# An elegant proof of self-testing for multipartite Bell inequalities

Ekta Panwar<EMAIL_ADDRESS>Institute of Theoretical Physics
and Astrophysics, Faculty of Mathematics, Physics and Informatics, University
of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland  Palash Pandya
Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics,
Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308
Gdańsk, Poland  Marcin Wieśniak  Institute of Theoretical Physics and
Astrophysics, Faculty of Mathematics, Physics and Informatics, University of
Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland  International Centre for
Theory of Quantum Technologies, University of Gdańsk, ul. Wita Stwosza 63,
80-308 Gdańsk, Poland

###### Abstract

The predictions of quantum theory are incompatible with local-causal
explanations. This phenomenon is called Bell non-locality and is witnessed by
violation of Bell-inequalities. The maximal violation of certain Bell-
inequalities can only be attained in an essentially unique manner. This
feature is referred to as _self-testing_ and constitutes the most accurate
form of certification of quantum devices. While self-testing in bipartite Bell
scenarios has been thoroughly studied, self-testing in the more complex
multipartite Bell scenarios remains largely unexplored. This work presents a
simple and broadly applicable self-testing argument for $N$-partite
correlation Bell inequalities with two binary outcome observables per party.
Our proof technique forms a generalization of the Mayer-Yao formulation and is
not restricted to linear Bell-inequalities, unlike the usual _sum of squares_
method. To showcase the versatility of our proof technique, we obtain self-
testing statements for $N$ party Mermin-Ardehali-Belinskii-Klyshko (MABK) and
Werner-Wolf-Weinfurter-Żukowski-Brukner (WWWŻB) family of linear Bell
inequalities, and Uffink’s family of $N$ party quadratic Bell-inequalities.

##  I Introduction

One of the most striking features of quantum theory is the deviation of its
predictions for Bell experiments from the predictions of classical theories
with local casual (hidden variable) explanations Bell (1964). This phenomenon
is captured by quantum violation of statistical inequalities, which are
satisfied by all local realistic theories, referred to as Bell-inequalities
Brunner et al. (2014). Experimental demonstrations of the loophole-free (e.g.,
Hensen et al. (2015); Giustina et al. (2015); Shalm et al. (2015)) violation
of such inequalities implies the possibility of sharing intrinsically random
private numbers among an arbitrary number of spatially separated parties which
power unconditionally secure private key distribution schemes (for more
information on Device-independent quantum cryptography see Vazirani and Vidick
(2014); Acín et al. (2006); Colbeck (2011); Ekert and Renner (2014); Ekert
(1991)). Such applications of Bell non-locality follow from the fact that the
extent of Bell inequality violation can uniquely identify the specific
entangled quantum states and measurements, a phenomenon referred to as self-
testing (see Mayers and Yao (1998, 2004) for initial contributions and the
recent review of the progress till now see Šupić and Bowles (2020)).

Self-testing statements are the most accurate form of certifications for
quantum systems. Self-testing schemes allow us to infer the underlying physics
of a quantum experiment, i.e., the state and the measurements (up to local
isometry) without any characterization of the internal workings of the
measurement devices, and based only on the observed statistics, i.e., treating
the measurement devices as black boxes with classical inputs and outputs.
Self-testing has found many application in several areas like device-
independent randomness generation Colbeck and Kent (2011); Colbeck (2011),
quantum cryptography Ekert and Renner (2014), entanglement detection Gheorghiu
et al. (2019), delegated quantum computing  Bowles et al. (2018); Sekatski et
al. (2018). While self-testing in the bipartite Bell scenarios has been
thoroughly studied, self-testing in more complex multipartite Bell scenarios
(see FIG. 1) remains largely unexplored.

In a multipartite setting, self-testing has been demonstrated for graph states
using stabilizer operators McKague (2014). Self-testing of multipartite graph
states and partially entangled _Greenberger-Horne-Zeilinger_ (GHZ) states has
been demonstrated using a stabilizer-based approach, and Bell inequalities
explicitly constructed for the state Baccari et al. (2020); Greenberger et al.
(2007). In general, multipartite Bell inequalities explicitly tailored for
self-testing of a multipartite entangled state can be obtained using linear
programming Pál et al. (2014); Bancal et al. (2015); Yang et al. (2014);
Sekatski et al. (2018). However, the self-testing statements must be verified
using the numerical technique (such as the Swap method), which tends to be
computationally expensive for multipartite scenarios with more than four
parties. Completely analytical self-testing statements have also been obtained
for multipartite states such as the W and the Dicke states by reprocessing
self-testing protocols of bipartite states Wu et al. (2014); Fadel (2017);
Šupić et al. (2018). Furthermore, parallel self-testing statements for
multipartite states can be obtained using categorical quantum mechanics
Buhrman and Massar (2005). Finally, the Mayers-Yao criterion can also be
utilized for self-testing of graph states when the underlying graph is a
triangular lattice Hayashi and Hajdušek (2018).

Mayers and Yao’s criterion states that if given a Bell inequality is maximally
violated, the shared quantum state must be equivalent to a reference state up
to local isometries, and the optimal observables are (almost) uniquely defined
with respect to the state. In this article, we prove Mayer-Yao-like self-
testing statements for multipartite Bell scenarios without relying on the
Bell-operator dependent sum-of-squares decomposition Bamps and Pironio (2015).
Consequently, our methodology immediately extends to all multipartite Bell
scenarios, where each spatially separated party has two observables. To
exemplify our proof technique, we obtain self-testing statements for $N$ party
Mermin-Ardehali-Belinskii-Klyshko (MABK) and Werner-Wolf-Weinfurter-Żukowski-
Brukner (WWWŻB) family of linear Bell inequalities. Moreover, our methodology
enables the recovery of self-testing statements for Bell functions which are
not only the mean value of a linear operator. Specifically, to showcase the
versatility of our methodology, we obtain self-testing statements for the
maximal violation of $N$ party Uffink’s quadratic Bell inequalities, which
form tight witnesses of genuine multipartite non-locality Uffink (2002).

The paper is organized as follows. In section II we present the requisite
preliminaries and specify the families of multipartite Bell inequalities we
consider in this article. Section III, we develop the mathematical
preliminaries of the self-testing scheme. In particular, we show that the
observables of each party in two setting binary outcome multipartite Bell
scenarios can always be simultaneously represented as anti-diagonal matrices.
In Section IV, we utilize this anti-diagonal matrix representation to obtain
self-testing statements for the maximal violations of $N$ party MABK and
tripartite WWWŻB family of linear (on correlators) Bell inequalities. While
the former serves to certify our scheme, the latter demonstrates the spectrum
of situations one can encounter. Finally, in Section IV we obtain self-testing
statements for the maximal violation of $N$ party Uffinik’s quadratic Bell
inequalities. In section V, we conclude by providing a brief summary of our
work and specifying potential applications.

##  II Two-setting $N$-party correlation Bell inequalities

This section presents the requisite preliminaries and specifies the families
of Bell inequalities considered in this article. Specifically, here we
consider multipartite Bell scenarios entailing $N$ spatially separated (hence
non-signaling) parties. We restrict ourselves to Bell scenarios where each
party $j\in\\{1,\ldots,N\\}$ has two binary outcome observables
$A^{(j)},A^{\prime(j)}$ for simplicity and brevity 111However, our technique
can easily be applied to proving more complex scenarios.. In contrast to the
well-studied bipartite Bell scenarios, multipartite scenarios are
substantially richer in complexity. While the notion of multipartite locality
is an obvious extension of bipartite locality, multipartite behaviors can be
non-local in many distinct ways. Apart from this, the most significant
impediment in obtaining self-testing statements for multipartite Bell
scenarios is that they do not admit simplifying characterizations such as
Schmidt decomposition, unlike the bipartite Bell scenarios.

Typically, Bell inequalities comprise of a Bell expression and a corresponding
local causal bound. The violation of Bell inequalities witnesses the non-
locality of the underlying behaviors. In the rest of this section, we
introduce the families of multipartite Bell inequalities, for which we
demonstrate self-testing statements using our proof technique.

###  II.1  Linear inequalities

The most frequently used Bell-inequalities comprise of Bell expressions which
are linear functionals of observed probabilities. Our self-testing argument is
immediately applicable for any Bell inequality whose Bell expression is the
mean of any linear combination of $N$ party operators of the form
${\bigotimes_{i=1}^{N}O^{(i)}}$, where
$O^{(i)}\in\\{A^{(i)},A^{\prime(i)}\\}$. Among these linear Bell inequalities,
we consider the _Werner-Wolf-Weinfurter-Żukowski-Brukner_ (WWWŻB) families of
correlation Bell inequalities for which there exists a well-defined systematic
characterization Scarani (2012). The Bell operator for WWWŻB family of Bell
inequalities has the following general form,

$\displaystyle\mathcal{W}_{N}=\frac{1}{2^{N}}\sum_{s_{1},...,s_{N}=\pm
1}S(s_{1},...,s_{N})\bigotimes_{j=1}^{N}(A^{(j)}+s_{j}A^{\prime(j)}),$ (1)

where $S(s_{1},...s_{N})=\pm 1$. Every tight (WWWŻB) inequality is given by
$\langle\mathcal{W}_{N}\rangle\leq_{\mathcal{L}}1$. Where $\mathcal{L}$ stands
for the local hidden variable polytope.

Next, we introduce a sub-family of (WWWŻB) Bell inequalities the _Mermin-
Ardehali-Belinskii-Klyshko_ (MABK) family of inequalities, featuring one
inequality for any number $N$ of parities Belinskiĭ and Klyshko (1993);
Ardehali (1992); Mermin (1990). Moreover, in the following subsection, we
introduce non-linear (quadratic) Bell inequalities composed of these MABK
inequalities. The $N$-party MABK operators can be obtained recursively as,

$\displaystyle\mathcal{M}_{N}$
$\displaystyle=\frac{1}{2}\Big{(}\mathcal{M}_{N-1}\otimes(A^{(j)}+A^{\prime(j)})$
(2) $\displaystyle\hskip
21.0pt+\mathcal{M^{\prime}}_{N-1}\otimes(A^{(j)}-A^{\prime(j)})\Big{)}$
$\displaystyle=\frac{1}{2}\Big{(}(\frac{1-i}{{2}})^{N-1}\bigotimes_{j=1}^{N}(A^{(j)}+iA^{\prime(j)})$
$\displaystyle\hskip
21.0pt+(\frac{1+i}{{2}})^{N-1}\bigotimes_{j=1}^{N}(A^{(j)}-iA^{\prime(j)})\Big{)},$

where $\mathcal{M}^{\prime}_{N-1}$ is the same expression as
$\mathcal{M}_{N-1}$ but with all $A^{(j)}$ and $A^{\prime(j)}$ interchanged.
The corresponding Bell inequalities are of the form,
$\langle\mathcal{M}_{N}\rangle\leq_{\mathcal{L}}1\leq_{\mathcal{Q}_{N-1}}2^{\frac{N-2}{2}}\leq_{\mathcal{Q}}2^{\frac{N-1}{2}},$
where $\mathcal{L},\mathcal{Q}_{N-1},\mathcal{Q}$ stand for the convex
polytope of local hidden variable correlations, the convex set of biseparable
quantum correlations, and the convex set of quantum behaviors, respectively.
The maximal quantum value can only be attained when the local observables
anti-commutes and the parties share the maximally entangled $N$-partite GHZ
state.

Finally, we present yet another relevant sub-family of the (WWWŻB)
inequalities, referred to as _Svetlichny_ inequalities Uffink (2002), which
were explicitly conceived to witness genuine $N$-partite non-locality.
Moreover, the quadratic Bell inequalities featured in the following subsection
can also be composed of $N$-partite Svetlichny-Bell inequalities. The
Svetlichny operator can be formed of MABK operators (LABEL:MABK) in the
following way,

$\mathcal{S}^{\pm}_{N}=\begin{cases}\displaystyle 2^{k-1}((-1)^{\frac{k(k\pm
1)}{2}}\mathcal{M}_{N}^{\pm},\\\ (\text{for }N=2k);\\\ 2^{k\pm
1}((-1)^{\frac{k(k\pm 1)}{2}}\mathcal{M}_{N}\mp(-1)^{\frac{k(k\mp
1)}{2}}\mathcal{M^{\prime}}_{N})),\\\ (\text{for }N=2k+1).\end{cases}$ (3)

The notation $\mathcal{M}^{+}_{N}$ is equivalent to $\mathcal{M}_{N}$ and
$\mathcal{M}^{-}_{N}$ is equivalent to $\mathcal{M^{\prime}}_{N}$. The
corresponding Svetlichny inequalities are of the form,
$\langle\mathcal{S}^{\pm}_{N}\rangle\leq_{\mathcal{Q}_{N-1}}2^{N-1}\leq_{\mathcal{Q}}2^{N-\frac{1}{2}}.$

Figure 1: This graphic is a schematic representation of the correlations in
multipartite (involving arbitrary number $N$ of spatially separated parties)
Bell scenarios. Just like the bipartite Bell scenarios, the correlations which
admit local hidden variable explanations form a convex polytope $\mathcal{L}$
(shiny blue small horizontal square), whose facets are the $N$-party MABK
inequalities (LABEL:MABK) (red edges). However, the convex set of biseparable
quantum correlations $\mathcal{Q}_{N-1}$ (sky blue disk) does not form a
polytope. Consequently, the linear inequalities such as the MABK (LABEL:MABK)
(pink edges of the large horizontal square) and Svetlichny inequalities (3)
(green edges of the tilted square) do not form tight witness of genuine
multipartite quantum nonlocality. As the boundary of biseparable quantum
correlations (black circle) is non-linear, the Uffink’s quadratic inequalities
(4) form tighter witnesses of genuine multipartite quantum nonlocality.

###  II.2 Quadratic inequalities

In bipartite Bell scenarios, without loss of generality, it is enough to
consider linear Bell inequalities to witness non-locality, as the set of
behaviors which admit a local casual explanation form a convex polytope. In
contrast to the bipartite case, non-linear Bell inequalities form tighter
witnesses of genuine multipartite non-locality. In multipartite Bell
scenarios, the convex set of bi-separable quantum behaviors does not form a
polytope.

In particular, the Uffink’s quadratic Bell inequalities form stronger
witnesses of genuine multipartite non-locality than the linear inequalities.
There are two distinct families of $N$ party quadratic Bell inequalities
formed of the MABK (LABEL:MABK) and Svetlichny (3), families of linear
inequalities, $\mathcal{U}^{\mathcal{M}}_{N}$ and
$\mathcal{U}^{\mathcal{S}}_{N}$, respectively, which have the form

$\mathcal{U}^{\mathcal{M}}_{N}=\left\langle\mathcal{M}_{N}\right\rangle^{2}+\left\langle\mathcal{M}^{\prime}_{N}\right\rangle^{2}\leq_{\mathcal{Q}_{N-1}}2^{N-2}\leq_{\mathcal{Q}}2^{N-1},$
(4)
$\mathcal{U}^{\mathcal{S}}_{N}=\left\langle\mathcal{S}_{N}\right\rangle^{2}+\left\langle\mathcal{S}^{\prime}_{N}\right\rangle^{2}\leq_{\mathcal{Q}_{N-1}}2^{2N-2}\leq_{\mathcal{Q}}2^{2N-1},$
(5)

##  III Characterizing local observables

In this section, we obtain a characterization for the binary outcome local
observables $A=A^{(j)}$ and $A^{\prime}=A^{\prime(j)}$ on an arbitrary Hilbert
space $\mathcal{H}=\mathcal{H}^{j}$ for any given party
$j\in\\{1,\ldots,N\\}$. In the light of Naimark’s dilation theorem, without
loss of generality we can take the observables $A$ and $A^{\prime}$ to be
projective, i.e., $A^{2}=A^{\prime 2}=\mathbb{I}$. As for Bell operators
composed of projective observables the optimal state is simply the eigenvector
corresponding to the maximum eigenvalue, the shared state can always be taken
to be pure. First, via the following lemma we demonstrate that second
observable $A^{\prime}$ can be split into two observables, one which commutes
with the first observable $A$ and one which anti-commutes with $A$,

######  Lemma III.1.

Given any two binary outcome projective observables $A$ and $A^{\prime}$,
$A^{\prime}$ can be decomposed as the sum of two observables $-\mathds{1}\leq
A^{\prime}_{-}\leq\mathds{1}$ and $-\mathds{1}\leq
A^{\prime}_{+}\leq\mathds{1}$, such that $\commutator{A}{A^{\prime}_{+}}=0$,
$\anticommutator{A}{A^{\prime}_{-}}=0$,
$\anticommutator{A^{\prime}_{+}}{A^{\prime}_{-}}=0$ and
$(A^{\prime}_{+})^{2}+(A^{\prime}_{-})^{2}=\mathds{1}$.

###### Proof.

Without loss of generality, the binary outcome projective observable $A$ can
be represented as a diagonal matrix with positive and negative eigenvalues
grouped together,

$A=\left(\begin{array}[]{cc}\openone_{m}&0\\\
0&-\openone_{n}\end{array}\right),$ (6)

where $\mathds{1}_{m}$ is the $m\times m$ identity operator. With respect to
$A$, the binary outcome projective observable $A^{\prime}$ has the following
generic matrix representation,

$A^{\prime}=\left(\begin{array}[]{cc}D_{1}&D_{2}\\\
D_{2}^{\dagger}&D_{3}\end{array}\right).$ (7)

such that,

$\displaystyle A^{\prime}_{-}$
$\displaystyle=\left(\begin{array}[]{cc}0&D_{2}\\\
D_{2}^{\dagger}&0\end{array}\right)\,,$ (10) $\displaystyle A^{\prime}_{+}$
$\displaystyle=\left(\begin{array}[]{cc}D_{1}&0\\\
0&D_{3}\end{array}\right)\,.$ (13)

Clearly, $\commutator{A}{A^{\prime}_{+}}=0$, and
$\anticommutator{A}{A^{\prime}_{-}}=0$. As $A^{\prime}$ is projective we have,

$\displaystyle(A^{\prime})^{2}=$
$\displaystyle(A^{\prime}_{+})^{2}+(A^{\prime}_{-})^{2}+\anticommutator{A^{\prime}_{+}}{A^{\prime}_{-}}$
(14) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}D_{1}^{2}+D_{2}D_{2}^{\dagger}&D_{1}D_{2}+D_{2}D_{3}\\\
D_{2}^{\dagger}D_{1}+D_{3}D_{2}^{\dagger}&D_{2}^{\dagger}D_{2}+D_{3}^{2}\end{array}\right)$
(17) $\displaystyle=$ $\displaystyle\,\mathds{1}\,,$ (18)

which requires the off-diagonal blocks to be zero,

$\displaystyle\\{A^{\prime}_{+},A^{\prime}_{-}\\}$
$\displaystyle=\left(\begin{array}[]{cc}0&D_{1}D_{2}+D_{2}D_{3}\\\
D_{2}^{\dagger}D_{1}+D_{3}D_{2}^{\dagger}&0\end{array}\right)$ (22)
$\displaystyle=0,$

and leaves $(A^{\prime}_{+})^{2}+(A^{\prime}_{-})^{2}=\mathds{1}$, which
completes the proof. ∎

Using the above Lemma, the relation between the eigenvalues of $A^{\prime}$,
$A^{\prime}_{-}$, and $A^{\prime}_{+}$ can be ascertained to be
$\lambda_{A^{\prime}}^{i}=\pm\sqrt{(\lambda^{i}_{A^{\prime}_{-}})^{2}+(\lambda^{i}_{A^{\prime}_{+}})^{2}}=\pm
1$, where $\lambda^{i}_{\cdot}$ denotes the $i^{\text{th}}$ eigenvalue
Ponomarenko and Selstad (2020). So that the eigenvalues can be without loss of
generality taken as $\pm\sin{\theta_{i}}$ and $\pm\cos{\theta_{i}}$ for
$A^{\prime}_{-}$ and $A^{\prime}_{+}$ respectively.

Next, we show that the spectra of any two anti-commuting observables must be
symmetric, and each observable maps the positive eigenspaces to the negative
eigenspaces of the other.

######  Lemma III.2.

Given any two anti-commuting observables $A$ and $A^{\prime}_{-}$, their
spectra are symmetric (i.e., if $\lambda$ is an eigenvalue of one of the
operators, then so is $-\lambda$), moreover, $A^{\prime}_{-}$ ($A$) is a
linear map between positive and negative eigenspaces of $A$ ($A^{\prime}_{-}$)
or nullifies its eigenvectors.

###### Proof.

Let $\ket{\psi}$ be an eigenvector of $A$ corresponding to the eigenvalue
$\lambda$ such that $A\ket{\psi}=\lambda\ket{\psi}$, then

$A(A^{\prime}_{-}\ket{\psi})=-A^{\prime}_{-}A\ket{\psi}=-\lambda
A^{\prime}_{-}\ket{\psi}.$ (23)

Thus $A^{\prime}_{-}\ket{\psi}$ is either a null vector or a new eigenvector
of $A$ with the sign of the eigenvalue flipped. Similarly, one can show the
same for eigenvalues of $A^{\prime}_{-}$. ∎

The above lemma further yields the following observations,

######  Corollary III.2.1.

Given any two anti-commuting observables $A$ and $A^{\prime}_{-}$, such that
$A^{2}=\mathds{1}$, then $A$ provides a bijective mapping between the
eigenspaces $E^{A^{\prime}_{-}}_{\lambda}$ and $E^{A^{\prime}_{-}}_{-\lambda}$
corresponding to the eigenvalues $\lambda^{A^{\prime}_{-}}$ and
$-\lambda^{A^{\prime}_{-}}$, respectively. Moreover, the direct sum of the
pair of eigenspaces
$E^{A^{\prime}_{-}}_{\pm\lambda_{i}}=E^{A^{\prime}_{-}}_{\lambda_{i}}\oplus
E^{A^{\prime}_{-}}_{-\lambda_{i}}$ is even dimensional, and so is the subspace
$\bigoplus_{i}E^{A^{\prime}_{-}}_{\pm\lambda_{i}}$.

###### Proof.

If $\lambda^{A^{\prime}_{-}}$ is an eigenvalue of $A^{\prime}_{-}$, with
eigenvector $\ket{\psi}$, then,

$AA^{\prime}_{-}\ket{\psi}=-A^{\prime}_{-}A\ket{\psi}=\lambda^{A^{\prime}_{-}}A\ket{\psi},$
(24)

and applying $A$ to the eigenvalue equation again,

$-AA^{\prime}_{-}A\ket{\psi}=\lambda^{A^{\prime}_{-}}A^{2}\ket{\psi}=\lambda^{A^{\prime}_{-}}\ket{\psi},$
(25)

we recover the same eigenvector again. In other words, $A$ maps the subspace
$E^{A^{\prime}_{-}}_{\lambda_{i}}\oplus E^{A^{\prime}_{-}}_{-\lambda_{i}}$ to
itself. This pairing of eigenvectors leads us to the conclusion that this
subspace is even dimensional. The diagonalizability of the operator,
$A^{\prime}_{-}$, implies that the effective Hilbert space can be decomposed
into the direct sum of its eigenspaces,

$\mathcal{H}=E^{A^{\prime}_{-}}_{\lambda_{1}}\oplus
E^{A^{\prime}_{-}}_{-\lambda_{1}}\cdots\oplus
E^{A^{\prime}_{-}}_{\lambda_{r}}\oplus
E^{A^{\prime}_{-}}_{-\lambda_{r}}\oplus\ker(A^{\prime}_{-}).$ (26)

It also follows from the anti-commutation relation that
$AA^{\prime}_{-}A^{-1}=-A^{\prime}_{-}$, and thus

$\det(A^{\prime}_{-})=(-1)^{d}\det(A^{\prime}_{-}),$ (27)

where $d=n+m$ is the dimension of the Hilbert space. This is only possible if
$\det(A^{\prime}_{-})=0$ or $A^{\prime}_{-}$ is even dimensional. In the case
when $\det(A^{\prime}_{-})=0$, the null subspace of $A^{\prime}_{-}$,
$\ker(A^{\prime}_{-})$, can be ignored as it does not contribute to the Bell
inequality violation. This implies that the effective subspace of such
operators can be truncated to an even-dimensional subspace. Finally, as
$A^{\prime}=A^{\prime}_{+}+A^{\prime}_{-}$, the effective subspaces of
$A^{\prime}_{+}$ and $A^{\prime}$ are also be truncated to the same even-
dimensional subspace as $A^{\prime}_{-}$. ∎

Moreover, the lemma III.2 yields the following succinct parameterization of
the two mutually anti-commuting components of any projective observable,

######  Corollary III.2.2.

Given two even-dimensional anti-commuting operators $A_{+}$ and $A_{-}$ such
that $(A^{\prime}_{+})^{2}+(A^{\prime}_{-})^{2}=\mathds{1}$, they can be
written in the form,

$\displaystyle A^{\prime}_{+|\theta_{i}}=\sin{\theta_{i}}B_{+|\theta_{i}}$
$\displaystyle A^{\prime}_{-|\theta_{i}}=\cos{\theta_{i}}B_{-|\theta_{i}}$
(28)

when restricted to the subspace $E^{A^{\prime}_{-}}_{+\theta_{i}}\oplus
E^{A^{\prime}_{-}}_{-\theta_{i}}$ with corresponding eigenvalues
$\pm\sin(\theta_{i})$ for $A_{+}$ and $\pm\cos(\theta_{i})$ for $A_{-}$, such
that the operators $B_{\pm|\theta_{i}}$ are traceless and projective.

###### Proof.

We have already shown the dimension of the combined eigenspaces
$E^{A^{\prime}_{-}}_{\lambda_{i}}\oplus E^{A^{\prime}_{-}}_{-\lambda_{i}}$ is
even. Then the two anti-commuting operators $A^{\prime}_{+}$ and
$A^{\prime}_{-}$ are restricted to this eigenspace, denoted by
$A^{\prime}_{+|\theta_{i}}$ and $A^{\prime}_{-|\theta_{i}}$, will only have
eigenvalues $\pm\sin{\theta_{i}}$ and $\pm\cos{\theta_{i}}$ respectively,
while still satisfying
$(A^{\prime}_{+|\theta_{i}})^{2}+(A^{\prime}_{-|\theta_{i}})^{2}=\mathds{1}$.
It is then possible to write the same relation in terms of scaled operators,
as

$\cos^{2}{\theta_{i}}B_{+|\theta_{i}}^{2}+\sin^{2}{\theta_{i}}B_{-|\theta_{i}}^{2}=\mathds{1}.$
(29)

As a result the operators $B_{\pm|\theta_{i}}$ have eigenvalues $\pm 1$ that
occur in pairs. Therefore, $\Tr(B_{\pm|\theta_{i}})=0$ and
$B_{\pm|\theta_{i}}^{2}=\mathds{1}$. ∎

Using the above results, in such an even-dimensional subspace the following
lemma holds.

######  Lemma III.3.

Given any two traceless and projective anti-commuting observables $B_{+}$ and
$B_{-}$, the observable $\cos{\alpha}B_{+}+\sin\alpha B_{-}$ is also traceless
and projective.

###### Proof.

Expanding the square,

$\displaystyle(\cos\alpha$ $\displaystyle B_{+}+\sin\alpha B_{-})^{2}$
$\displaystyle=$ $\displaystyle\cos^{2}\alpha B_{+}^{2}+\sin^{2}\alpha
B_{-}^{2}+\cos\alpha\sin\alpha\anticommutator{B_{+}}{B_{-}}$ $\displaystyle=$
$\displaystyle\mathds{1}.$ (30)

And the trace is simply,

$\cos\alpha\Tr(B_{+})+\sin\alpha\Tr(B_{-})=0.$ (31)

∎

The above lemma shows that whenever any inequality is maximally violated the
effective local dimension of any subsystem must be even dimensional. Now, we
present the main ingredient of our self-testing proof technique, i.e., anti-
diagonal representation for local observables of each party up to local
isometries.

######  Theorem III.4.

Given any three binary outcome traceless and projective observables $A$,
$B_{+}$, and $B_{-}$, such that $\commutator{A}{B_{+}}=0$,
$\anticommutator{A}{B_{-}}=0$, and $\anticommutator{B_{+}}{B_{-}}=0$, then
these operators have a simultaneous anti-diagonal matrix representation.

###### Proof.

As $\commutator{A}{B_{+}}=0$ we can take the dimension of the subspace for
which the eigenvalues of $A$ and $B_{+}$ are equal to be $2d_{1}$, and
$2d_{2}$ for the subspace where the eigenvalues differ. Consequently, without
loss of generality, by Lemma III.2 the operators $A$, $B_{+}$, and $B_{-}$
have the following matrix representations,

$\displaystyle A=$
$\displaystyle\left(\begin{array}[]{cccc}\mathds{1}_{d_{1}}&\cdot&\cdot&\cdot\\\
\cdot&\mathds{1}_{d_{2}}&\cdot&\cdot\\\
\cdot&\cdot&-\mathds{1}_{d_{2}}&\cdot\\\
\cdot&\cdot&\cdot&-\mathds{1}_{d_{1}}\end{array}\right),$ (36) $\displaystyle
B_{+}=$
$\displaystyle\left(\begin{array}[]{cccc}\mathds{1}_{d_{1}}&\cdot&\cdot&\cdot\\\
\cdot&-\mathds{1}_{d_{2}}&\cdot&\cdot\\\
\cdot&\cdot&\mathds{1}_{d_{2}}&\cdot\\\
\cdot&\cdot&\cdot&-\mathds{1}_{d_{1}}\end{array}\right),$ (41) $\displaystyle
B_{-}=$ $\displaystyle\left(\begin{array}[]{cccc}\cdot&\cdot&\cdot&U_{1}\\\
\cdot&\cdot&U_{2}&\cdot\\\ \cdot&U_{2}^{\dagger}&\cdot&\cdot\\\
U_{1}\dagger&\cdot&\cdot&\cdot\end{array}\right).$ (46)

Since $B_{-}$ is projective, $U_{1}$ and $U_{2}$ must be unitary. Thus,
without altering $A$ or $B_{+}$ we can take four unitaries $V_{1}$, $V_{2}$,
$V_{3}$, $V_{4}$, each acting on a different block, such that
$V_{1}U_{1}V_{4}^{\dagger}=J_{d_{1}}$ and
$V_{2}U_{2}V_{3}^{\dagger}=J_{d_{2}}$, where $J_{d}$ is the row-reversed
$d\times d$ identity matrix.

Consequently, we can now restrict ourselves to considering any one of the
$d_{1}+d_{2}$ two-dimensional subspaces, on which $A$ and $B_{+}$ are
represented by $\pm\sigma_{z}$, while $B_{-}$ is projected onto $\sigma_{x}$.
Finally, in each of these subspaces applying a rotation by $\frac{2\pi}{3}$
with respect to axis $(1,1,1)$ yields,

$U=\frac{1}{2}\left(\begin{array}[]{cc}-1+\iota&1+\iota\\\
-1+\iota&-1-\iota\end{array}\right),$ (47)

transforming $\sigma_{z}\rightarrow\sigma_{x}\rightarrow\sigma_{y}$, and
bringing all three operators to strictly antidiagonal form. ∎

Summarizing, the theorem III.4 along with lemmas III.2 and III.3 allow us to
take the first observable of each party $A^{(j)}$ to be equivalent to
$\sigma_{x}$ on all relevant two dimensional subspaces, while the second
operator $A^{\prime(j)}$ can be taken to be
$A^{\prime(j)}=\cos{\theta_{j}}\sigma_{x}+\sin{\theta_{j}}\sigma_{y}$. We
would like to stress the key significance of this theorem. As the Bell
operators considered in this article are tensor products of local observables
for each party, and via theorem III.4 each local observable can have anti-
diagonal matrix representation, the Bell operator itself has anti-diagonal
matrix representation. Consequently, only the states with maximal (in modulo)
antidiagonal elements, or their mixtures, can provide the maximal contribution
to the Bell violation. In particular, this implies the maximum violation two
setting $N$-partite correlation Bell inequalities necessitates $N$-qubit GHZ
states, or their mixtures, as we demonstrate in the following section.

##  IV Self-testing statements for multipartite inequalities

In this section, we use the tools developed in the previous section to obtain
self-testing statements for linear MABK family (LABEL:MABK) of $N$ party
inequalities, and sketch the proofs for the self-testing statements for all
distinct equivalence classes of tripartite WWWŻB facet inequalities (1).
Finally, we obtain self-testing statements for Uffnick’s family (4) of non-
linear $N$ party inequalities.

###  IV.1 $N$ party MABK inequalities

######  Theorem IV.1.

In order to achieve maximal quantum violation of a $N$-party MABK inequality,
$\langle\mathcal{M}_{N}\rangle=2^{\frac{N-1}{2}}$, the parties must share a
$N$ qubit GHZ state $\ket{GHZ_{N}}=\frac{1}{\sqrt{2}}(\ket{0}^{\otimes
N}+e^{\iota\phi_{N}}\ket{1}^{\otimes N})$ and perform maximally anti-commuting
projective measurements $A^{(j)}=\sigma_{x}$ and $A^{\prime(j)}=\sigma_{y}$
(upto local auxiliary systems and local isometries).

###### Proof.

From lemma III.2 and theorem III.4, without loss of generality, the local
observables of any party $j$ can be taken to be,

$\displaystyle A^{(j)}$ $\displaystyle=\sigma_{x},$ (48) $\displaystyle
A^{\prime(j)}$
$\displaystyle=\cos{\theta_{j}}\sigma_{x}+\sin{\theta_{j}}\sigma_{y},$

acting on the effective two dimensional subspace. This parametrization implies
that the MABK operator (LABEL:MABK) $\mathcal{M}_{N}$ has the following anti-
diagonal matrix representation,

$\displaystyle\mathcal{M}_{N}=\text{adiag}\begin{pmatrix}\frac{1}{2}\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1+\iota
e^{-\iota\theta_{j}})\\\ \hskip
21.0pt+(\frac{1+\iota}{2})^{N-1}\Pi_{j=1}^{N}(1-\iota
e^{-\iota\theta_{j}})\Big{)}\\\ \vdots\\\
\frac{1}{2}\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1+\iota
e^{\iota\theta_{j}})\\\ \hskip
21.0pt+(\frac{1+\iota}{2})^{N-1}\Pi_{j=1}^{N}(1-\iota
e^{\iota\theta_{j}})\Big{)}\end{pmatrix},$ (49)

where adiag represents a matrix with non-zero values only on the anti-
diagonal. It is easy to see for any combination, $\forall
j\in\\{1,\ldots,N\\}:\theta_{j}=\pm\frac{\pi}{2}$, one of the anti-diagonal
element attains the maximum absolute value of $2^{\frac{N-1}{2}}$ while the
others vanish.

Let us, for simplicity, consider the subspace spanned by $\\{\ket{0}^{\otimes
N},\ket{1}^{\otimes N}\\}$. On the subspace we consider the state
$\ket{\psi_{N}}=\alpha\ket{0}^{\otimes N}+\beta\ket{1}^{\otimes N}$ which
effectively yields a weighted sum of the top and bottom anti-diagonal elements
of the matrix (49). Consequently, the expectation value of the Hermitian
operator $\mathcal{M}_{N}$ for the state $\ket{\psi_{N}}$ has the expression,

$\displaystyle\bra{\psi_{N}}\mathcal{M}_{N}\ket{\psi_{N}}=$ $\displaystyle
2\real\\{\overline{\alpha}\frac{1}{2}\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1+\iota
e^{-\iota\theta_{j}})$ (50)
$\displaystyle+(\frac{1+\iota}{2})^{N-1}\Pi_{j=1}^{N}(1-\iota
e^{-\iota\theta_{j}})\Big{)}\beta\\},$

Using the fact that $\forall\alpha,\beta\in\mathbb{C}:\
\Re{\alpha\beta}\leq|\alpha\beta|=|\alpha||\beta|$ we bound
$\bra{\psi_{N}}\mathcal{M}_{N}\ket{\psi_{N}}$ from above in the following way,

$\displaystyle\bra{\psi_{N}}\mathcal{M}_{N}\ket{\psi_{N}}\leq$ $\displaystyle
2|\overline{\alpha}||\beta||\frac{1}{2}\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1+\iota
e^{-\iota\theta_{j}})$ (51)
$\displaystyle+(\frac{1+\iota}{2})^{N-1}\Pi_{j=1}^{N}(1-\iota
e^{-\iota\theta_{j}})\Big{)}|.$

As $|\alpha|^{2}+|\beta|^{2}=1$, the maximum value of the above expression can
only be attained for $|\alpha|=|\beta|=\frac{1}{\sqrt{2}}$, which picks out
$\ket{GHZ_{N}}$ as the shared state. Consequently, we retrieve the following
upper bound,

$\displaystyle\bra{\psi_{N}}\mathcal{M}_{N}\ket{\psi_{N}}\leq$
$\displaystyle|\frac{1}{2}\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1+\iota
e^{-\iota\theta_{j}})$ (52)
$\displaystyle+(\frac{1+\iota}{2})^{N-1}\Pi_{j=1}^{N}(1-\iota
e^{-\iota\theta_{j}})\Big{)}|.$

Now, as
$|(\frac{1+\iota}{2})^{N-1}|=|(\frac{1-\iota}{2})^{N-1}|=\frac{1}{2^{\frac{N-1}{2}}}$
and $(1+\iota
e^{-\iota\theta_{j}})=2\cos{(\frac{\pi}{4}+\frac{\theta_{j}}{2})}$ and
$(1-\iota e^{-\iota\theta_{j}})=2\sin{(\frac{\pi}{4}+\frac{\theta_{j}}{2})}$,
we can further upper bound the above expression as,

$\displaystyle\bra{\psi_{N}}\mathcal{M}_{N}\ket{\psi_{N}}\leq$ $\displaystyle
2^{\frac{N-1}{2}}\Big{(}\Pi_{j=1}^{N}|\cos{\frac{\pi}{4}+\frac{\theta_{j}}{2}}|$
$\displaystyle\hskip
25.0pt+\Pi_{j=1}^{N}|\sin{\frac{\pi}{4}+\frac{\theta_{j}}{2}}|\Big{)}.$

Dropping positive terms corresponding to any $N-2$ parties we retrieve the
simplified upper bound,

$\displaystyle\bra{\psi_{N}}\mathcal{M}_{N}\ket{\psi_{N}}$
$\displaystyle\leq{2^{\frac{N-1}{2}}}\Big{(}\left|\sin\left(\frac{2\theta_{i}+\pi}{4}\right)\sin\left(\frac{2\theta_{j}+\pi}{4}\right)\right|$
(53) $\displaystyle\hskip
21.0pt+\left|\cos\left(\frac{2\theta_{i}+\pi}{4}\right)\cos\left(\frac{2\theta_{j}+\pi}{4}\right)\right|\Big{)}$
$\displaystyle=2^{\frac{N-1}{2}}\max\Big{\\{}\Big{|}\cos(\frac{\theta_{i}+\theta_{j}+\pi}{2})\Big{|},$
$\displaystyle\hskip
27.0pt\Big{|}\cos(\frac{\theta_{i}-\theta_{j}}{2}n)\Big{|}\Big{\\}}$
$\displaystyle\leq 2^{\frac{N-1}{2}},$

As the choice of $i,j\in\\{1,\ldots,N\\}$ is completely arbitrary, the
inequality can only be saturated when $\forall
j\in\\{1,\ldots,N\\}:\theta_{j}=\pm\frac{\pi}{2}$, i.e., for each party the
local observables maximally anti-commute and the state that maximally violate
the quntum bound is $\ket{GHZ_{N}}$ up to auxiliarly degrees of freedom on
which the measurements act trivially, and local basis transformations. This
implies that the actual shared state could be of the form
$\ket{GHZ_{N}}\bigotimes\ket{\Psi}$ where the arbitrary state $\ket{\Psi}$ on
auxiliary degrees of freedom does not contribute to the operational Bell
violation, and thus is referred to as the junk state.

∎

The proof technique readily applies to the Svetlichny family of $N$-party
inequalities as they are composed of the $N$-party MABK inequalities.
Furthermore, the proof technique enables self-testing of a much broader class
of $N$-party WWWŻB inequalities. To demonstrate this, we sketch the proofs for
self-testing of tripartite WWWŻB inequalities (1).

###  IV.2 Tripartite WWWŻB inequalities

For the tripartite scenarios where each party has two binary outcomes
observables the local real polytope is characterized by $256$ facet
correlation inequalities which can be grouped into four equivalent classes of
non-trivial inequalities,

The first class is composed of correlation inequalities equivalent (up-to
relabeling) to the Mermin’s inequality
$\left\langle\mathcal{M}_{3}\right\rangle\leq 1$,

$\displaystyle\frac{1}{2}\Big{(}$
$\displaystyle\left\langle(A^{(1)}A^{(2)}A^{\prime(3)}\right\rangle+\left\langle
A^{(1)}A^{\prime}{(2)}A^{(3)}\right\rangle$ $\displaystyle+\left\langle
A^{\prime(1)}A^{(2)}A^{(3)}\right\rangle-\left\langle
A^{\prime(1)}A^{\prime(2)}A^{\prime(3)})\right\rangle\Big{)}\leq 1.$

For these inequalities the proof of Theorem IV.1 directly applies. As a
consequence, we retrieve the tripartite GHZ state $\ket{GHZ_{3}}$ as well as
maximally anti-commuting local observables as necessary ingredients for the
maximal quantum violation 2.

The second equivalence class of tripartite inequalities is that of unbalanced
inequalities, which have the following generic form.

$\displaystyle\Big{(}$ $\displaystyle 3\left\langle
A^{(1)}A^{(2)}A^{(3)}\right\rangle+\left\langle
A^{(1)}A^{(2)}A^{\prime(3)}\right\rangle$ $\displaystyle+\left\langle
A^{(1)}A^{\prime(2)}A^{(3)}\right\rangle+\left\langle
A^{\prime(1)}A^{(2)}A^{(3)}\right\rangle$ $\displaystyle-\left\langle
A^{(1)}A^{\prime(2)}A^{\prime(3)}\right\rangle-\left\langle
A^{\prime(1)}A^{(2)}A^{\prime(3)}\right\rangle$ $\displaystyle-\left\langle
A^{\prime(1)}A^{\prime(2)}A^{(3)}\right\rangle+\left\langle
A^{\prime(1)}A^{\prime(2)}A^{\prime(3)}\right\rangle\Big{)}\leq 4.$

The corresponding operator has a anti-diagonal matrix representation with
values of the form,

$\displaystyle 4+(-1+e^{\iota k_{1}\theta_{1}})(-1+e^{\iota
k_{2}\theta_{2}})(-1+e^{\iota k_{3}\theta_{3}}),$ (54)

Where $(k_{1},k_{2},k_{3}\in\\{-1,1\\})$. Unlike the MABK class of inequalties
the absolute values of these anti-diagonal terms are maximised when
$\cos\theta_{j}=-\frac{1}{3}$. Thus, yet again, the tripartite GHZ state
$\ket{GHZ_{3}}$ is distinguished, but non maximally anti-commuting operators
are required for maximal quantum violation of these inequalities is
$\frac{20}{3}$.

Next, we have the equivalence class of extended CHSH inequalities of the form,

$\displaystyle\frac{1}{2}\Big{(}$ $\displaystyle\left\langle
A^{(1)}A^{(2)}A^{(3)}\right\rangle+\left\langle
A^{\prime(1)}A^{(2)}A^{(3)}\right\rangle$ $\displaystyle+\left\langle
A^{(1)}A^{\prime(2)}A^{\prime(3)}\right\rangle-\left\langle
A^{\prime(1)}A^{\prime(2)}A^{\prime(3)}\right\rangle\Big{)}\leq 1.$

For these inequalities, the operator in the anti-diagonal matrix
representation has elements of the form,

$\displaystyle\frac{1}{2}(1+e^{\iota
k_{1}\theta_{1}}+e^{\iota(k_{2}\theta_{2}+k_{3}\theta_{3})}-e^{\iota(k_{1}\theta_{1}+k_{2}\theta_{2}+k_{3}\theta_{3}))}.$
(55)

Consequently,the corresponding absolute values are
$2\sqrt{1\pm\sin\theta_{1}\sin(k_{2}\theta_{2}\pm k_{3}\theta_{3})}$. Clearly,
for the maximal quantum violation the operators $A^{(1)}$ and $A^{\prime(1)}$
must maximally anti-commute, i.e., $\theta_{1}=\pm\frac{\pi}{2}$. However, the
maximal quantum violation only requires the sum $k_{2}\theta_{2}\pm
k_{3}\theta_{3}=\pm\frac{\pi}{2}$, i.e., the optimal $A^{\prime(2)}$ is
defined only in reference to $A^{\prime(3)}$. Clearly, for the maximal quantum
violation $2\sqrt{2}$ the shared state must be equivalent to the bipartite
maximally entangled state $\ket{GHZ_{2}}$ but these inequalities do not
satisfy the Mayers-Yao self-testing criterion .

Finally, we have the equivalence class of CHSH-like inequalities of the form,

$\displaystyle\frac{1}{2}\Big{(}$ $\displaystyle\left\langle
A^{(1)}A^{(2)}A^{(3)}\right\rangle+\left\langle
A^{(1)}A^{\prime(2)}A^{(3)}\right\rangle$ $\displaystyle+\left\langle
A^{\prime(1)}A^{(2)}A^{(3)}\right\rangle-\left\langle
A^{\prime(1)}A^{\prime(2)}A^{(3)}\right\rangle\Big{)}\leq 1.$

These inequalities are equivalent to the CHSH or $\mathcal{M}_{2}$ inequality
for which the proof of Theorem IV.1 directly applies. The nonzero elements of
the anti-diagonal matrix representation of the corresponding Bell operator are
of the form,

$\displaystyle(1+e^{\iota k_{1}\theta_{1}}+e^{\iota
k_{2}\theta_{2}}-e^{\iota(k_{1}\theta_{1}+k_{2}\theta_{2}}).$ (56)

The modulo of these values is $2\sqrt{1\pm\sin\theta_{1}\sin\theta_{2}}$,
which implies that for quantum violation $\sqrt{2}$ the observables for the
first pair of parties must maximally anti-commute, i.e.,
$\pm\theta_{1}=\pm\theta_{2}=\frac{\pi}{2}$ and shared state must be
equivalent to the bipartite maximally entangled state $\ket{GHZ_{2}}$.

Apart from the linear inequalities considered here, it is easy to see that our
proof technique, which relies on the anti-diagonal matrix representation of
the Bell operator, directly yields the self-testing statements for all two
settings binary outcome linear (on correlators) multipartite Bell
inequalities. Next, we demonstrate that our proof technique can be applied to
obtain self-testing statements for a large class of non-linear multipartite
Bell inequalities.

###  IV.3 $N$ party Uffink’s quadratic inequalities

As the convex set of bi-seprable multipartite quantum correlations does not
form a polytope, linear inequalities like Svetlichny inequalities do not form
tight efficient witnesses of genuine multipartite entanglement. on the other
hand, the non-linear inequalities such as the Uffink’s family of $N\geq 3$
party quadratic (on correlators) inequalities (4) better capture the boundary
of the quantum set of bi-separable correlations, and hence form better
witnesses of genuine multipartite quantum non-locality. Relying on the simple
observation that for any two real numbers $x_{1},x_{2}\in\mathbb{R}$,
$x_{1}^{2}+x_{2}^{2}=|x_{1}+\iota x_{2}|^{2}$, we linearize the Uffink’s
$N\geq 3$ party quadratic inequalities and obtain self-testing statements
based on anti-diagonal matrix representation of an Hermitian operator.

######  Theorem IV.2.

In order to achieve maximal quantum violation of a $N$ party Uffnick’s
inequality, $\left\langle\mathcal{U}_{N}\right\rangle=2^{N+1}$ , the parties
must share a $N\geq 3$ qubit GHZ state
$\ket{GHZ_{N}}=\frac{1}{\sqrt{2}}(\ket{0}^{\otimes
N}+e^{\iota\phi_{N}}\ket{1}^{\otimes N})$ and perform maximally anti-commuting
projective measurements $A^{(j)}=\sigma_{x}$ and $A^{\prime(j)}=\sigma_{y}$
(upto local auxiliary systems and local isometries).

###### Proof.

We start with a linearization of the Bell criterion. Note that as both
$\mathcal{U}_{N}$ and $\mathcal{U}^{\prime}_{N}$ operators are Hermitian,
$\left\langle\mathcal{U}_{N}\right\rangle,\left\langle\mathcal{U}^{\prime}_{N}\right\rangle\in\mathbb{R}$,
we have the following identity,

$\mathcal{U}^{\mathcal{M}}_{N}=\left\langle\mathcal{U}_{N}\right\rangle^{2}+\left\langle\mathcal{U}^{\prime}_{N}\right\rangle^{2}=|\left\langle\mathcal{U}_{N}\right\rangle+\iota\left\langle\mathcal{U}^{\prime}_{N}\right\rangle|^{2}.$
(57)

Our problem is hence reduced to studying the modulo of the mean value of non-
Hermitian operator, $|\mathcal{\tilde{U}}_{N}|$,

$\displaystyle\mathcal{\tilde{U}}_{N}=$
$\displaystyle(\mathcal{U}_{N}+\iota\mathcal{U}^{\prime}_{N})$
$\displaystyle=$
$\displaystyle\left(\frac{1-\iota}{2}\right)^{N-1}\bigotimes_{j=1}^{N}(A^{[j]}+\iota
A^{\prime[j]}).$ (58)

From lemma III.2 and theorem III.4, without loss of generality, the local
observables of any party $j$ can be taken to be,

$\displaystyle A^{(j)}$ $\displaystyle=\sigma_{x},$ (59) $\displaystyle
A^{\prime(j)}$
$\displaystyle=\cos{\theta_{j}}\sigma_{x}+\sin{\theta_{j}}\sigma_{y},$

acting on the effective two dimensional subspace. This parametrization implies
that the uffink operator $\mathcal{\tilde{U}}_{N}$ (IV.3) has the following
anti-diagonal matrix representation,

$\displaystyle\mathcal{\tilde{U}}_{N}=\text{adiag}\begin{pmatrix}\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1+\iota
e^{-\iota\theta_{j}}\Big{)}\\\ \vdots\\\
\Big{(}(\frac{1-\iota}{2})^{N-1}\Pi_{j=1}^{N}(1-\iota
e^{\iota\theta_{j}}\Big{)}\end{pmatrix},$ (60)

where adiag represent a matrix with values on the anti-diagonal. It is easy to
see for any combination, $\forall
j\in\\{1,\ldots,N\\}:\theta_{j}=\pm\frac{\pi}{2}$, one of the anti-diagonal
element attains the maximum absolute value of $2^{\frac{N-1}{2}}$ while the
others vanish.

Yet again, for simplicity 222Other subspaces are fully equivalent, with the
sings standing in front of phases being the only difference., we consider on
the subspace spanned by $\\{\ket{0}^{\otimes N},\ket{1}^{\otimes N}\\}$. On
this subspace, the modulo of the complex expectation value of
$\mathcal{\tilde{U}}_{N}$ with respect to the state
$\ket{\psi_{N}}=\alpha\ket{0}^{\otimes N}+\beta\ket{1}^{\otimes N}$ has the
expression,

$\displaystyle|\bra{\psi_{N}}\tilde{\mathcal{U}}_{N}\ket{\psi_{N}}|$
$\displaystyle=\frac{1}{2^{\frac{N-1}{2}}}|\overline{\alpha}\Pi_{j=1}^{N}(1+\iota
e^{-\iota\theta_{j}})\beta$ (61) $\displaystyle\hskip
25.0pt+\overline{\beta}\Pi_{j=1}^{N}(1-\iota e^{\iota\theta_{j}})\alpha|$
$\displaystyle\leq\frac{1}{2^{\frac{N-1}{2}}}|\alpha||\beta|\Big{(}\Pi_{j=1}^{N}|(1+\iota
e^{-\iota\theta_{j}})|$ $\displaystyle\hskip 25.0pt+\Pi_{j=1}^{N}|(1-\iota
e^{\iota\theta_{j}})|\Big{)}.$

As $|\alpha|^{2}+|\beta|^{2}=1$ the maximum value of the above expression can
only be attained for $|\alpha|=|\beta|=\frac{1}{\sqrt{2}}$, which picks out
$\ket{GHZ_{N}}$ as the shared state. Consequently, we retrieve the following
upper bound,

$\displaystyle|\bra{\psi_{N}}\tilde{\mathcal{U}}_{N}\ket{\psi_{N}}|$
$\displaystyle\leq\frac{1}{2^{\frac{N+1}{2}}}\Big{(}\Pi_{j=1}^{N}|(1+\iota
e^{-\iota\theta_{j}})|$ $\displaystyle\left.\hskip
25.0pt+\Pi_{j=1}^{N}|(1-\iota e^{\iota\theta_{j}})|\right)$
$\displaystyle=2^{\frac{N-1}{2}}\left(\Pi_{j=1}^{N}\left|\cos{\left(\frac{\pi}{4}+\frac{\theta_{j}}{2}\right)}\right|\right.$
$\displaystyle\left.\hskip
25.0pt+\Pi_{j=1}^{N}\left|\sin{\left(\frac{\pi}{4}+\frac{\theta_{j}}{2}\right)}\right|\right).$

Now, dropping the positive terms corresponding to any $N-2$ parties we obtain,

$\displaystyle\hskip
30.0pt|\bra{\psi_{N}}\tilde{\mathcal{U}}_{N}\ket{\psi_{N}}|$
$\displaystyle\leq{2^{\frac{N-1}{2}}}\Big{(}\left|\sin\left(\frac{2\theta_{i}+\pi}{4}\right)\sin\left(\frac{2\theta_{j}+\pi}{4}\right)\right|$
$\displaystyle\hskip
21.0pt+\left|\cos\left(\frac{2\theta_{i}+\pi}{4}\right)\cos\left(\frac{2\theta_{j}+\pi}{4}\right)\right|\Big{)}$
$\displaystyle=2^{\frac{N-1}{2}}\max\Big{\\{}\Big{|}\cos(\frac{\theta_{i}+\theta_{j}+\pi}{2})\Big{|},$
$\displaystyle\hskip
30.0pt\Big{|}\cos(\frac{\theta_{i}-\theta_{j}}{2}n)\Big{|}\Big{\\}}$
$\displaystyle\leq 2^{\frac{N-1}{2}},$ (62)

Now, as the choice of $i,j$ is completely arbitrary the third inequality can
only be saturated when $\forall
j\in\\{1,\ldots,N\\}:\theta_{j}=\pm\frac{\pi}{2}$, i.e., for each party the
local observables maximally anti-commute. Due to the quadratic form of the
Uffink’s inequalities, the state that maximally violates them is not unique,
but it must be pure and LOCC-equivalent of $\ket{GHZ_{N}}$, i.e, the state is
allowed to orbit under the action of local unitary groups.

It remains to discuss the role of the junk state. Imagine that we have
different realizations $\\{R_{j}\\}_{j}$ of the scenario such that they all
give
$\left\langle\tilde{\mathcal{U}}_{N}\right\rangle_{R_{j}}={2^{\frac{N-1}{2}}}e^{i\varphi_{j}}$.
Realizations may differ in the orientation of the $\ket{GHZ_{N}}$ state and
observables for each party, i.e., without loss of generality, the global state
can always be written as $\sum_{j}\gamma_{j}\ket{R_{j}}\ket{GHZ_{N}}$, with
$\\{\ket{R_{j}}\\}$ being orthogonal states of the rest of the system. The
follows from the necessity that the observables are adjusted to the
realization, requires that each orientation $\ket{R_{j}}$ is perfectly
distinguishable from any other orientation $\ket{R_{j^{\prime}}}$ by any
individual party. In turn, this implies that by distinguishing between them,
the parties can conditionally and locally transform each realization to the
same form, ending up in the state
$\left(\sum_{j}\gamma_{j}\ket{R_{j}}\right)\ket{GHZ_{N}}$. ∎

##  V Conclusions

Quantum correlations that violate Bell inequalities can certify the shared
entangled state and local measurements in an entirely device-independent
manner. This feature of quantum non-local correlations is referred to as self-
testing. In this work, we presented a simple and broadly applicable proof
technique to obtain self-testing statements for the maximum quantum violation
of a large relevant class of multipartite (involving arbitrary number $N$ of
spatially separated parties) Bell inequalities. Unlike the relatively
straightforward bipartite Bell scenarios, the multipartite scenarios are
substantially richer in complexity owing to the various kinds of multipartite
non-locality.

In contrast to the traditionally employed sum-of-squares-like proof
techniques, which rely on the specific structure of the Bell inequality, the
proof technique presented in this work applies to any Bell inequality
(independent of its specific form) as it only relies on the anti-diagonal
matrix representation of the Bell operator. In section III we show that the
observables of each party in two setting binary outcome multipartite Bell
scenarios can always be simultaneously represented as anti-diagonal matrices
(theorem III.4). Consequently, in such scenarios, the Hermitian Bell operators
corresponding to all linear (on correlators) Bell inequalities also have an
anti-diagonal matrix representation. In the section IV, to demonstrate our
proof technique, we obtain proofs of self-testing statements for the MABK
family of $N$ party inequalities (theorem IV.1) 333This constitutes a
reproduction of the self-testing statements for MABK inequalities from Ref.
Kaniewski (2017), and hence serves as a preliminary certification of our proof
technique., followed by self-testing statements for tripartite WWWŻB
inequalities (section IV.2). These inequalities exemplify all possible
exceptions to perfect self-testing statements such as non-unique optimal
states or observables, and non-anticommuting pairs of optimal measurements. To
further demonstrate the versatility of our proof technique which relies only
on the anti-diagonal matrix representation of the Bell operator and not even
on its Hermiticity, we obtain self-testing statements for Uffink’s family of
quadratic Bell inequalities (theorem IV.2). These quadratic Bell inequalities
do not allow for sum-of-squares-like techniques which rely on the linearity of
the Bell inequalities, and hence form the distinguishing application of our
proof technique.

As self-testing statements completely certify the local sub-systems of each
party, they can readily be used to certify the parties’ private randomness,
which in turn fuels cryptographic applications like device-independent
randomness certification and conference key distribution schemes. As Uffink’s
quadratic inequalities form tighter witnesses of genuine multipartite non-
locality compared to the linear inequalities, the corresponding self-testing
statements can fuel information processing and communication tasks that
require the participation of all parties, such as quantum secret sharing.

##  VI Acknowledgements

We would like to thank Anubhav Chaturvedi for insightful discussions. This
work is a part of NCN Grant No. 2017/26/E/ST2/01008. MW acknowledges partial
support by the Foundation for Polish Science (IRAP project, ICTQT, Contract
No. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme).

##  VII Note

During the preparation of the current work, we have become aware of Sarkar and
Augusiak (2021). While it presents similar results, our proof technique, which
constitutes the main conceptual contribution of this work, is substantially
different from their proof technique. In particular, the key contrasting
feature of our proof technique is its applicability to non-linear Bell
inequalities.

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