\sqrt[3]{\frac{M_{1}}{2}}$, the amount of one-sided measurement
independence is $F=1-p=1-(\sqrt{2}-1)^{3}$ in agreement with the analytical
measurement independence of $92\%$. Similarly with bilocal scenario, one can
also see the maximum 3-local violation(PR correlation) can be simulated by
only sacrificing the full measurement independence of one party($p=1$). Now we
generalize the idea of imposing measurement dependence for further complex
network structure with increasing source number in a star network and fig:3
shows the need of measurement dependence for only one party to simulate the
maximal quantum violation for each structure with increasing source number.
When the source number is greater than five, nearly full dependence is needed
because of the structural complexity.

Figure 3: Percentage of Measurement Dependence vs Sources Numbers. Figure 4:
Schematic diagram of a bilocal network in a practical way. Here middle party
is a $Boss$, who is powerful here and $A$ and $C$ are the two parties. The two
buildings are the sources of quantum states. The wavy line from $Boss$ to $A$
represents the influence that $A$ got from $Boss$ to relax the measurement
independence. Figure 5: This is the 3-local network in a practical way. Here
also middle party is a $Boss$, who is powerful here and $A$, $B$ and $C$ are
the three parties. The sources are three. Here the red line from $Boss$ to $A$
represents the influence.

##  V Practical Aspects

Thus in the network nonlocality scenario, the assumption of measurement
independence is very strong. This work can be implemeted in a practical way of
security breaches. We name it $Boss$ $Breaches$. Fig:(4) and fig:(5) represent
a schematic diagram of a practical way of describing a network with multiple
parties and multiple sources. Here the buildings represent companies which are
the two quantum sources, distributing quantum states between A, Boss and C in
the preferred way. Now we are thinking of a situation where the Boss is a
corrupt person and he is the only one who connects A and C in a security
protocol through quantum sources and they are running any specific security
protocol based on the maximum quantum violation of Bilocal Inequality(2). Now
the perfect picture is where A and C make measurements on their particle
received from two different companies randomly and the Boss is neutral and
makes a single input with four output measurements and with both purely
quantum sources the correlation reaches the maximum value of the inequality.
Now think of a corrupt situation where the corrupt Boss wants to run the same
security protocol with the same maximum violation of the inequality but this
time only A is with the Boss and A has a special power to relax the
measurement randomness and will do the same at the Boss’s order. Now our
results corresponding to the Bilocal scenario confirm that whenever Boss wants
to twist the security he asks A to relax the measurement randomness and asks
him to relax it to $82\%$ so that the Boss can simulate classically, with his
own local variable, the whole correlation(SNN) which violates the Bilocal
Inequality maximally and C, the Companies, even A do not know that the
violation is coming from the quantum states or the classical simulation. This
is what we can say about a security fallacy where only one party just listens
to the Boss and the Boss can control the whole correlation classically without
any other person’s knowledge.

Now this situation can also happen in a stronger correlation structure. If we
consider the scenario with three parties connected again with the corrupt Boss
and the companies are distributing quantum states to the parties and Boss
(fig:5), the relaxation of $92\%$ on the measurement randomness on only one
party can give the Boss power to simulate classically the whole maximum
quantum correlation(FNN) in a 3-local scenario again without the knowledge of
the other. The classical model scripted in the tables can certify this
statement. This is a practically rich way to understand the network theory and
the relaxation of the measurement independence of any one party. We can think
of another way to use this theory and it is randomness certification. Using a
typical Bell scenario randomness certification has been done many times but in
a multi-party multi-sources quantum network scenario, the assumption of
measurement dependence can be used for randomness certification in a more
complicated manner.

##  VI Conclusion

This work captures that if any security protocol is running based on the
maximal quantum violation of network inequality for the Bilocal scenario and
star network scenario, then the middle party has the strength to activate his
local classical model without informing anyone and still can get the maximal
violation just by asking one end party not to choose the measurement inputs
randomly. This strong statement we can make here just because we showed here
how effective it is if we can lose some amount of measurement independence in
the quantum network scenario that we can simulate, classically the network
nonlocal correlation. In the trivial quantum network scenario, Bilocal
Network, Network Nonlocality comes by violating the Bilocal non-linear
inequality when the two sources are non-local in naturecava12 . According to
thegisin17 ; andreoli ; Kundu20 , the nature of the two sources can vary,
which is true for n-locality also. InfullNN , by producing maximum quantum
violation for a slightly complex network scenario, a three-source star
network, by one local and one N-S correlation and they framed the network
nonlocality in two classes, Standard Network Nonlocality(SNN) and Full Network
Nonlocality(FNN). The former correlation is the correlation which can come
from a situation where any one source can be the local source, but the latter
is the correlation which confirms that all sources are nonlocal in nature.
This work shows that in a bilocal scenario, the SNN can be revealed by a no-
signalling classical model with 82$\%$ measurement dependence on only one
party. Whereas the FNN, the correlation in the three-source four-party
scenario when all the sources produce nonlocal states, can also be simulated
by no-signalling classical model with 92$\%$ measurement dependence on only
one party. This is a very strong result and to the best of our knowledge, this
is the first work of finding optimal experimental dependence in network
structure for both SNN and FNN. We here also incorporate the result of how the
measurement dependence on only one party depends on the number of sources in a
star network scenario. Simulating n-locality with $n$ sources network will be
great work to do in future. This work is solely a step forward to better
understand the correlation of SNN and FNN and is also very important to
proceed with the idea in the more complex and non-trivial network structure.

Recently inI  the effect of relaxing the assumption of source independence in
network scenario are discussed. An interesting extension of this work can be
the study of the joint effect of source dependence and measurement dependence
in a quantum network scenario.

##  VII Acknowledgements

The authors AK and DS acknowledge the work as part of QUest initiatives by DST
India.

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##  Appendix A modified inequality with measurement dependence assumption for
Bilocal scenario

The factorizable form of the $I$ and $J$ from the Bilocal inequality;

$\displaystyle I=\frac{1}{4}|\int\int
d\lambda_{1}d\lambda_{2}\rho(\lambda_{1},\lambda_{2})V(A_{x}|\lambda_{1})V(B_{0})V(C_{z}|\lambda_{2})+V(A_{x}|\lambda_{1})V(B_{0})V(C_{z^{\prime}}|\lambda_{2})$
$\displaystyle+V(A_{x^{\prime}}|\lambda_{1})V(B_{0})V(C_{z}|\lambda_{2})+V(A_{x^{\prime}}|\lambda_{1})V(B_{0})V(C_{z^{\prime}}|\lambda_{2})|$
(7)

$\displaystyle J=\frac{1}{4}|\int\int
d\lambda_{1}d\lambda_{2}\rho(\lambda_{1},\lambda_{2})V(A_{x}|\lambda_{1})V(B_{1})V(C_{z}|\lambda_{2})-V(A_{x}|\lambda_{1})V(B_{1})V(C_{z^{\prime}}|\lambda_{2})$
$\displaystyle-V(A_{x^{\prime}}|\lambda_{1})V(B_{1})V(C_{z}|\lambda_{2})+V(A_{x^{\prime}}|\lambda_{1})V(B_{1})V(C_{z^{\prime}}|\lambda_{2})|$
(8)

Here for given value of $\lambda(\lambda_{1},\lambda_{2})$ each observable
takes a definite value, i.e.,

$V(A_{x}|\lambda_{1}),V(B_{0}|\lambda_{1}\lambda_{2}),V(C_{z}|\lambda_{2})$

and

$V(A_{x^{\prime}}|\lambda_{1}),V(B_{1}|\lambda_{1}\lambda_{2}),V(C_{z^{\prime}}|\lambda_{2}).$

With fixed value ${+1,-1}$. Now from equ:[7 and 8] we get;

$\displaystyle
I\leq\frac{1}{4}\int|[\rho(\lambda_{1}|x)V(A_{x})V(B_{0})V(C_{z})\rho(\lambda_{2}|z)+\rho(\lambda_{1}|x)V(A_{x})V(B_{0})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})$
$\displaystyle+\rho(\lambda_{1}|x^{\prime})V(A_{x^{\prime}})V(B_{0})V(C_{z})\rho(\lambda_{2}|z)+\rho(\lambda_{1}|x^{\prime})V(A_{x^{\prime}})V(B_{0})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})]|$
(9)

and

$\displaystyle
J\leq\frac{1}{4}\int|[\rho(\lambda_{1}|x)V(A_{x})V(B_{1})V(C_{z})\rho(\lambda_{2}|z)-\rho(\lambda_{1}|x)V(A_{x})V(B_{1})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})$
$\displaystyle-\rho(\lambda_{1}|x^{\prime})V(A_{x^{\prime}})V(B_{1})V(C_{z})\rho(\lambda_{2}|z)+\rho(\lambda_{1}|x^{\prime})V(A_{x^{\prime}})V(B_{1})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})]|$
(10)

as the absolute value of an integral is less than or equal to an integral of
the absolute value of the integrand($|\int f(x)dx|\leq\int|f(x)|dx$). From
equ:[9] and equ:[10], taking out the density function, we get

$I\leq\frac{1}{4}\int|\rho(\lambda_{1}|x)[V(A_{x})V(B_{0})V(C_{z})\rho(\lambda_{2}|z)+V(A_{x})V(B_{0})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})]$
$+\rho(\lambda_{1}|x^{\prime})[V(A_{x^{\prime}})V(B_{0})V(C_{z})\rho(\lambda_{2}|z)+V(A_{x^{\prime}})V(B_{0})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})]|$

and

$J\leq\frac{1}{4}\int|\rho(\lambda_{1}|x)[V(A_{X})V(B_{1})V(C_{z})\rho(\lambda_{2}|z)-V(A_{x})V(B_{1})V(C_{z^{\prime}})\rho(\lambda_{2}|z^{\prime})]$
$+\rho(\lambda_{1}|x^{\prime})[V(A_{x^{\prime}})V(B_{1})V(C_{z}^{\prime})\rho(\lambda_{2}|z^{\prime})-V(A_{x^{\prime}})V(B_{1})V(C_{z})\rho(\lambda_{2}|z)]|.$

It goes as,

$I\leq\frac{1}{4}\int|\rho(\lambda_{1}|x)V(A_{x})V(B_{0})V(C_{z})[\rho(\lambda_{2}|z)+\frac{V(C_{z^{\prime}})}{V(C_{z})}\rho(\lambda_{2}|z^{\prime})]$
$+\rho(\lambda_{1}|x^{\prime})V(A_{x^{\prime}})V(B_{0})V(C_{z})[\rho(\lambda_{2}|z)+\frac{V(C_{z^{\prime}})}{V(C_{z})}\rho(\lambda_{2}|z^{\prime})]|$

and

$J\leq\frac{1}{4}\int|\rho(\lambda_{1}|x)V(A_{x})V(B_{1})V(C_{z})[\rho(\lambda_{2}|z)-\frac{V(C_{z^{\prime}})}{V(C_{z})}\rho(\lambda_{2}|z^{\prime})]$
$+\rho(\lambda_{1}|x^{\prime})[V(A_{x^{\prime}})V(B_{1})V(C_{z^{\prime}})[\frac{V(C_{z^{\prime}})}{V(C_{z})}\rho(\lambda_{2}|z^{\prime})-\rho(\lambda_{2}|z)]|.$

Since Alice’s and Bob’s observables attains a value $\pm 1$, we can write,

$I\leq\frac{1}{4}\int|[\rho(\lambda_{1}|X)+k^{\prime}\rho(\lambda_{1}|X^{\prime})][\rho(\lambda_{2}|Z)+k\rho(\lambda_{2}|Z^{\prime})]|$

and

$J\leq\frac{1}{4}\int|[\rho(\lambda_{1}|X)-k^{\prime}\rho(\lambda_{1}|X^{\prime})][\rho(\lambda_{2}|Z)-k\rho(\lambda_{2}|Z^{\prime})]|,$

where $k=\frac{V(C_{z^{\prime}})}{V(C_{z})}$ and
$k^{\prime}=\frac{V(A_{x^{\prime}}}{V(A_{x})}$ takes a value $+1$ or $-1$ for
a given hidden variable. We can see the symmetry of the integrand of $I$ and
$J$ won’t effect the changing of the value of $k$ and $k^{\prime}$. So by
changing the value of each $k$ and $k^{\prime}$ we get;

$4+M_{1}M_{2}$

and

$M_{1}M_{2}+4.$

We here only consider one-sided measurement dependence so we can use $M_{1}=2$
or $M_{2}=2$. So the $S_{biloc}=\sqrt{I}+\sqrt{J}\leq
1+\sqrt{\frac{M_{1}}{2}}$ for Alice part of $1+\sqrt{\frac{M_{2}}{2}}$ for Bob
part.

##  Appendix B modified inequality for star network scenario

For trivial star network scenario: Consider trivial star network for three
sources and four party. The 3-local inequality is,

$S_{3}=\sum_{j=1}^{4}|I_{j}|^{\frac{1}{3}}\leq 2$

where

$I_{1}=\frac{1}{8}(\langle A^{1}_{X}A^{2}_{X}A^{3}_{X}B^{1}\rangle+\langle
A^{1}_{X}A^{2}_{X}A^{3}_{X^{\prime}}B^{1}\rangle+\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X}B^{1}\rangle+\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{1}\rangle+$ $\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X}B^{1}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X^{\prime}}B^{1}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X}B^{1}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{1}\rangle)$

$I_{2}=\frac{1}{8}(\langle A^{1}_{X}A^{2}_{X}A^{3}_{X}B^{2}\rangle+\langle
A^{1}_{X}A^{2}_{X}A^{3}_{X^{\prime}}B^{2}\rangle-\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X}B^{2}\rangle-\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{2}\rangle$ $-\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X}B^{2}\rangle-\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X^{\prime}}B^{2}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X}B^{2}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{2}\rangle)$

$I_{3}=\frac{1}{8}(\langle A^{1}_{X}A^{2}_{X}A^{3}_{X}B^{3}\rangle-\langle
A^{1}_{X}A^{2}_{X}A^{3}_{X^{\prime}}B^{3}\rangle+\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X}B^{3}\rangle-\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{3}\rangle$ $-\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X}B^{3}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X^{\prime}}B^{3}\rangle-\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X}B^{3}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{3}\rangle)$

$I_{4}=\frac{1}{8}(\langle A^{1}_{X}A^{2}_{X}A^{3}_{X}B^{4}\rangle-\langle
A^{1}_{X}A^{2}_{X}A^{3}_{X^{\prime}}B^{4}\rangle-\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X}B^{4}\rangle+\langle
A^{1}_{X}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{4}\rangle$ $+\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X}B^{4}\rangle-\langle
A^{1}_{X^{\prime}}A^{2}_{X}A^{3}_{X^{\prime}}B^{4}\rangle-\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X}B^{4}\rangle+\langle
A^{1}_{X^{\prime}}A^{2}_{X^{\prime}}A^{3}_{X^{\prime}}B^{4}\rangle)$

Using similar approach we get,

$I_{1}\leq\frac{1}{8}\int[\rho(\lambda_{1}|A_{X}^{1})+k\rho(\lambda_{1}|A_{X^{\prime}}^{1})][\rho(\lambda_{2}|A_{X}^{2})+k\rho(\lambda_{2}|A_{X^{\prime}}^{2})][\rho(\lambda_{3}|A_{X}^{3})+k\rho(\lambda_{3}|A_{X^{\prime}}^{3})]$

$I_{2}\leq\frac{1}{8}\int[\rho(\lambda_{1}|A_{X}^{1})+k\rho(\lambda_{1}|A_{X^{\prime}}^{1})][\rho(\lambda_{2}|A_{X}^{2})-k\rho(\lambda_{2}|A_{X^{\prime}}^{2})][\rho(\lambda_{3}|A_{X}^{3})-k\rho(\lambda_{3}|A_{X^{\prime}}^{3})]$

$I_{3}\leq\frac{1}{8}\int[\rho(\lambda_{1}|A_{X}^{1})-k\rho(\lambda_{1}|A_{X^{\prime}}^{1})][\rho(\lambda_{2}|A_{X}^{2})+k\rho(\lambda_{2}|A_{X^{\prime}}^{2})][\rho(\lambda_{3}|A_{X}^{3})-k\rho(\lambda_{3}|A_{X^{\prime}}^{3})]$

$I_{3}\leq\frac{1}{8}\int[\rho(\lambda_{1}|A_{X}^{1})-k\rho(\lambda_{1}|A_{X^{\prime}}^{1})][\rho(\lambda_{2}|A_{X}^{2})-k\rho(\lambda_{2}|A_{X^{\prime}}^{2})][\rho(\lambda_{3}|A_{X}^{3})+k\rho(\lambda_{3}|A_{X^{\prime}}^{3})]$

Where $k$ signifies the same quantity which for a given local variable either
takes +1 or -1. For every possible value of $k$ we can have,

$\sqrt[3]{|I_{1}|}+\sqrt[3]{|I_{2}|}+\sqrt[3]{|I_{3}|}+\sqrt[3]{|I_{4}|}\leq
1+\sqrt[3]{\frac{2M_{2}M_{3}}{8}}+\sqrt[3]{\frac{M_{1}2M_{3}}{8}}+\sqrt[3]{\frac{M_{1}M_{2}2}{8}}$

For any one sided measurement dependence, we can get,

$\sqrt[3]{|I_{1}|}+\sqrt[3]{|I_{2}|}+\sqrt[3]{|I_{3}|}+\sqrt[3]{|I_{4}|}\leq
2+\sqrt[3]{4M_{1}},$

by considering only one party dependence.