Measurement dependence in tripartite non-locality

Abstract

The assumption of measurement independence is essential for the derivation of Bell’s inequalities under local, realistic conditions. Violations of these inequalities indicate that the assumption of measurement independence must be relaxed to some extent in order to obtain locally realistic models. The extent to which this assumption needs to be relaxed to achieve violations of certain bipartite Bell inequalities has been studied in Hall (Phys Rev Lett 105:250404, 2010) and Friedman (Phys Rev A 99:012121, 2019). In this paper, we investigate the minimal degree of relaxation required to simulate violations of various known tripartite Bell inequalities. We also provide local deterministic models that achieve these violations.

Appendices

Appendix A: relaxed Mermin inequality

In tripartite system with two measurement settings for each party and each measurement has two possible outcomes (as mentioned in Sect. (2.1)) standard tripartite non-locality is detected via the violation of Mermin inequality [42] which has the expression,

$$\begin{aligned} \vert S_\textit{M} \vert = \vert \langle A_1B_0C_0 \rangle +\langle A_0B_1C_0 \rangle +\langle A_0B_0C_1 \rangle -\langle A_1B_1C_1 \rangle \vert \le 2. \end{aligned}$$

(A1)

Hence, $S_\textit{M} = \langle A_1 B_0 C_0 \rangle + \langle A_0 B_1 C_0 \rangle +\langle A_0 B_0 C_1 \rangle - \langle A_1 B_1 C_1 \rangle$.

Following the ideas of [19, 21], here in a deterministic no-signaling model, we apply arbitrary measurement dependence in either of the parties measurement settings. In this model, the measurement outcomes are noted as $u(A_{x},\;\lambda ),\;u(A_{x'},\;\lambda )$ each taking values from the set $\lbrace 1, -1\rbrace$ for Alice’s outcome. Identical interpretation follows for $v(B_{y},\;\lambda ),\;v(B_{y'},\;\lambda )$ and $w(C_{z},\;\lambda ),\;w(C_{z'},\;\lambda )$ accordingly.

Let us assume the three parties perform their spin measurements along the directions $A_{\hat{m}}$, $B_{\hat{m}}$ and $C_{\hat{m}}$, respectively, where the measurement direction for the party X ($= A, \,B,\,C$) is denoted as $X_{\hat{m}}\; \equiv \, (\sin \theta _{X_{\hat{m}}} \cos \phi _{X_{\hat{m}}}, \sin \theta _{X_{\hat{m}}}\sin \phi _{X_{\hat{m}}}, \cos \theta _{X_{\hat{m}}})$, with $\theta _{X_{\hat{m}}} \in [0, \pi ]$ and $\phi _{X_{\hat{m}}} \in [0, 2\pi ]$. The correlation is defined as $\langle A_1 B_0 C_0 \rangle$ = $\int {\rho (\lambda |x^{'},\;y,\;z)u(A_{x'},\;\lambda )\; v(B_y,\;\lambda )\; w(C_z,\;\lambda )\;\textrm{d}\lambda }$, and similarly other terms of the expression follows as well. Using these quantities the parameter $S_{\textit{M}}$ then gives,

$S_{\textit{M}} = \int {\rho (\lambda |x',\;y,\;z)\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }$+ $\int \rho (\lambda |x,\;y',\;z)\;u(A_x,\lambda )\;v(B_{y'},\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda$ + $\int {\rho (\lambda |x,\;y,\;z')\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$ - $\int {\rho (\lambda |x',\;y',\;z')\; u(A_{x'},\;\lambda )\; v(B_{y'},\;\lambda )\; w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

We next introduce the measurement dependence factors in $S_{\textit{M}}$ by using simple mathematical process as follow,

$S_{\textit{M}} = \int {\rho (\lambda |x',\;y,\;z)\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda } - \int \rho (\lambda |x,\;y,\;z)\;u(A_x,\lambda )\; v(B_y,\;\lambda )\; w(C_z,\;\lambda )\;\textrm{d }\lambda$ + $\int \rho (\lambda |x,\;y,\;z)\;u(A_x,\;\lambda )v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda$ + $\int \rho (\lambda |x,\;y',\;z)\; u(A_x,\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda + \int \rho (\lambda |x,\;y,\;z')\; u(A_x,\;\lambda )v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\textrm{d}\lambda$ - $\int \rho (\lambda |x',\;y,\;z')\;u(A_{x'},\;\lambda )v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$ + $\int \rho (\lambda |x',\;y,\;z')\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda - \int \rho (\lambda |x',\;y',\;z')\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$.

$= \int {[\rho (\lambda |x',\;y,\;z)\;u(A_{x'},\;\lambda )-\rho (\lambda |x,\;y,\;z)\;u(A_x,\;\lambda )]\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }$ + $\int {[\rho (\lambda |x,\;y,\;z)\;v(B_y,\;\lambda )+\rho (\lambda |x,\;y',\;z)\;v(B_{y'},\;\lambda )]\;u(A_x,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int {[\rho (\lambda |x,\;y,\;z')\;u(A_x,\;\lambda ) -\rho (\lambda |x',\;y,\;z')\;u(A_{x'},\;\lambda )]\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }+\int {[\rho (\lambda |x',\;y,\;z')\;v(B_y,\;\lambda )-\rho (\lambda |x',\;y',\;z')\;v(B_{y'},\;\lambda )]\;u(A_{x'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

$=\int {[\rho (\lambda |x',\;y,\;z)-\rho (\lambda |x,\;y,\;z)\;\frac{u(A_x,\;\lambda )}{u(A_{x'},\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int {[\rho (\lambda |x,\;y,\;z)+\rho (\lambda |x,\;y',\;z)\;\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )}]\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }$ -$\int [\rho (\lambda |x',\;y,\;z')-\rho (\lambda |x,\;y,\;z')\;\frac{u(A_{x},\;\lambda )}{u(A_x',\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda +\int {[\rho (\lambda |x',\;y,\;z')-\rho (\lambda |x',\;y',\;z')\;\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }.$

In this scenario, we are considering measurement dependence for Alice or Bob’s measurement settings over the variable $\lambda$, hence the response functions $\frac{u(A_x,\;\lambda )}{u(A_{x'},\;\lambda )}$ and $\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )}$ are function of $\lambda$. Let us define $\frac{u(A_x,\;\lambda )}{u(A_{x'},\;\lambda )}$ = $S_{1}(\rho (\lambda \vert x,\;y,\;z)-\rho (\lambda \vert x',\;y,\;z))$ and $\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )}$ = $S_{2}(\rho (\lambda \vert x',\;y,\;z')-\rho (\lambda \vert x',\;y',\;z'))$ for given values of $\lambda$, where $S_{i}$($\alpha$) ( i = 1, 2, 3) is defined below (for simplicity we will use the notation $S_{1}$ and $S_{2}$ onward )

$$\begin{aligned} S_{i}(\alpha )= Sign(\alpha )= {\left\{ \begin{array}{ll} 1, &{} \text {if } \alpha \ge 0\\ -1, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

Hence, $S_{M} = \int [\rho (\lambda |x',\;y,\;z)- S_{1} \rho (\lambda |x,\;y,\;z)] \; u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\textrm{d}\lambda$ + $\int [\rho (\lambda |x,\;y,\;z)+ S_{2}\rho (\lambda |x,\;y',\;z)] \;u(A_{x},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\textrm{d}\lambda$ + $\int [ \rho (\lambda |x,\;y,\;z')S_{1}- \rho (\lambda |x',\;y,\;z')] \;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z',\;\lambda )\, \textrm{d}\lambda$ + $\int [ \rho (\lambda |x',\;y,\;z')-S_{2} \rho (\lambda |x',\;y',\;z')] \;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z',\;\lambda ) \;\textrm{d}\lambda .$

= $\int [\rho (\lambda |x',\;y,\;z)-S_{1}\rho (\lambda |x',\;y,\;z)+S_{1}\rho (\lambda |x',\;y,\;z) - S_{1} \rho (\lambda |x,\;y,\;z)] \; u(A_{x'},\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\,\textrm{d}\lambda$ + $\int [\rho (\lambda |x,\;y,\;z)-S_{2}\rho (\lambda |x,\;y,\;z)+S_{2}\rho (\lambda |x,\;y,\;z)+ S_{2}\rho (\lambda |x,\;y',\;z)] \;u(A_{x},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\textrm{d}\lambda$

+ $\int [ \rho (\lambda |x,\;y,\;z')S_{1}-S_{1}\rho (\lambda |x',\;y,\;z')+S_{1}\rho (\lambda |x',\;y,\;z') - \rho (\lambda |x',\;y,\;z')] \;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z',\;\lambda )\, \textrm{d}\lambda$ + $\int [ \rho (\lambda |x',\;y,\;z')-S_{2}\rho (\lambda |x',\;y,\;z')+S_{2}\rho (\lambda |x',\;y,\;z')-S_{2} \rho (\lambda |x',\;y',\;z')] \;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z',\;\lambda ) \;\textrm{d}\lambda .$

Hence, $\vert S_{M} \vert \le \int \vert S_{1} \vert \,\vert \rho (\lambda |x',\;y,\;z)-\rho (\lambda |x,\;y,\;z) \vert \,\textrm{d}\lambda + \int \vert S_{1} \vert \,\vert \rho (\lambda |x,\;y,\;z')-\rho (\lambda |x',\;y,\;z') \vert \,\textrm{d}\lambda + \int \vert S_{2} \vert \,\vert \rho (\lambda |x,\;y,\;z)+\rho (\lambda |x,\;y',\;z) \vert \,\textrm{d}\lambda + \int \vert S_{2} \vert \,\vert \rho (\lambda |x',\;y,\;z')-\rho (\lambda |x',\;y',\;z') \vert \,\textrm{d}\lambda + \int \vert 1-S_{1} \vert \rho (\lambda |x',\;y,\;z)\, \textrm{d}\lambda + \int \vert S_{1}-1 \vert \rho (\lambda |x',\;y,\;z')\, \textrm{d}\lambda +\int \vert 1-S_{2} \vert \rho (\lambda |x,\;y,\;z)\, \textrm{d}\lambda + \int \vert 1-S_{2} \vert \rho (\lambda |x',\;y,\;z')\, \textrm{d}\lambda .$

= $T_{1} + T_{2}.$

where $T_{1} = \int \vert S_{1} \vert \,\vert \rho (\lambda |x',\;y,\;z)-\rho (\lambda |x,\;y,\;z) \vert \,\textrm{d}\lambda + \int \vert S_{1} \vert \,\vert \rho (\lambda |x,\;y,\;z')-\rho (\lambda |x',\;y,\;z') \vert \,\textrm{d}\lambda + \int \vert S_{2} \vert \,\vert \rho (\lambda |x,\;y,\;z)+\rho (\lambda |x,\;y',\;z) \vert \,\textrm{d}\lambda + \int \vert S_{2} \vert \,\vert \rho (\lambda |x',\;y,\;z')-\rho (\lambda |x',\;y',\;z') \vert \,\textrm{d}\lambda ,$

=$2 + M_{1} + M_{2}$

and $T_{2} = \int \vert 1-S_{1} \vert \rho (\lambda |x',\;y,\;z)\, \textrm{d}\lambda + \int \vert S_{1}-1 \vert \rho (\lambda |x',\;y,\;z')\, \textrm{d}\lambda +\int \vert 1-S_{2} \vert \rho (\lambda |x,\;y,\;z)\, \textrm{d}\lambda + \int \vert 1-S_{2} \vert \rho (\lambda |x',\;y,\;z')\, \textrm{d}\lambda .$

Case 1 Let us assume $\lambda$ such that both $S_{1}$ and $S_{2}$ becomes positive, then it results $T_{2} = 0.$

Case 2 On the contrary, if $\lambda$ is arbitrarily chosen, then $S_{1}$ and $S_{2}$ can take +1 or −1 in any occurrence of $\lambda$. In such a scenario, $\vert 1-S_{1/2} \vert$ will produce 0 or 2, and consequently, $T_{2}$ will range between (0,8).

Hence, from the above analysis, an algebraic bound which can saturate Mermin inequality is possible, when $T_{2}$ reduce to 0, therefore

$$\begin{aligned} \vert S_{\textit{M}}\vert \le 2 + M_{2} + 2M_{1}. \end{aligned}$$

(A2)

Following the same arguments and reordering the terms of $S_{\textit{M}}$, we have,

$$\begin{aligned} \vert S_{\textit{M}} \vert \le 2 + 2 M_{1} + M_{3}. \end{aligned}$$

(A3)

Again we can rearrange the terms of $S_{\textit{M}}$ such that we have the following structure

$S_{\textit{M}} = \int [\rho (\lambda |x',\;y,\;z)-\rho (\lambda |x',\;y,\;z')\;\frac{w(C_{z'},\;\lambda )}{w(C_{z},\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\textrm{d}\lambda +\int [\rho (\lambda |x',\;y,\;z')-\rho (\lambda |x',\;y',\;z')\;\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\textrm{d}\lambda$ +$\int [\rho (\lambda |x,\;y',\;z)-\rho (\lambda |x,\;y',\;z')\;\frac{w(C_{z'},\;\lambda )}{w(C_z,\;\lambda )}]\;u(A_x,\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z},\;\lambda )\;\textrm{d}\lambda$ +$\int {[\rho (\lambda |x,\;y',\;z')+\rho (\lambda |x,\;y,\;z')\;\frac{v(B_{y},\;\lambda )}{v(B_{y'},\;\lambda )}]\;u(A_{x},\;\lambda )\;v(B_{y'},\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }.$

Applying the same procedure as used in above paragraph, we have the saturated bound

$$\begin{aligned} \vert S_{\textit{M}} \vert \le 2 + 2 M_{3} + M_{2}. \end{aligned}$$

(A4)

Reshuffling the terms in above expression, it ensue

$S_{\textit{M}}$ = $\int [\rho (\lambda |x',\;y,\;z)-\rho (\lambda |x',\;y,\;z')\;\frac{w(C_{z'},\;\lambda )}{w(C_{z},\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\textrm{d}\lambda$ +$\int {[\rho (\lambda |x',\;y,\;z')+\rho (\lambda |x,\;y,\;z')\;\frac{u(A_{x},\;\lambda )}{u(A_{x'},\;\lambda )}]\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$ +$\int {[\rho (\lambda |x,\;y',\;z)-\rho (\lambda |x,\;y',\;z')\;\frac{w(C_{z'},\;\lambda )}{w(C_z,\;\lambda )}]\;u(A_x,\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z},\;\lambda )\;\textrm{d}\lambda }$ + $\int {[\rho (\lambda |x,\;y',\;z')-\rho (\lambda |x',\;y',\;z')\;\frac{u(A_{x'},\;\lambda )}{u(A_{x},\;\lambda )}]\;u(A_{x},\;\lambda )\;v(B_{y'},\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }.$

Consequently, by the same reasoning as before the modified bound is

$$\begin{aligned} \vert S_{\textit{M}} \vert \le 2 + 2 M_{3} + M_{1}. \end{aligned}$$

(A5)

Since $\vert S_{\textit{M}} \vert$ is less than or equal to the right-hand side of Eqs. (A2), (A3), (A4) and (A5), then it must be upper bounded by the minimum of these bounds, i.e.,

$$\begin{aligned} \vert S_{\textit{M}} \vert \le 2 + \lbrace 2M_{1}+ \min ( M_{2}, M_{3} ),\; 2M_{3}+ \min ( M_{2}, M_{1}) \rbrace . \end{aligned}$$

(A6)

Meanwhile, $\vert S_{\textit{M}}\vert \le 4$ [52, 53], therefore we arrive at

$$\begin{aligned} \vert S_{\textit{M}} \vert \le 2 + \lbrace 2M_{1}+ \min ( M_{2}, M_{3} ),\; 2M_{3}+ \min ( M_{2}, M_{1}),\;2 \rbrace . \end{aligned}$$

(A7)

Clearly, the formalism make no distinction between the observers detectors, we can carry out similar set of manipulation, reversing the arrangement of inputs for the parties, we finally obtain,

$$\begin{aligned} \vert S_{\textit{M}} \vert \le 2+ \min \lbrace 2M_{1}+ \min ( M_{2}, M_{3} ),\; 2M_{3}+ \min ( M_{2}, M_{1}),\;2M_{2}+ \min ( M_{1}, M_{3} ), \;2 \rbrace . \end{aligned}$$

(A8)

Meanwhile, it is worth mentioning that the above bounds may not tight, as we have gathered extra inputs in formulating the modified bounds.

Appendix B: relaxed Svetlichny inequality

Here our motivation is to establish the effect of measurement dependence in one-sided measurement settings for Svetlichny inequality [41], which has the following representation,

$$\begin{aligned} \vert S_{\textit{S}} \vert= & {} \langle A_0B_0C_0 \rangle +\langle A_0B_0C_1 \rangle +\langle A_1B_0C_0 \rangle -\langle A_1B_0C_1\rangle +\langle A_0 B_1C_0 \rangle - \langle A_0 B_1C_1 \rangle \nonumber \\{} & {} -\langle A_1B_1C_0 \rangle -\langle A_1B_1C_1 \rangle \vert \le 4. \end{aligned}$$

(B1)

Therefore, $S_{\textit{S}} = \langle A_0 B_0 C_0 \rangle +\langle A_0 B_0 C_1 \rangle$+$\langle A_1 B_0 C_0 \rangle - \langle A_1 B_0 C_1 \rangle +\langle A_0 B_1 C_0 \rangle - \langle A_0 B_1 C_1 \rangle - \langle A_1 B_1 C_0 \rangle - \langle A_1 B_1 C_1 \rangle$.

where $\langle A_0 B_1 C_0 \rangle =\int {\rho (\lambda |x,\;y',\;z)\;u(A_x,\;B_{y'},\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda }$, with $-1\le u(A_x,\;B_{y'},\;\lambda )\le 1;\;-1\le v(C_z,\lambda ) \le 1$,

and similarly we define other terms of $S_{\textit{S}}$.

Let us assume the three parties perform their spin measurements along the directions $A_{\hat{m}}$, $B_{\hat{m}}$ and $C_{\hat{m}}$, respectively, where the measurement direction for the party X ($= A, \,B,\,C$) is denoted as $X_{\hat{m}}\; \equiv \, (\sin \theta _{X_{\hat{m}}} \cos \phi _{X_{\hat{m}}}, \sin \theta _{X_{\hat{m}}}\sin \phi _{X_{\hat{m}}}, \cos \theta _{X_{\hat{m}}})$, with $\theta _{X_{\hat{m}}} \in [0, \pi ]$ and $\phi _{X_{\hat{m}}} \in [0, 2\pi ]$. Here also following the footprint of [19, 21], we have derived an modified form of Svetlichny inequality in a deterministic no-signaling scenario.

Therefore

$S_{\textit{S}} =\int {\rho (\lambda |x,\;y,\;z)\;u(A_x,\;B_y,\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda }+\int \rho (\lambda |x',\;y,\;z)\;u(A_{x'},\;B_y,\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda$ +$\int {\rho (\lambda |x,\;y,\;z')\;u(A_x,\;B_y,\;\lambda )\;v(C_{z'},\lambda )\;\textrm{d}\lambda }$ - $\int \rho (\lambda |x',\;y,\;z')\;u(A_{x'}\;B_y,\;\lambda )\;v(C_{z'},\;\lambda )\;\textrm{d}\lambda$ +$\int {\rho (\lambda |x,\;y',\;z)\;u(A_x,\;B_{y'},\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda } - \int {\rho (\lambda |x',\;y',\;z)\;u(A_{x'},\;B_{y'},\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda }$ − $\int \rho (\lambda |x,\;y',\;z')\;u(A_x,\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )\;\textrm{d}\lambda$ - $\int {\rho (\lambda |x',\;y',\;z')\;u(A_{x'},\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

= $\int {u(A_x,\;B_y,\;\lambda )\;v(C_z,\;\lambda )\;[\rho (\lambda |x,\;y,\;z)+\rho (\lambda |x',\;y,\;z)\frac{u(A_{x'},\;B_y,\;\lambda )}{u(A_x,\;B_y,\;\lambda )}]\;\textrm{d}\lambda }$ +$\int u(A_x,\;B_y,\;\lambda )\;v(C_{z'},\lambda )\;[\rho (\lambda |x,\;y,\;z')-\rho (\lambda |x',\;y,\;z')\frac{u(A_{x'},\;B_y,\;\lambda )}{u(A_x,\;B_y,\;\lambda )}]\;\textrm{d}\lambda$ +$\int {u(A_x,\;B_{y'},\;\lambda )\;v(C_z,\;\lambda )\;[\rho (\lambda |x,\;y',\;z)-\rho (\lambda |x',\;y',\;z)\;\frac{u(A_{x'},\;B_{y'},\;\lambda )}{u(A_x,\;B_{y'},\;\lambda )}]\;\textrm{d}\lambda }$ − $\int {u(A_x,\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )\;[\rho (\lambda |x,\;y',\;z')+\rho (\lambda |x',\;y',\;z')\;\frac{u(A_{x'},\;B_{y'},\;\lambda )}{u(A_x,\;B_{y'},\;\lambda )}]\;\textrm{d}\lambda }$.

Since we have ascribed measurement dependence on Alice measurement settings, hence the response function $\frac{u(A_{x'},\;B_y,\;\lambda )}{u(A_x,\;B_y,\;\lambda )}$ and $\frac{u(A_{x'},\;B_{y'},\;\lambda )}{u(A_x,\;B_{y'},\;\lambda )}$ are function of the hidden variable $\lambda$.

We set, $\frac{ u(A_{x'},\;B_y,\;\lambda )}{u(A_x,\;B_y,\;\lambda )} = S_{1}(\rho (\lambda \vert x,\;y,\;z)-\rho (\lambda \vert x',\;y,\;z))$ and $\frac{u(A_{x'},\;B_{y'},\;\lambda )}{u(A_x,\;B_{y'},\;\lambda )} = S_{2}(\rho (\lambda \vert x,\;y'\;z')-\rho (\lambda \vert x',\;y',\;z'))$,.

$\therefore$ $|S_{S}|$ $\le$ $\int \vert {u(A_x,\;B_y,\;\lambda )\;v(C_z,\;\lambda )\vert \,\vert [\rho (\lambda |x,\;y,\;z)+ S_{1}\;\rho (\lambda |x',\;y,\;z)]\vert \,\textrm{d}\lambda }$ + $\int \vert {u(A_x,\;B_y,\;\lambda )\;v(C_{z'},\;\lambda )\vert \,\vert [\rho (\lambda |x,\;y,\;z')- S_{1}\;\rho (\lambda |x',\;y,\;z')]\vert \,\textrm{d}\lambda }$+$\int \vert u(A_x,\;B_{y'},\lambda )\;v(C_z,\;\lambda )\vert \,\vert [\rho (\lambda |x,\;y',\;z)-S_{2}\;\rho (\lambda |x',\;y',\;z)]\vert \,\textrm{d}\lambda + \int \vert u(A_x,\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )\vert \vert [\rho (\lambda |x,\;y',\;z')+S_{2}\;\rho (\lambda |x',\;y',\;z')]\vert \, \textrm{d}\lambda$.

$\le \int \vert \rho (\lambda |x,\;y,\;z)-S_{1} \rho (\lambda |x,\;y,\;z)+S_{1} \rho (\lambda |x,\;y,\;z)+S_{1} \rho (\lambda |x',\;y,\;z) \vert \textrm{d}\lambda + \int \vert \rho (\lambda |x,\;y,\;z')-S_{1} \rho (\lambda |x,\;y,\;z')+S_{1} \rho (\lambda |x,\;y,\;z')-S_{1} \rho (\lambda |x',\;y,\;z') \vert \textrm{d}\lambda + \int \vert \rho (\lambda |x,\;y',\;z)-S_{2} \rho (\lambda |x,\;y',\;z)+S_{2} \rho (\lambda |x,\;y',\;z)-S_{2} \rho (\lambda |x',\;y',\;z) \vert \textrm{d}\lambda + \int \vert \rho (\lambda |x,\;y',\;z')-S_{2} \rho (\lambda |x,\;y',\;z')+S_{2} \rho (\lambda |x,\;y',\;z')+S_{2} \rho (\lambda |x',\;y',\;z') \vert \textrm{d}\lambda .$

$\le \int \vert S_{1} \vert \vert \rho (\lambda |x,\;y,\;z)+ \rho (\lambda |x',\;y,\;z) \vert \textrm{d}\lambda + \int \vert S_{1} \vert \vert \rho (\lambda |x,\;y,\;z')- \rho (\lambda |x',\;y,\;z') \vert \textrm{d}\lambda + \int \vert S_{2} \vert \vert \rho (\lambda |x,\;y',\;z)- \rho (\lambda |x',\;y',\;z) \vert \textrm{d}\lambda + \int \vert S_{2} \vert \vert \rho (\lambda |x,\;y',\;z')+ \rho (\lambda |x',\;y',\;z') \vert \textrm{d}\lambda + \int \vert 1-S_{1} \vert \, [\vert \rho (\lambda |x,\;y,\;z) \vert + \vert \rho (\lambda |x,\;y,\;z') \vert ]\, \textrm{d}\lambda + \int \vert 1-S_{2} \vert \,[\vert \rho (\lambda |x,\;y',\;z) \vert + \vert \rho (\lambda |x,\;y',\;z') \vert ]\, \textrm{d}\lambda .$

=$T_{1} + T_{2}.$

where $T_{1} = \int \vert S_{1} \vert \vert \rho (\lambda |x,\;y,\;z)+ \rho (\lambda |x',\;y,\;z) \vert \textrm{d}\lambda + \int \vert S_{1} \vert \vert \rho (\lambda |x,\;y,\;z')- \rho (\lambda |x',\;y,\;z') \vert \textrm{d}\lambda + \int \vert S_{2} \vert \vert \rho (\lambda |x,\;y',\;z)- \rho (\lambda |x',\;y',\;z) \vert \textrm{d}\lambda + \int \vert S_{2} \vert \vert \rho (\lambda |x,\;y',\;z')+ \rho (\lambda |x',\;y',\;z') \vert \textrm{d}\lambda .$ $= 4+2M_{1},$

and $T_{2} = \int \vert 1-S_{1} \vert \, [\vert \rho (\lambda |x,\;y,\;z) \vert + \vert \rho (\lambda |x,\;y,\;z') \vert ]\, \textrm{d}\lambda + \int \vert 1-S_{2} \vert \,[\vert \rho (\lambda |x,\;y',\;z) \vert + \vert \rho (\lambda |x,\;y',\;z') \vert ]\, \textrm{d}\lambda .$

Case 1 Let us consider $\lambda$ such that both $S_{1}$ and $S_{2}$ becomes positive, then it results $T_{2} = 0.$

Case 2 However, if $\lambda$ is arbitrarily chosen, then $S_{1}$ and $S_{2}$ can take +1 or -1 in any occurrence of $\lambda$. In such a scenario, $\vert 1-S_{1/2} \vert$ will produce 0 or 2, and consequently, $T_{2}$ will range between (0,8).

Hence, from the above analysis, an algebraic bound which can saturate Svetlichny inequality is possible, when $T_{2}$ reduce to 0; therefore, we have

$$\begin{aligned} |S_{\textit{S}}| \le 4+2 M_{1}. \end{aligned}$$

(B2)

However, since the formalism make no distinction between the observers detectors, we can carry out similar set of manipulation, and aiming at Bob and Charlie’s measurement settings, it provides accordingly,

$$\begin{aligned}{} & {} |S_{\textit{S}}| \le 4+2 M_{2}. \end{aligned}$$

(B3)

$$\begin{aligned}{} & {} |S_{\textit{S}}| \le 4+2 M_{3}. \end{aligned}$$

(B4)

Finally, as $\vert S_{\textit{S}} \vert$ is less than or equal to the right-hand sides of Eqs. (B2), (B3) and (B4), hence it must be bounded above by the minimum of these bounds i.e.,

$$\begin{aligned} |S_{\textit{S}}| \le 4 + 2\min \lbrace M_{1},\, M_{2},\, M_{3} \rbrace . \end{aligned}$$

(B5)

Appendix C: relaxed NS$_{2}$ inequality

Already we have seen the effect of arbitrary one-sided measurement dependence for Mermin and Svetlichny inequality in last two subsections. Here we have drive modified $NS_{2}$ inequality, with the ideas form [19, 21]. The $NS_{2}$ inequality [43] has the following form,

$$\begin{aligned} \vert S_{NS}\vert = |\langle A_0B_0 \rangle +\langle A_0C_0 \rangle +\langle B_0C_1 \rangle -\langle A_1B_1C_0\rangle +\langle A_1B_1C_1\rangle |\le 3. \end{aligned}$$

(C1)

where $\langle A_0 B_0 C_0 \rangle$ = $\int {\rho (\lambda |x,\;y,\;z)\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }$.

$\langle A_1 B_1 C_1 \rangle$ = $\int {\rho (\lambda |x',\;y',\;z')\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

Therefore the quantity $S_{NS}$ is

$S_{NS} =\langle A_0 B_0 \rangle + \langle A_0 C_0 \rangle +\langle B_0 C_1 \rangle - \langle A_1 B_1 C_0 \rangle +\langle A_1 B_1 C_1 \rangle$.

= $\int {\rho (\lambda |x,\;y)\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |x,\;z)\;u(A_x,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int \rho (\lambda |y,\;z')\;v(B_y,\;\lambda )\;w(C_{z'},\lambda )\;\textrm{d}\lambda$ - $\int \rho (\lambda |x',\;y',\;z)\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_z,\lambda )\;\textrm{d}\lambda +\int {\rho (\lambda |x',\;y',\;z')\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

= $\int {\rho (\lambda |x,\;y)\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |x,\;z)\;u(A_x,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int \rho (\lambda |y,\;z')\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$ - $\int \rho (\lambda |x',\;y',\;z)\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_z,\lambda )\;\textrm{d}\lambda +\int {\rho (\lambda \vert x',\;y',\;z')\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\lambda )\; \textrm{d}\lambda }$.

= $\int {\rho (\lambda |x,\;y)\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |x,\;z)\;u(A_x,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |y,\;z')\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$ + $\int [\rho (\lambda |x',\;y',\;z')-\rho (\lambda |x',\;y',\;z)\;\frac{w(C_z,\;\lambda )}{w(C_{z'},\;\lambda )}]u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$.

Considering the hidden variable $\lambda$, we set the response function $\frac{w(C_z,\;\lambda )}{w(C_{z'},\;\lambda )}$ identically as Appendix A. The quantity $\frac{w(C_z,\;\lambda )}{w(C_{z'},\;\lambda )}$ = $S_{3} ( \rho (\lambda \vert x',\,y',\,z)-\rho (\lambda \vert x',\,y',\,z')$).

Hence, $\vert S_{\textit{NS}}\vert \le$ $\int \vert {\rho (\lambda |x,\;y)\vert \;\textrm{d}\lambda }+\int \vert {\rho (\lambda |x,\;z)\vert \;d\lambda }$ + $\int \vert {\rho (\lambda |y,\;z'))\vert \;\textrm{d}\lambda }$ + $\int \vert {[\rho (\lambda |x',\;y',\;z')-S_{3}\,\rho (\lambda |x',\;y',\;z)]\vert \;\textrm{d}\lambda }\;\;$.

Therefore, the tight bound of NS$_{2}$ inequality is given by

$$\begin{aligned} |S_{\textit{NS}}|\le 3+M_{3}. \end{aligned}$$

(C2)

However, the terms of $S_{\textit{NS}}$ are rearranged so that we obtain

$S_{\textit{NS}} = \int {\rho (\lambda |x,\;y)\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |x,\;z)\;u(A_x,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |y,\;z')\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$ + $\int [\rho (\lambda |x',\;y',\;z')-\rho (\lambda |x,\;y',\;z')\;\frac{u(A_x,\;\lambda )}{u(A_{x'},\;\lambda )}]u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$. + $\int [\rho (\lambda |x',\;y',\;z)-\rho (\lambda |x,\;y',\;z)\;\frac{u(A_{x},\;\lambda )}{u(A_{x'},\;\lambda )}]u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z},\;\lambda )\;\textrm{d}\lambda$ + $\int [\rho (\lambda |x,\;y',\;z')-\rho (\lambda |x,\;y',\;z)\;\frac{w(C_{z},\;\lambda )}{w(C_{z'},\;\lambda )}]u(A_{x},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$

Therefore, in a scenario where Alice or Charlie’s measurement settings depend on the hidden variable $\lambda$, we set $\frac{u(A_x,\;\lambda )}{u(A_{x'},\;\lambda )} = S_{1} ( \rho (\lambda \vert x,\;y',\;z)-\rho (\lambda \vert x',\;y',\;z)$ and $\frac{w(C_{z},\;\lambda )}{w(C_{z'},\;\lambda )} = S_{3} ( \rho (\lambda \vert x',\,y',\,z)-\rho (\lambda \vert x',\,y',\,z')$.

Ergo, $S_{\textit{NS}}$ = $\int {\rho (\lambda |x,\;y)\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;\textrm{d}\lambda }+\int \rho (\lambda |x,\;z)\;u(A_x,\;\lambda )\;w(C_z,\;\lambda )\textrm{d}\lambda +\int {\rho (\lambda |y,\;z')\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$ + $\int [\rho (\lambda |x',\;y',\;z')- S_{1}\, \rho (\lambda |x,\;y',\;z')]u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda$.+ $\int [\rho (\lambda |x,\;y',\;z)-S_{1} \rho (\lambda |x',\;y',\;z)\;u(A_{x},\;\lambda )v(B_{y'},\;\lambda )\;w(C_{z},\;\lambda )\;\textrm{d}\lambda$ +$\int [\rho (\lambda |x,\;y',\;z')- S_{3}\, \rho (\lambda |x,\;y',\;z)]\;u(A_{x},\;\lambda )\;v(B_{y'},\;\lambda )w(C_{z'},\;\lambda )\;\textrm{d}\lambda$

According as $S_{1}$ and $S_{3}$ are the outcomes depends on $\lambda$, we are inquiring only such bound which can be saturated the NS$_{2}$ bound for a given $\lambda$. Hence, the average value reduce to

$$\begin{aligned} |S_{\textit{NS}}|\le 3+ 2M_{1} + M_{3}. \end{aligned}$$

(C3)

Clearly, this formalism makes no difference between the observers detectors (Alice and Bob), we can carry out similar set of manipulation, reversing the treatment of inputs of the parties, we obtain

$$\begin{aligned} |S_{\textit{NS}}|\le 3+ 2M_{2} + M_{3}. \end{aligned}$$

(C4)

According as $\vert S_{\textit{NS}} \vert$ is less than or equal to the right-hand sides of Eqs. (C2), (C3) and (C4), hence it must be bounded above by the minimum of these bounds, i.e.,

$$\begin{aligned} |S_{\textit{NS}}|\le 3+ \min \lbrace M_{3},\, 2M_{1} + M_{3},\, 2M_{2} + M_{3} \rbrace . \end{aligned}$$

(C5)

Consequently,

$$\begin{aligned} |S_{\textit{NS}}|\le 3+ M_{3}. \end{aligned}$$

(C6)

Appendix D:BI-party relaxation of Mermin inequality

In this section we investigate the effect of arbitrary measurement relaxation [17, 18, 21] in any two-party measurement setting in the same tripartite measurement scenario for Mermin inequality[42].

The Mermin inequality (MI) is

|$\langle A_1 B_0 C_0\rangle +\langle A_0 B_1 C_0\rangle + \langle A_0 B_0 C_1\rangle -\langle A_1 B_1 C_1 \rangle$| $\le$ 2.

We use the same notations as used in Sect. (3.1). Here we follow the same formalism that has been used in (A)

Consider, $S_{M}$ = $\langle A_1 B_0 C_0 \rangle +\langle A_0 B_1 C_0 \rangle +\langle A_0 B_0 C_1 \rangle$ − $\langle A_1 B_1 C_1 \rangle$.

= $\int {\rho (\lambda |x',\;y,\;z)\;u(A_{x'},\;\lambda )\;v(B_y,\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda }+\int \rho (\lambda |x,\;y',\;z)\;u(A_x,\;\lambda )v(B_{y'},\;\lambda )\;w(C_z,\;\lambda )\;\textrm{d}\lambda$ +$\int {\rho (\lambda |x,\;y,\;z')\;u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda } - \int {\rho (\lambda |x',\;y',\;z')\;u(A_{x'},\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

= $\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y,\;z') - \rho (\lambda \vert x',\;y',\;z')\; \frac{u(A_{x'},\;\lambda )}{u(A_x,\;\lambda )}\,\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )}]\; u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )$

+$\int \textrm{d}\lambda \;[\rho (\lambda \vert x,\;y',\;z)+\rho (\lambda \vert x',\;y,\;z) \,\frac{u(A_{x'},\;\lambda )}{u(A_x,\;\lambda )}\,\frac{v(B_y,\;\lambda )}{v(B_{y'},\lambda )}]\; u(A_x,\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_z,\;\lambda )$.

Since the outcomes depends on the variables $\lambda$ in a measurement dependence scenario, henceforth we set, $\frac{u(A_{x'},\;\lambda )}{u(A_x,\;\lambda )} = S_{1}(\rho ( \lambda \vert x,\,y,\,z') - \rho ( \lambda \vert x',\,y',\,z'))$ and $\frac{v(B_{y'},\;\lambda )}{v(B_y,\;\lambda )} = S_{2}(\rho ( \lambda \vert x',\,y,\,z') - \rho ( \lambda \vert x,\,y',\,z'))$

Hence, $S_{M}$ = $\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y,\;z') - S_{1}\, S_{2}\, \rho (\lambda \vert x',\;y',\;z')]\; u(A_x,\;\lambda )\;v(B_y,\;\lambda )\;w(C_{z'},\;\lambda )$

+$\int \textrm{d}\lambda \;[\rho (\lambda \vert x,\;y',\;z)+ \frac{S_{1}}{S_{2}} \,\rho (\lambda \vert x',\;y,\;z)]\, u(A_x,\;\lambda )\;v(B_{y'},\;\lambda )\;w(C_z,\;\lambda )$.

We are inquiring such bounds which can saturate the MI, consequently we have

$$\begin{aligned} \vert S_{\textit{M}}\vert \le 2+ M_{12}. \end{aligned}$$

(D1)

However, since the formalism makes no distinction between the observers detectors, we can carry out similar set of manipulation, reversing the treatment of inputs of the parties, we obtain

$$\begin{aligned}{} & {} \vert S_{\textit{M}}\vert \le 2+ M_{13} \end{aligned}$$

(D2)

$$\begin{aligned}{} & {} \vert S_{\textit{M}}\vert \le 2+ M_{23} \end{aligned}$$

(D3)

Finally, as $\vert S_{\textit{M}} \vert$ is less than or equal to the right-hand sides of Eqs. (D1), (D2) and (D3), hence it must be bounded above by the minimum of these bounds, i.e.,

$$\begin{aligned} \vert S\vert \le 2+ \min \lbrace M_{12},\; M_{23},\; M_{13} \rbrace . \end{aligned}$$

(D4)

Appendix E:BI-party relaxation of Svetlichny inequality

In this last appendix we have founded the modified Svetlichny inequality, when any two parties have restricted measurement settings in a tripartite non-locality scenario. The required inequality is known as [41]

$\vert S_{\textit{S}} \vert$ = |$\langle A_0 B_0 C_0 \rangle - \langle A_1 B_1 C_0 \rangle +\langle A_0 B_0 C_1 \rangle - \langle A_0 B_1 C_1 \rangle$+$\langle A_1 B_0 C_0 \rangle +\langle A_0 B_1 C_0 \rangle - \langle A_1 B_0 C_1 \rangle - \langle A_1 B_1 C_1 \rangle$| $\le$ 4.

Consequently, $S_{\textit{S}}$ = $\langle A_0 B_0 C_0 \rangle +\langle A_0 B_0 C_1 \rangle$+$\langle A_1 B_0 C_0 \rangle - \langle A_1 B_0 C_1 \rangle +\langle A_0 B_1 C_0 \rangle - \langle A_0 B_1 C_1 \rangle - \langle A_1 B_1 C_0 \rangle - \langle A_1 B_1 C_1 \rangle$.

Here, we have used the same formalism as used in one-sided measurement dependence scenario for Svetlichny inequality in (B), and applying the ideas of [19, 21] we obtain the following simplification,

$S_{\textit{S}}$ = $\int {\rho (\lambda |x,\;y,\;z)\;u(A_x,\;B_y,\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda } - \int \rho (\lambda |x',\;y',\;z)\;u(A_{x'},\;B_{y'},\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda$ +$\int {\rho (\lambda |x,\;y,\;z')\;u(A_x,\;B_y,\;\lambda )\;v(C_{z'},\;\lambda )\;\textrm{d}\lambda }$ − $\int \rho (\lambda |x,\;y',\;z')\;u(A_x,\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )\;\textrm{d}\lambda$ +$\int {\rho (\lambda |x',\;y,\;z)\;u(A_{x'},\;B_y,\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda }+\int {\rho (\lambda |x,\;y',\;z)\;u(A_x,\;B_{y'},\;\lambda )\;v(C_z,\;\lambda )\;\textrm{d}\lambda }$ - $\int \rho (\lambda |x',\;y,\;z')\;u(A_{x'},\;B_y,\;\lambda )\;v(C_{z'},\lambda )\;\textrm{d}\lambda - \int {\rho (\lambda |x',\;y',\;z')\;u(A_{x'},\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )\;\textrm{d}\lambda }$.

= $\int \textrm{d}\lambda \;[\rho (\lambda \vert x,\;y,\;z) - \rho (\lambda \vert x',\;y',\;z) \;\frac{u(A_{x'},\;B_{y'},\;\lambda )}{u(A_x,\;B_y,\;\lambda )}]\; u(A_x,\;B_y,\;\lambda )\;v(C_z,\;\lambda )$ +$\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y,\;z') - \rho (\lambda \vert x',\;y',\;z')\; \frac{u(A_{x'},\;B_{y'},\;\lambda )}{u(A_x,\;B_y,\;\lambda )}] u(A_x,\;B_y,\;\lambda )\;v(C_{z'},\;\lambda )$ - $\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y',\;z')+\rho (\lambda \vert x',\;y,\;z') \;\frac{u(A_{x'},\;B_y,\;\lambda )}{u(A_x,\;B_{y'},\;\lambda )}] \;u(A_x,\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )$ +$\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y',\;z)+\rho (\lambda \vert x',\;y,\;z)\; \frac{u(A_x',\;B_{y},\;\lambda )}{u(A_{x},\;B_y',\;\lambda )}]\; u(A_{x},\;B_y',\;\lambda )\;v(C_z,\;\lambda )$.

Since in a measurement dependence scenario, the outcomes depends on the variables $\lambda$ henceforth we set $\frac{u(A_{x'},\,B_{y'},\,\lambda )}{u(A_x,\,B_y,\,\lambda )}$ = $S_{1}(\rho (\lambda \vert x,\,y,\,z')-\rho (\lambda \vert x',\,y',\,z'))$ and $\frac{u(A_{x'},\,B_y,\,\lambda )}{u(A_x,\,B_{y'},\,\lambda )}$ = $S_{2}(\rho ( \lambda \vert x',\,y,\,z) - \rho ( \lambda \vert x,\,y',\,z))$.

Therefore, $S_{S}$ = $\int \textrm{d}\lambda \;[\rho (\lambda \vert x,\;y,\;z) - S_{1}\,\rho (\lambda \vert x',\;y',\;z)]\; u(A_x,\;B_y,\;\lambda )\;v(C_z,\;\lambda )$ +$\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y,\;z') - S_{1}\, \rho (\lambda \vert x',\;y',\;z')] u(A_x,\;B_y,\;\lambda )\;v(C_{z'},\;\lambda )$ - $\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y',\;z')+S_{2}\, \rho (\lambda \vert x',\;y,\;z')] \;u(A_x,\;B_{y'},\;\lambda )\;v(C_{z'},\;\lambda )$ +$\int \textrm{d}\lambda \; [\rho (\lambda \vert x,\;y',\;z)+ S_{2}\, \rho (\lambda \vert x',\;y,\;z)]\; u(A_{x},\;B_y',\;\lambda )\;v(C_z,\;\lambda )$.

We are exploring such bounds which can saturate the SI; consequently we have,

$$\begin{aligned} \vert S_{\textit{S}}\vert \le 4+2 M_{12}. \end{aligned}$$

(E1)

However, since the formalism makes no distinction between the observers detectors, we execute similar set of manipulation, altering the behavior of inputs of the parties, we attain

$$\begin{aligned}{} & {} \vert S_{\textit{S}}\vert \le 4+2 M_{13}. \end{aligned}$$

(E2)

$$\begin{aligned}{} & {} \vert S_{\textit{S}}\vert \le 4+2 M_{23}. \end{aligned}$$

(E3)

Finally, as $\vert S_{\textit{S}} \vert$ is less than or equal to the right-hand sides of Eqs. (E1), (E2) and (E3), hence it must be upper bounded by the minimum of these items, i.e.,

$$\begin{aligned} \vert S_{\textit{S}}\vert \le 4 + 2\min \lbrace M_{12},\;M_{23},\;M_{13}\rbrace . \end{aligned}$$

(E4)