# Self-testing of genuine multipartite non-local and non-maximally entangled
states

Ranendu Adhikary<EMAIL_ADDRESS>Electronics and Communication Sciences
Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India.

###### Abstract

Self-testing enables the characterization of quantum systems with minimal
assumptions on their internal working as such it represents the strongest form
of certification for quantum systems. In the existing self-testing literature,
self-testing states that are not maximally entangled, but exhibit genuine
multipartite nonlocality, have remained an open problem. This is particularly
important because, for many-body systems, genuine multipartite nonlocality has
been recognized as the strongest form of multipartite quantum correlation. In
this work, we present a Cabello-like paradox for scenarios involving an
arbitrary number of parties. This paradox is a tool for detecting genuine
multipartite nonlocality, allowing for the specific identification and self-
testing of states that defy the paradox’s limits the most, which turn out to
be non-maximally multipartite entangled states. While recent results [Šupić et
al., Nature Physics, 2023] suggest network self-testing as a means to self-
test all quantum states, here we operate within the standard self-testing
framework to self-test genuine multipartite non-local and non-maximally
entangled states.

## Introduction.—

Since the 1920s, quantum mechanics has not only revolutionized our
understanding of fundamental physics but has also driven technological
progress. This field has been instrumental in the development of everyday
technologies, such as lasers, transistors, and the Global Positioning System
(GPS). In recent years, advancements in quantum theory have led to the
emergence of innovative applications like quantum computing, quantum networks,
and quantum-encrypted communication. At the forefront of these advancements is
the phenomenon of quantum entanglement [1, 2], a unique aspect of quantum
theory that was recognized by the 2022 Nobel Prize for its critical role in
quantum information theory and the foundations of quantum mechanics. The
exploration of quantum entanglement’s complex theory remains a dynamic and
rich area of research, full of unanswered questions that, once solved, could
pave the way for new technological applications [3, 4, 5].

As the scale of quantum devices increases, certifying their internal workings
becomes increasingly complex due to the vastness of the associated Hilbert
space [6]. To fully describe an $n$-qubit pure state, one needs $2^{n}-1$
complex parameters. Quantum state tomography, a common approach for system
certification, involves obtaining the description of a quantum state by
performing measurements on an ensemble of identical quantum states. However,
estimating an unknown quantum state within a dimension $d$ to an error
$\varepsilon$ (in $l_{1}$ norm) requires
$\Theta\left(\frac{d^{2}}{\varepsilon^{2}}\right)$ copies of that state [7].
This requirement makes tomography impractical for large systems. Additionally,
tomography is device-dependent, assuming characterized measurements, which is
not always feasible in scenarios involving third-party devices.

The necessity to overcome these challenges has led to the development of self-
testing protocols [8]. Crucially, these protocols are device-independent,
meaning they do not require any prior knowledge or characterization of the
quantum devices being tested. Moreover, the number of measurements scales
linearly with number of qubits. This makes them highly valuable for scalable
and assumption-minimal certification of quantum systems [9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 9, 20, 21, 22, 23, 24, 25, 26, 25, 27, 28, 29, 30, 31,
32, 33]. In their study, Mayers and Yao demonstrated that certain statistical
correlations exist which necessitate that black boxes replicating these
correlations must inherently produce the maximally entangled state
$|\phi_{+}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. Consequently,
these Bell correlations serve as a self-test for the state $|\phi_{+}\rangle$.
The entanglement in the quantum state allows for correlations that allow the
underlying entangled state to be self-tested.

A natural question arises: which entangled states can be self-tested? In a
breakthrough paper, Coladangelo et al. demonstrated that all pure bipartite
states are amenable to self-testing [31]. However, in the broader context of
multipartite states, our understanding remains limited. This lack of
comprehensive knowledge is not unexpected, given that entanglement presents
more complexity and challenges in characterization when it involves multiple
parties. In particular, self-testing for entangled states in the multipartite
setting that admit genuine multipartite nonlocality [34] has remained poorly
understood. Genuine multipartite nonlocality is a stronger form of nonlocality
and understanding entangled states that manifest them can deepen our
understanding of quantum theory. For multipartite systems, genuine
multipartite nonlocality has been recognized as the strongest form of
multipartite quantum correlation [35, 36]. Previous works focused on self-
testing for maximally entangled states [33] that admit genuine multipartite
nonlocality and it is unclear whether the non-maximally entangled states that
admit genuine multipartite nonlocality can be self-tested.

In this paper, we answer the above question in the affirmative. We present a
Cabello-like paradox for a scenario involving an arbitrary number of
participants, each having access to two measurements that can yield any
arbitrary number of outcomes. Our paradox helps witness genuine multipartite
nonlocality. The state maximally violating the constraints of our paradox can
be self-tested. Regarding robust self-testing, we provide a partial robustness
result in the three-party scenario where we consider the non-ideal state to be
a three-qubit state. Moreover, it is important to note that the aforementioned
state is non-maximally entangled. We highlight that unlike Ref [32], our work
does not require quantum networks. We operate in the standard Bell-test based
self-testing framework and do not need any additional assumptions unlike Ref
[32].

## Genuine tripartite non-local correlation.—

Consider three distant observers denoted as $\mathcal{A}_{1}$,
$\mathcal{A}_{2}$ and $\mathcal{A}_{3}$ engaged in rounds of measurements on
tripartite quantum systems. In each round, every party selects a local
measurement denoted as $\hat{x}_{i}$, and subsequently records the resulting
outcome as $x_{i}$. The resultant joint conditional probability distribution,
denoted as
$P_{A_{1}A_{2}A_{3}}(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})$, is
classified as local if it can be factorized. This factorization holds under
the additional condition of possessing knowledge about a potentially concealed
shared classical cause $\lambda$:

$\begin{split}&P_{\mathcal{L}}(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})\\\
&=\sum_{\lambda}\nu(\lambda)P_{A_{1}}(x_{1}|\hat{x}_{1},\lambda)P_{A_{2}}(x_{2}|\hat{x}_{2},\lambda)P_{A_{3}}(x_{3}|\hat{x}_{3},\lambda)\end{split}$
(1)

The discrete random variable representing the common cause is denoted as
$\lambda$, with a distribution $\nu(\lambda)\geq 0$ and the normalization
condition $\sum_{\lambda}\nu(\lambda)=1$. The conditional probability
distribution $P_{A_{i}}(x_{i}|\hat{x}_{i},\lambda)$ characterizes the behavior
of party $\mathcal{A}_{i}$. A distribution
$P(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})$ is considered non-local
if it cannot be decomposed as described in equation (1). Given the measurement
setups, we work under the assumption of the NS principle [37]. This implies
that party $\mathcal{A}_{1}$ is unable to communicate with other parties
through the choice of measurement. Specifically, the equality
$P_{A_{2}A_{3}}(x_{2}x_{3}|\hat{x}_{2}\hat{x}_{3})=P_{A_{2}A_{3}}(x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})=\sum_{x_{1}}P(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3}),\forall
x_{1}$ holds, and the analogous statements apply to parties $\mathcal{A}_{2}$
and $\mathcal{A}_{3}$.

Suppose we relax the locality assumption such that any pair of parties can
group and share non-local resources. This type of hybrid local and non-local
model leads to joint conditional probability distributions,

$\begin{split}&P_{2\ versus\
1}(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})\\\
&=\sum_{\lambda_{1}}\nu_{1}(\lambda_{1})P_{A_{1}A_{2}}(x_{1}x_{2}|\hat{x}_{1}\hat{x}_{2},\lambda_{1})P_{A_{3}}(x_{3}|\hat{x}_{3},\lambda_{1})\\\
&+\sum_{\lambda_{2}}\nu_{2}(\lambda_{2})P_{A_{2}A_{3}}(x_{2}x_{3}|\hat{x}_{2}\hat{x}_{3},\lambda_{2})P_{A_{1}}(x_{1}|\hat{x}_{1},\lambda_{2})\\\
&+\sum_{\lambda_{3}}\nu_{3}(\lambda_{3})P_{A_{1}A_{3}}(x_{1}x_{3}|\hat{x}_{1}\hat{x}_{3},\lambda_{3})P_{A_{2}}(x_{2}|\hat{x}_{2},\lambda_{3})\end{split}$
(2)

with $\nu_{i}(\lambda_{i})\geq 0$ and
$\sum_{i,\lambda_{i}}\nu_{i}(\lambda_{i})=1$. Distributions denoted as
$P(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})$ that cannot be
expressed in the form described in equation (2) are termed as genuine
tripartite non-local distributions. According to findings in [38], the
original concept of multipartite non-locality introduced by Svetlichny [34]
encounters practical challenges when operationalized. To address these
challenges, it is assumed that the NS principle [39] also holds at the level
of distributions $P_{A_{i}A_{j}}(x_{i}x_{j}|\hat{x}_{i}\hat{x}_{j},\lambda)$,
implying that the conditional probabilities
$P(x_{i}|\hat{x}_{i},\lambda)=P(x_{i}|\hat{x}_{i}\hat{x}_{j},\lambda)=\sum_{x_{j}}P(x_{i}x_{j}|\hat{x}_{i}\hat{x}_{j},\lambda),\forall\hat{x}_{j}$,
are well defined for all $\lambda$.

Let us consider a scenario where all three parties Alice, Bob, and Charlie
have two possible measurement choices $\hat{x}_{1}\in\\{A_{0},A_{1}\\}$,
$\hat{x}_{2}\in\\{B_{0},B_{1}\\}$ and $\hat{x}_{3}\in\\{C_{0},C_{1}\\}$
respectively. Each measurement has two possible outcomes, say $\\{+,-\\}$. We
define a tripartite correlation
$P(x_{1},x_{2},x_{3}|\hat{x}_{1},\hat{x}_{2},\hat{x}_{3})$ that satisfies the
following equality constraint,

$\begin{split}&P_{AB}(++|A_{1}B_{0})=0,\\\ &P_{BC}(++|B_{1}C_{0})=0,\\\
&P_{AC}(++|A_{0}C_{1})=0.\\\ \end{split}$ (3)

Let us define $p_{3}\equiv P(+++|A_{0}B_{0}C_{0})$ and $q_{3}\equiv
P(---|A_{1}B_{1}C_{1})$. We define their difference to be
$\mathcal{S}_{3}:=p_{3}-q_{3}$. We will first show that no local theory can
satisfy $\mathcal{S}_{3}>0$. Further, we will see that it is not just any non-
local correlation; rather, it is a genuine non-local correlation.

######  Theorem 1.

If a non-local correlation satisfies the conditions in (3) with
$\mathcal{S}_{3}>0$, then it is a genuine tripartite non-local correlation.

###### Proof.

We will first show that no local theory can satisfy conditions (3) with
$\mathcal{S}_{3}>0$. Let us define a local hidden variable $\lambda$ that
captures the full description of the system. This $\lambda$ takes values from
a set $\Omega$. Moreover, we assert that the comprehensive depiction of the
joint system is denoted by $\nu(\lambda)$. In a local description, the joint
probability can be expressed as follows:
$P(x_{1},x_{2},x_{3}|\hat{x}_{1},\hat{x}_{2},\hat{x}_{3})=\sum_{\lambda\in\Omega}\nu(\lambda)\prod_{j=1}^{3}p(x_{j}|\hat{x}_{j},\lambda)$,
where $\nu(\lambda)\geq 0$, $\sum_{\lambda}\nu(\lambda)=1$. Let’s assume that
$\mathcal{S}_{3}>0$, which indicates that $p_{3}$ is greater than $0$. This
implies the existence of a nonempty subset $\Lambda\subseteq\Omega$, such that
for all $\lambda\in\Lambda$,
$P(+|A_{0},\lambda)=P(+|B_{0},\lambda)=P(+|C_{0},\lambda)=1$. By equation (3)
we have $P(-|A_{1},\lambda)=P(-|B_{1},\lambda)=P(-|C_{1},\lambda)=1$.
Therefore, we can deduce that
$P(-,-,-|A_{1},B_{1},C_{1},\Lambda)=P(+,+,+|A_{0},B_{0},C_{0})$. However,
since $\Lambda\subseteq\Omega$, it follows that
$P(-,-,-|A_{1},B_{1},C_{1},\Lambda)\leq P(-,-,-|A_{1},B_{1},C_{1})$. In other
words, $\mathcal{S}_{3}\leq 0$, which contradicts the initial assumption of
$\mathcal{S}_{3}>0$. Therefore, we can conclude that if any correlation
satisfies (3) and gives $\mathcal{S}_{3}>0$, then it must be non-local.

Now we will show that the non-locality we have is a genuine one. If this
correlation were not genuine, it could be broken down according to the
expression provided in equation (2). Given that $\mathcal{S}_{3}>0$, at least
one component within the decomposition must yield a non-zero value for
$P(+++|A_{0}B_{0}C_{0})$. Let’s consider this specific component to be the
initial one:
$P_{AB}(x_{1}x_{2}|\hat{x}_{1}\hat{x}_{2},\lambda_{1})P_{C}(x_{3}|\hat{x}_{3},\lambda_{1})$.
Consequently, $P_{AB}(++|A_{0}B_{0},\lambda_{1})P_{C}(+|C_{0},\lambda_{1})\neq
0$. The condition $P_{BC}(++|B_{1}C_{0})=0$ in equation (3) leads to the
conclusion that $P_{B}(+|B_{1},\lambda_{1})P_{C}(+|C_{0},\lambda_{1})$ must be
zero. This, in turn, implies that $P_{B}(+|B_{1},\lambda_{1})=0$, given that
$P_{C}(+|C_{0},\lambda_{1})$ cannot be zero. Therefore, we deduce that
$P_{B}(-|B_{1},\lambda_{1})=1$. Now as the NS principle is assumed at the
level of distribution $P_{AB}(x_{1}x_{2}|\hat{x}_{1}\hat{x}_{2},\lambda_{1})$,
a deterministic marginal of
$P_{AB}(x_{1}x_{2}|\hat{x}_{1}\hat{x}_{2},\lambda_{1})$ indicates that
$P_{AB}(x_{1}x_{2}|\hat{x}_{1}\hat{x}_{2},\lambda_{1})=P_{A}(x_{1}|\hat{x}_{1},\lambda_{1})P_{B}(x_{2}|\hat{x}_{2},\lambda_{1})$.
Similarly, all the other components within the decomposition become fully
product-based. As a result, the distributions
$P_{ABC}(x_{1}x_{2}x_{3}|\hat{x}_{1}\hat{x}_{2}\hat{x}_{3})$ transform into
fully local distribution, which contradicts the initial premise considering
$\mathcal{S}_{3}>0$. ∎

## Genuine $N$-partite non-local correlation.—

Let’s consider a scenario where $N$ subsystems are distributed among $N$
separate parties $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\ldots$, and
$\mathcal{A}_{N}$. Each party, $\mathcal{A}_{i}$, can measure one of two
observables: either $\hat{u}_{i}$ or $\hat{v}_{i}$, on their local subsystem.
The measurement outcomes, denoted as $x_{i}$, can take values from 1 to
$d_{i}$, where $d_{i}$ represents the dimension of the Hilbert space
associated with the $i$-th subsystem $\mathcal{A}_{i}$. We are interested in
examining the joint probabilities
$P(x_{1},x_{2},\dots,x_{N}|\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{N})$, where
$\hat{x}_{i}$ can be either $\hat{u}_{i}$ or $\hat{v}_{i}$. Let us consider
the following set of conditions:

$\begin{split}&P_{A_{i}A_{i+1}}(v_{r},1|\hat{v}_{i},\hat{u}_{i+1})=0,\\\
&\forall r,v_{r}\neq d_{i}.\end{split}$ (4)

We consider the subsystem $\mathcal{A}_{N+1}$ to be $\mathcal{A}_{1}$. Similar
to the tripartite case, we define $p_{N}\equiv
P(1,1,\dots,1|\hat{u}_{1},\hat{u}_{2},\dots,\hat{u}_{N})$, $q_{N}\equiv
P(d_{1},d_{2},\dots,d_{N}|\hat{v}_{1},\hat{v}_{2},\dots,\hat{v}_{N})$ and
their difference to be $\mathcal{S}_{N}:=p_{N}-q_{N}$. We will show that only
non-local correlation can satisfy (4) with $\mathcal{S}_{N}>0$ and
furthermore, this non-locality is genuine.

######  Theorem 2.

If a non-local correlation satisfies the conditions in (4) with
$\mathcal{S}_{N}>0$, then it is a genuine multipartite non-local correlation.

###### Proof.

We will first show that no local theory can satisfy conditions (4) with
$\mathcal{S}_{N}>0$. Let us define a local hidden variable $\lambda$ that
captures the full description of the system. This $\lambda$ takes values from
a set $\Omega$. Moreover, we assert that the comprehensive depiction of the
joint system is denoted by $\nu(\lambda)$. In a local description, there exist
conditional probabilities $p_{A_{j}}(u_{j}|\hat{u}_{j},\lambda)$ and
$p_{A_{j}}(v_{j}|\hat{v}_{j},\lambda)$ such that the joint probability can be
expressed as follows:
$P(x_{1},x_{2},\dots,x_{N}|\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{N})=\sum_{\lambda\in\Omega}\nu(\lambda)\prod_{j=1}^{N}p_{A_{j}}(x_{j}|\hat{x}_{j},\lambda)$,
where $\hat{x}_{j}\in\\{\hat{u}_{j},\hat{v}_{j}\\}$. Now, let’s assume that
$\mathcal{S}_{N}>0$, indicating that $p_{N}>0$. This implies the existence of
a nonempty subset $\Lambda\subseteq\Omega$, such that for all
$\lambda\in\Lambda$, $P_{A_{i}}(1|\hat{u}_{i},\lambda)=1$ for all $i$.
Consequently, for all $r$ where $v_{r}\neq d_{i}$, we have
$P_{A_{i}}(v_{r}|\hat{v}_{i},\lambda)=0$, which further implies
$P(d_{i}|\hat{v}_{i},\lambda)=1$. Therefore, we can deduce that
$P(d_{1},d_{2},\dots,d_{N}|\hat{v}_{1},\hat{v}_{2},\dots,\hat{v}_{N},\Lambda)=P(1,1,\dots,1|\hat{u}_{1},\hat{u}_{2},\dots,\hat{u}_{N})=p_{N}$.
However, since $\Lambda\subseteq\Omega$, it follows that
$P(d_{1},d_{2},\dots,d_{N}|\hat{v}_{1},\hat{v}_{2},\dots,\hat{v}_{N},\Lambda)\leq
P(d_{1},d_{2},\dots,d_{N}|\hat{v}_{1},\hat{v}_{2},\dots,\hat{v}_{N})$. In
other words, $\mathcal{S}_{N}\leq 0$, which contradicts the initial assumption
of $\mathcal{S}_{N}$. Therefore, we can conclude that if any correlation
satisfies (4) and gives $\mathcal{S}_{N}>0$, then it must be non-local.

Now we will show the genuineness of the non-locality. Consider a $N$-partite
probability distribution
$P(x_{1},x_{2},\dots,x_{N}|\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{N})$
satisfying conditions in (4) with $\mathcal{S}_{N}>0$, which is not genuine.
Hence, it is a convex mixture of bi-local correlation. Let’s consider a
partition $(1,2,\dots,m)$ versus $(m+1,m+2,\dots,N)$. Consequently, there must
be at least one element within the convex combination that contributes non-
zero value to the condition
$P(1,1,\dots,1|\hat{u}_{1},\hat{u}_{2},\dots,\hat{u}_{N})>0$, given
$p_{N}>q_{N}\geq 0$. Let’s assume such an element takes the form
$P_{1}(x_{1},\dots,x_{m}|\hat{x}_{1},\dots,\hat{x}_{m},\lambda)P_{2}(x_{m+1},\dots,x_{N}|\hat{x}_{m+1},\dots,\hat{x}_{N},\lambda)$.
As $p_{N}>0$ we must have,

$\begin{split}P_{1}(1,\dots,1|\hat{u}_{1},\dots,\hat{u}_{m},\lambda)P_{2}(1,\dots,1|\hat{u}_{m+1},\dots,\hat{u}_{N},\lambda)>0\end{split}$
(5)

From the equation (4), we have $P_{A_{m}A_{m+1}}(v_{m}\neq
d_{m},1|\hat{v}_{m},\hat{u}_{m+1})=0$ and $P_{A_{N}A_{1}}(v_{N}\neq
d_{N},1|\hat{v}_{N},\hat{u}_{1})=0$. Hence, we have two important relations,

$\begin{split}&P_{1}(v_{m}\neq
d_{m}|\hat{v}_{m},\lambda)P_{2}(1|\hat{u}_{m+1},\lambda)=0,\\\
&P_{2}(v_{N}\neq
d_{N}|\hat{v}_{N},\lambda)P_{1}(1|\hat{u}_{1},\lambda)=0.\end{split}$ (6)

The form of $P_{1}$: Since by (5), we have
$P_{2}(1,\dots,1|\hat{u}_{m+1},\dots,\hat{u}_{N},\lambda)\neq 0$, one
definitely have $P_{2}(1|\hat{u}_{m+1},\lambda)\neq 0$. We also get
$P_{1}(v_{m}\neq d_{m}|\hat{v}_{m},\lambda)P_{2}(1|\hat{u}_{m+1},\lambda)=0$
from equation (6). Hence, we get $P_{1}(v_{m}\neq
d_{m}|\hat{v}_{m},\lambda)=0$ and $P_{1}(d_{m}|\hat{v}_{m},\lambda)=1$. Now as
the NS principle is assumed at the level of distribution
$P_{1}(x_{1}x_{2}\dots x_{m}|\hat{x}_{1}\hat{x}_{2}\dots\hat{x}_{m},\lambda)$,
a deterministic marginal of $P_{1}(x_{1}x_{2}\dots
x_{m}|\hat{x}_{1}\hat{x}_{2}\dots\hat{x}_{m},\lambda)$ indicates that
$P_{1}(x_{1}x_{2}\dots x_{m}|\hat{x}_{1}\hat{x}_{2}\dots\hat{x}_{m},\lambda)$
factorizes to certain
$P^{\prime}_{1}(x_{1},\dots,x_{m-1}|\hat{x}_{1},\dots,\hat{x}_{m-1},\lambda)P^{m}_{1}(x_{m}|\hat{x}_{m},\lambda)$.

From the equation (4), we also have $P_{A_{m-1}A_{m}}(v_{m-1}\neq
d_{m-1},1|\hat{v}_{m-1},\hat{u}_{m})=0$. As $P_{1}(x_{1}x_{2}\dots
x_{m}|\hat{x}_{1}\hat{x}_{2}\dots\hat{x}_{m},\lambda)$ factorizes we have
$P^{\prime}_{1}(v_{m-1}\neq
d_{m-1}|\hat{v}_{m-1},\lambda)P^{m}_{1}(1|\hat{u}_{m},\lambda)=0$, however
again using equation (5) we have $P^{m}_{1}(1|\hat{u}_{m},\lambda)\neq 0$ and
it implies that $P^{\prime}_{1}(d_{m-1}|\hat{v}_{m-1},\lambda)=1$. Using the
same argument as above $P^{\prime}_{1}$ factorizes. Hence eventually it is
easy to see that $P_{1}$ fully factorizes.

The form of $P_{2}$: We have $P_{1}(1|\hat{u}_{1},\lambda)\neq 0$ and
$P_{2}(v_{N}\neq d_{N}|\hat{v}_{N},\lambda)P_{1}(1|\hat{u}_{1},\lambda)=0$
from equation (5) and (6) respectively. Hence we have $P_{2}(v_{N}\neq
d_{N}|\hat{v}_{N},\lambda)=0$, thus $P_{2}(d_{N}|\hat{v}_{N},\lambda)=1$.
Using the same argument for factorization of $P_{1}$, $P_{2}$ also fully
factorizes.

Thus the distribution
$P_{1}(x_{1},\dots,x_{m}|\hat{x}_{1},\dots,\hat{x}_{m},\lambda)P_{2}(x_{m+1},\dots,x_{N}|\hat{x}_{m+1},\dots,\hat{x}_{N},\lambda)$
fully factorizes. Since the proof is the same for all cuts, the distribution
$P(x_{1},x_{2},\dots,x_{N}|\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{N})$ becomes
fully local. Such probability distribution cannot satisfy the equation (4) for
$\mathcal{S}_{N}>0$. Hence we are done. ∎

## Self-testing of genuine tripartite non-local states.—

We will now demonstrate self-testing for the tripartite case. Alice, Bob, and
Charlie each choose measurements $\hat{x}_{1}\in\\{A_{0},A_{1}\\}$,
$\hat{x}_{2}\in\\{B_{0},B_{1}\\}$, and $\hat{x}_{3}\in\\{C_{0},C_{1}\\}$,
respectively. Their binary outcomes are denoted as $x_{1}\in\\{\pm 1\\}$,
$x_{2}\in\\{\pm 1\\}$, and $x_{3}\in\\{\pm 1\\}$. We define a tripartite
correlation $P(x_{1},x_{2},x_{3}|\hat{x}_{1},\hat{x}_{2},\hat{x}_{3})$ that
satisfies the equality constraint 3. To illustrate genuine non-locality,
consider that Alice, Bob, and Charlie share the following general pure three-
qubit state,

$\begin{split}&|\Psi\rangle_{ABC}\\\
&=a_{000}|000\rangle+a_{001}|001\rangle+a_{010}|010\rangle+a_{100}|100\rangle\\\
&+a_{011}|011\rangle+a_{101}|101\rangle+a_{110}|110\rangle+a_{111}|111\rangle,\end{split}$
(7)

such that $\sum_{i,j,k\in\\{0,1\\}}|a_{ijk}|^{2}=1$. We consider
$\hat{x}_{1}=|u_{x_{1}}^{+}\rangle\langle
u_{x_{1}}^{+}|-|u_{x_{1}}^{-}\rangle\langle u_{x_{1}}^{-}|$,
$\hat{x}_{2}=|v_{x_{2}}^{+}\rangle\langle
v_{x_{2}}^{+}|-|v_{x_{2}}^{-}\rangle\langle v_{x_{2}}^{-}|$ and
$\hat{x}_{3}=|w_{x_{3}}^{+}\rangle\langle
w_{x_{3}}^{+}|-|w_{x_{3}}^{-}\rangle\langle w_{x_{3}}^{-}|$ to be the
measurement of Alice, Bob and Charlie respectively. We can choose the
following measurements with $\hat{x}_{1}\in\\{A_{0},A_{1}\\}$ and
$\hat{x}_{2}\in\\{B_{0},B_{1}\\}$ and $\hat{x}_{3}\in\\{C_{0},C_{1}\\}$ as,

$\begin{split}A_{0}&\equiv\left\\{\begin{array}[]{l}|u_{A_{0}}^{+}\rangle=|0\rangle,\\\
|u_{A_{0}}^{-}\rangle=|1\rangle\end{array}\right\\},\quad\\\
A_{1}&\equiv\left\\{\begin{array}[]{l}|u_{A_{1}}^{+}\rangle=\cos(\frac{\alpha_{1}}{2})|0\rangle+e^{i\phi_{1}}\sin(\frac{\alpha_{1}}{2})|1\rangle,\\\
|u_{A_{1}}^{-}\rangle=-\sin(\frac{\alpha_{1}}{2})|0\rangle+e^{i\phi_{1}}\cos(\frac{\alpha_{1}}{2})|1\rangle\end{array}\right\\},\quad\\\
B_{0}&\equiv\left\\{\begin{array}[]{l}|v_{B_{0}}^{+}\rangle=|0\rangle,\\\
|v_{B_{0}}^{-}\rangle=|1\rangle\end{array}\right\\},\quad\\\
B_{1}&\equiv\left\\{\begin{array}[]{l}|v_{B_{1}}^{+}\rangle=\cos(\frac{\alpha_{2}}{2})|0\rangle+e^{i\phi_{2}}\sin(\frac{\alpha_{2}}{2})|1\rangle,\\\
|v_{B_{1}}^{-}\rangle=-\sin(\frac{\alpha_{2}}{2})|0\rangle+e^{i\phi_{2}}\cos(\frac{\alpha_{2}}{2})|1\rangle\end{array}\right\\},\quad\\\
C_{0}&\equiv\left\\{\begin{array}[]{l}|w_{C_{0}}^{+}\rangle=|0\rangle,\\\
|w_{C_{0}}^{-}\rangle=|1\rangle\end{array}\right\\},\quad\\\
C_{1}&\equiv\left\\{\begin{array}[]{l}|w_{C_{1}}^{+}\rangle=\cos(\frac{\alpha_{3}}{2})|0\rangle+e^{i\phi_{3}}\sin(\frac{\alpha_{3}}{2})|1\rangle,\\\
|w_{C_{1}}^{-}\rangle=-\sin(\frac{\alpha_{3}}{2})|0\rangle+e^{i\phi_{3}}\cos(\frac{\alpha_{3}}{2})|1\rangle\end{array}\right\\}\end{split}$
(8)

where $0<\alpha_{1},\alpha_{2},\alpha_{3}<\pi$ and
$0\leq\phi_{1},\phi_{2},\phi_{3}<2\pi$. Without loss of generality we can
choose observable $A_{0}=B_{0}=C_{0}=\sigma_{z}\equiv|0\rangle\langle
0|-|1\rangle\langle 1|$ as long as we consider the general form of the state
$|\Psi\rangle_{ABC}$ and of the three observables $A_{1}$, $B_{1}$, $C_{1}$.

The class of pure three-qubit state that satisfies the condition (3) is of the
form:

$\begin{split}&|\Psi\rangle_{ABC}\\\
&=a|000\rangle-a(e^{i\phi_{3}}\cot\frac{\alpha_{3}}{2}|001\rangle+e^{i\phi_{2}}\cot\frac{\alpha_{2}}{2}|010\rangle\\\
&+e^{i\phi_{1}}\cot\frac{\alpha_{1}}{2}|100\rangle)+a(e^{i(\phi_{2}+\phi_{3})}\cot\frac{\alpha_{2}}{2}\cot\frac{\alpha_{3}}{2}|011\rangle\\\
&+e^{i(\phi_{1}+\phi_{3})}\cot\frac{\alpha_{1}}{2}\cot\frac{\alpha_{3}}{2}|101\rangle+e^{i(\phi_{1}+\phi_{2})}\cot\frac{\alpha_{1}}{2}\cot\frac{\alpha_{2}}{2}|110\rangle)\\\
&+e^{i\delta}\sqrt{\begin{split}&1-a^{2}(1+\cot^{2}\frac{\alpha_{1}}{2}+\cot^{2}\frac{\alpha_{2}}{2}\\\
&+\cot^{2}\frac{\alpha_{3}}{2}+\cot^{2}\frac{\alpha_{2}}{2}\cot^{2}\frac{\alpha_{3}}{2}\\\
&+\cot^{2}\frac{\alpha_{1}}{2}\cot^{2}\frac{\alpha_{3}}{2}+\cot^{2}\frac{\alpha_{1}}{2}\cot^{2}\frac{\alpha_{2}}{2})\end{split}}|111\rangle\end{split}$
(9)

where $a$ and $\delta$ satisfy $0\leq a\leq 1$ and $0\leq\delta<2\pi$. Then,
maximizing $\mathcal{S}_{3}$ over these parameters yields,
$\frac{1}{8}\left(2\left(7a\sqrt{1-7a^{2}}-17a^{2}\right)-1\right)\approx
0.02035$ for $\alpha_{1}=\alpha_{2}=\alpha_{3}=\frac{\pi}{2}$,
$a=\frac{1}{2}\sqrt{\frac{316-17\sqrt{158}}{1106}}\approx 0.15207$ and
$\delta=\phi_{1}+\phi_{2}+\phi_{3}$. The maximum value of $\mathcal{S}_{3}$ is
attainable by the following pure three-qubit state,

$\begin{split}&|\Psi_{max}^{\star 3}\rangle_{ABC}\\\
&=a|000\rangle-a(e^{i\phi_{3}}|001\rangle+e^{i\phi_{2}}|010\rangle+e^{i\phi_{1}}|100\rangle)\\\
&+a(e^{i(\phi_{2}+\phi_{3})}|011\rangle+e^{i(\phi_{1}+\phi_{3})}|101\rangle+e^{i(\phi_{1}+\phi_{2})}|110\rangle)\\\
&+e^{i\delta}\sqrt{1-7a^{2}}|111\rangle\end{split}$

We will first show that the maximum achievable value of $\mathcal{S}_{3}$ over
$\mathbb{C}^{n}\otimes\mathbb{C}^{n}\otimes\mathbb{C}^{n}$ is same as over
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Further, we will
see that the quantum state $|\Psi_{max}^{\star 3}\rangle_{ABC}$ can be self-
tested.

######  Theorem 3.

The maximum achievable value of $\mathcal{S}_{3}$ among all three-qubit states
represents the optimal value attainable within tri-partite quantum states of
any finite dimension.

###### Proof.

Here we only present the outline of the proof as the details are quite similar
to the proof given in [40].

Consider a general tripartite state $\rho$ shared among Alice, Bob, and
Charlie. The operator $\Pi_{x|\hat{x}}$ represents the outcome $x$ obtained by
Alice when she measures the observable $\hat{x}$. Similarly, we define
$\Pi_{y|\hat{y}}$ and $\Pi_{z|\hat{z}}$ for Bob and Charlie respectively.
Thus,

$\begin{split}P(x,y,z|\hat{x},\hat{y},\hat{z})=\operatorname{Tr}[\rho(\Pi_{x|\hat{x}}\otimes\Pi_{y|\hat{y}}\otimes\Pi_{z|\hat{z}})].\end{split}$
(10)

Considering no constraints on dimensionality, we employ Neumark’s dilation
theorem, restricting our focus to projective measurements. Thus, the
observables associated with Alice, Bob, and Charlie consist of Hermitian
operators having eigenvalues of $\pm 1$, depicted as:

$\begin{split}\hat{x}&=(+1)\Pi_{+|\hat{x}}+(-1)\Pi_{-|\hat{x}},\quad\hat{x}\in\\{A_{0},A_{1}\\}\\\
\hat{y}&=(+1)\Pi_{+|\hat{y}}+(-1)\Pi_{-|\hat{y}},\quad\hat{y}\in\\{B_{0},B_{1}\\}\\\
\hat{z}&=(+1)\Pi_{+|\hat{z}}+(-1)\Pi_{-|\hat{z}},\quad\hat{z}\in\\{C_{0},C_{1}\\}\end{split}$
(11)

Now using the lemma stated in [41] applies to Alice’s observable
$\\{A_{0},A_{1}\\}$ inducing a decomposition $H_{A}=\bigoplus_{i}H_{A}^{i}$,
Bob’s observable $\\{B_{0},B_{1}\\}$ inducing a decomposition
$H_{B}=\bigoplus_{j}H_{B}^{j}$, and Charlie’s observable $\\{C_{0},C_{1}\\}$
inducing a decomposition $H_{C}=\bigoplus_{k}H_{C}^{k}$. Consequently, the
following expression is derived:

$\begin{split}P(x,y,z|\hat{x},\hat{y},\hat{z})&=\sum_{i,j,k}\lambda_{ijk}\operatorname{Tr}[\rho_{ijk}(\Pi_{x|\hat{x}}\otimes\Pi_{y|\hat{y}}\otimes\Pi_{z|\hat{z}})]\\\
&\equiv\sum_{i,j,k}\lambda_{ijk}P_{ijk}(x,y,z|\hat{x},\hat{y},\hat{z})\end{split}$
(12)

where
$\rho_{ijk}=\frac{\Pi_{i}\otimes\Pi_{j}\otimes\Pi_{k}\rho\Pi_{i}\otimes\Pi_{j}\otimes\Pi_{k}}{\lambda_{ijk}}$
is, at most, a three-qubit state. The coefficients $\lambda_{ijk}$ are
determined as
$\lambda_{ijk}=\operatorname{Tr}[\rho(\Pi_{i}\otimes\Pi_{j}\otimes\Pi_{k})]$,
with $\lambda_{ijk}\geq 0$ $\forall i,j,k$ and $\sum_{i,j,k}\lambda_{ijk}=1$.

If the joint probability $P(x,y,z|\hat{x},\hat{y},\hat{z})$ meets the
conditions outlined in (3), then it implies that the joint probability
$P_{ijk}(x,y,z|\hat{x},\hat{y},\hat{z})$ will also adhere to the constraint
equations. Consequently, the maximum attainable value of $\mathcal{S}_{3}$
across all three-qubit states denotes the optimal value achievable within tri-
partite quantum states.

∎

######  Theorem 4.

If the maximum value of $\mathcal{S}_{3}$ is observed, then the state of the
system is equivalent up to local isometries to
$|\sigma\rangle_{ABC}\otimes|\Psi_{max}^{\star
3}\rangle_{A^{\prime}B^{\prime}C^{\prime}}$, where $|\Psi_{max}^{\star
3}\rangle$ attains the maximum value of $\mathcal{S}_{3}$ and $|\sigma\rangle$
is an arbitrary tripartite state.

###### Proof.

We can choose eigenstates of the observables $A_{0}$, $B_{0}$, and $C_{0}$ to
be computational eigenbasis:

$\begin{split}\Pi_{+|A_{0}}^{i}&=|2i\rangle\langle
2i|,\quad\Pi_{-|A_{0}}^{i}=|2i+1\rangle\langle 2i+1|,\\\
\Pi_{+|B_{0}}^{j}&=|2j\rangle\langle
2j|,\quad\Pi_{-|B_{0}}^{j}=|2j+1\rangle\langle 2j+1|,\\\
\Pi_{+|C_{0}}^{k}&=|2k\rangle\langle
2k|,\quad\Pi_{-|C_{0}}^{k}=|2k+1\rangle\langle 2k+1|\end{split}$

where $i$, $j$, and $k$ belong to the set $\\{0,1,2,\dots\\}$. Now, the
difference $p_{ijk}-q_{ijk}$, where $p_{ijk}=P_{ijk}(+,+,+|A_{0},B_{0},C_{0})$
and $q_{ijk}=P_{ijk}(-,-,-|A_{1},B_{1},C_{1})$, in the subspace
$H_{A}^{i}\otimes H_{B}^{j}\otimes H_{C}^{k}$ can attain the maximum value if
and only if $\rho_{ijk}=|\Psi_{max}^{\star
3}\rangle_{ijk}\langle\Psi_{max}^{\star 3}|$, where $|\Psi_{max}^{\star
3}\rangle_{ijk}$ represents the three-qubit state. Therefore, the unknown
state $|\chi\rangle$ can give the maximum value of $\mathcal{S}_{3}$, if and
only if,

$|\chi\rangle=\bigoplus_{i,j,k}\sqrt{\lambda_{ijk}}|\Psi_{max}^{\star
3}\rangle_{ijk}.$

Hence, if we choose the local isometries in the following way,

$\displaystyle\Phi_{A}=\Phi_{B}=\Phi_{C}=\Phi,$
$\displaystyle\Phi|2m,0\rangle_{XX^{\prime}}\rightarrow|2m,0\rangle_{XX^{\prime}},$
$\displaystyle\Phi|2m+1,0\rangle_{XX^{\prime}}\rightarrow|2m,1\rangle_{XX^{\prime}},$

where $XX^{\prime}\in\\{AA^{\prime},BB^{\prime},CC^{\prime}\\}$, then we have,

$\begin{split}&(\Phi_{A}\otimes\Phi_{B}\otimes\Phi_{C})|\chi\rangle_{ABC}|000\rangle_{A^{\prime}B^{\prime}C^{\prime}}\\\
&=|\sigma\rangle_{ABC}\otimes|\Psi_{max}^{\star
3}\rangle_{A^{\prime}B^{\prime}C^{\prime}},\end{split}$

where $|\sigma\rangle_{ABC}$ some junk state. ∎

## Self-testing of genuine $N$-partite non-local states.—

Now we will provide self-testing of genuine $N$-partite non-local
correlations. We will consider binary outcomes $\\{+,-\\}$ for all observables
$\hat{x}_{i}$. So our conditions become:

$\begin{split}&P_{A_{i}A_{i+1}}(+,+|\hat{v}_{i},\hat{u}_{i+1})=0,\\\ &\forall
i\leq N.\end{split}$ (13)

We consider $p_{N}\equiv
P(+,+,\dots,+|\hat{u}_{1},\hat{u}_{2},\dots,\hat{u}_{N})$ and $q_{N}\equiv
P(-,-,\dots,-|\hat{v}_{1},\hat{v}_{2},\dots,\hat{v}_{N})$. We have already
seen that no local theory will satisfy $\mathcal{S}_{N}(:=p_{N}-q_{N})>0$. Let
us consider the following general $N$-qubit pure state shared between $N$
parties $\mathcal{A}_{1},\mathcal{A}_{2},\dots,$ and $\mathcal{A}_{N}$:

$\begin{split}&|\Psi\rangle_{A_{1}...A_{N}}\\\ &=a_{0...0}|0...0\rangle\\\
&+\sum_{\begin{subarray}{c}x_{i}=1\\\ j\neq
i,x_{j}=0\end{subarray}}a_{x_{1},\dots,x_{i},\dots,x_{N}}|x_{1},\dots,x_{i},\dots,x_{N}\rangle\\\
&+\sum_{\begin{subarray}{c}x_{i_{1}}=1,x_{i_{2}}=1\\\
j\neq\\{i_{1},i_{2}\\},x_{j}=0\end{subarray}}a_{x_{1},\dots,x_{i_{1}},\dots,x_{i_{2}},\dots,x_{N}}|x_{1},\dots,x_{i_{1}},\dots,x_{i_{2}}\dots,x_{N}\rangle\\\
&+\dots\\\ &+\sum_{\begin{subarray}{c}x_{i}=0\\\ j\neq
i,x_{j}=1\end{subarray}}a_{x_{1},\dots,x_{i},\dots,x_{N}}|x_{1},\dots,x_{i},\dots,x_{N}\rangle\\\
&+a_{1...1}|1...1\rangle,\end{split}$ (14)

such that $\sum_{i_{1},\dots i_{N}\in\\{0,1\\}}|a_{i_{1},\dots i_{N}}|^{2}=1$,
and let the projective measurement of $A_{i}$ be
$\hat{x}_{i}^{j}=|A_{i}^{x^{j},+}\rangle\langle
A_{i}^{x^{j},+}|-|A_{i}^{x^{j},-}\rangle\langle A_{i}^{x^{j},-}|$. One can
choose the basis for the measurements $\hat{x}_{i}^{j}$ where $j\in\\{0,1\\}$
as follows:

$\begin{split}\hat{x}_{i}^{0}&\equiv\left\\{\begin{array}[]{l}|A_{i}^{x^{0},+}\rangle=|0\rangle,\\\
|A_{i}^{x^{0},-}\rangle=|1\rangle,\end{array}\right\\}\quad\\\
\hat{x}_{i}^{1}&\equiv\left\\{\begin{array}[]{l}|A_{i}^{x^{1},+}\rangle=\cos(\frac{\alpha_{i}}{2})|0\rangle+e^{i\phi_{i}}\sin(\frac{\alpha_{i}}{2})|1\rangle,\\\
|A_{i}^{x^{1},-}\rangle=-\sin(\frac{\alpha_{i}}{2})|0\rangle+e^{i\phi_{i}}\cos(\frac{\alpha_{i}}{2})|1\rangle,\end{array}\right\\}\end{split}$
(15)

where $0<\alpha_{i}<\pi$ and $0\leq\phi_{i}<2\pi$. Now the state that will
satisfy the equation (13) has amplitude of the following form:

$\begin{split}&a_{0...0}=a,\\\ &\forall i,x_{i}=1,j\neq i,x_{j}=0,\\\
&a_{x_{1},\dots,x_{i},\dots,x_{N}}=-a\cdot
e^{i\phi_{i}}\cdot\cot{\frac{\alpha_{i}}{2}}\\\ &\forall
i_{1},i_{2},x_{i_{1}}=1,x_{i_{2}}=1,j\neq\\{i_{1},i_{2}\\},x_{j}=0,\\\
&a_{x_{1},\dots,x_{i_{1}},\dots,x_{i_{2}},\dots,x_{N}}=a\cdot
e^{i(\phi_{i_{1}}+\phi_{i_{2}})}\cdot\cot{\frac{\alpha_{i_{1}}}{2}}\cdot\cot{\frac{\alpha_{i_{2}}}{2}},\\\
&\dots\dots,\\\
&a_{1...1}=e^{i\delta}\cdot\sqrt{\begin{split}1-&a^{2}_{0...0}-\sum_{\begin{subarray}{c}x_{i}=1\\\
j\neq i,x_{j}=0\end{subarray}}a^{2}_{x_{1},\dots,x_{i},\dots,x_{N}}\\\
&-\sum_{\begin{subarray}{c}x_{i_{1}}=1,x_{i_{2}}=1\\\
j\neq\\{i_{1},i_{2}\\},x_{j}=0\end{subarray}}a^{2}_{x_{1},\dots,x_{i_{1}},\dots,x_{i_{2}},\dots,x_{N}}\\\
&-\dots-\sum_{\begin{subarray}{c}x_{i}=0\\\ j\neq
i,x_{j}=1\end{subarray}}a^{2}_{x_{1},\dots,x_{i},\dots,x_{N}}\end{split}}\end{split}$
(16)

where $a$ and $\delta$ satisfy $0\leq a\leq 1$ and $0\leq\delta<2\pi$.

We will find the maximum value of $\mathcal{S}_{N}$ using equation (16) like
we have already found the maximum value for $\mathcal{S}_{3}$ for triparty
case. The result in [41] works for any $N$-party with two input and two output
scenarios. Hence, our calculations for triparty case easily generalize for
$N$-party also. Hence, we have self-testing of the state which gives the
maximum value for $\mathcal{S}_{N}$.

######  Theorem 5.

If the maximum value of $\mathcal{S}_{N}$ is observed, then the state of the
system is equivalent up to local isometries to
$|\sigma_{N}\rangle_{A_{1}\ldots A_{N}}\otimes|\Psi_{max}^{\star
N}\rangle_{A^{\prime}_{1}\ldots A^{\prime}_{N}}$, where $|\Psi_{max}^{\star
N}\rangle$ attains the maximum value of $\mathcal{S}_{N}$ and
$|\sigma_{N}\rangle$ is an arbitrary $N$-partite state.

###### Proof.

We omit the proof as it is similar to the proof of Theorem 4. ∎

## Numerical experiments: Non-ideal constraints and entanglement cost.—

In an ideal case for three party scenario, the paradox demands that marginal
probabilities should be equal to zero. But in a real experiment, this demand
is very difficult to ensure. Hence a modified paradox that accounts real
experiment can be of the form,

$\begin{split}&P_{AB}(++|A_{1}B_{0})\leq\epsilon,\\\
&P_{BC}(++|B_{1}C_{0})\leq\epsilon,\\\ &P_{AC}(++|A_{0}C_{1})\leq\epsilon.\\\
\end{split}$ (17)

where $\epsilon\geq 0$ is some noise parameter. Without loss of generality, we
can always choose the same noise parameter for each of the marginals. The
local bound for the modified paradox takes the form,

$\mathcal{S}_{3}^{\epsilon}\equiv
P(+,+,+|A_{0},B_{0},C_{0})-P(-,-,-|A_{1},B_{1},C_{1})\leq 3\epsilon.$

Now we will show that even with this modified paradox the maximum value of
$\mathcal{S}_{3}^{\epsilon}$ still can be achieved by performing projective
measurements on pure three-qubits states. The analytical technique being
introduced is not suitable for this scenario, necessitating a numerical
demonstration. To begin, we will ascertain the upper limit of
$\mathcal{S}_{3}^{\epsilon}$ by employing the NPA method pioneered by
Navascues, Pironio, and Acin. Next, we show that for a particular class of
three-qubit state, there exist projective measurements that achieve this upper
bound with a very high order of accuracy. Note that, we only need to consider
the noise parameter to be $0\leq\epsilon<1/3$. We obtain the upper bound on
$\mathcal{S}_{3}^{\epsilon}$ by maximizing it over $\mathcal{Q}_{3}$ (NPA
Hierarchy of level three). Next, we examined a specific category of pure
states involving three qubits

$\begin{split}&|\psi\rangle_{ABC}=a_{000}|000\rangle+a_{001}\left(e^{-i\phi_{2}}|010\rangle+e^{-i\phi_{1}}|100\rangle+e^{-i\phi_{3}}|001\rangle\right)\\\
&+a_{011}\left(e^{-i(\phi_{2}+\phi_{3})}|011\rangle+e^{-i(\phi_{1}+\phi_{3})}|101\rangle+e^{-i(\phi_{1}+\phi_{2})}|110\rangle\right)\\\
&+a_{111}e^{-i(\phi_{1}+\phi_{2}+\phi_{3})}|111\rangle,\end{split}$ (18)

We performed numerical maximization of $\mathcal{S}_{3}^{\epsilon}$ across all
states of the form 18 and measurement parameters, subject to the constraints
outlined in equation (17). This process yielded $\mathcal{S}_{3}^{\epsilon}$
values for various noise parameter settings, ranging from $0$ to
$\frac{1}{3}$. Fig. 1 illustrates the maximum $\mathcal{S}_{3}^{\epsilon}$
values under $\mathcal{Q}_{3}$, three-qubit quantum states, and local model.
Notably, we observed convergence between the maximum
$\mathcal{S}_{3}^{\epsilon}$ values under $\mathcal{Q}_{3}$ and three-qubit
quantum states, with an accuracy of approximately $10^{-6}$ within the error
margin $0\leq\epsilon\leq 0.08$.

We have also calculated different entanglement monotones for these states
against $GHZ$ and $W$ states in three and four parties, respectively. These
results are depicted in Fig. 2. These states require very few resources to
$GHZ$ and $W$ states.

Figure 1: Robustness of our self-testing scheme. Fig 1 depicts a plot of the
maximum value of $\mathcal{S}_{3}^{\epsilon}$ of the paradox for triparty
against the noise parameter $\epsilon$. The three-qubit system matches
$\mathcal{Q}_{3}$ in the error bound range $0\leq\epsilon\leq 0.08$. Fig 1
contains the variation of trace distance between ideal state and non-ideal
state according to the range of $\epsilon$. Here
$p_{\epsilon}=0.5*\lVert|\Psi_{max}^{\star}\rangle_{ABC}-|\Psi^{\epsilon}\rangle_{ABC}\rVert_{1}$,
where $|\Psi^{\epsilon}\rangle_{ABC}$ varies with $\epsilon$. We see that the
distance is upper bounded by $\epsilon^{\frac{1}{14}}.$

| GHZ | W | $|\Psi_{max}^{\star 3}\rangle$
---|---|---|---
Concurrence | 1 | 0.94 | 0.56
Negativity | 0.5 | 0.47 | 0.28
Log Negativity | 1 | 0.96 | 0.64
Entanglement Entropy | 1 | 0.92 | 0.42
Rényi Entanglement Entropy | 1 | 0.96 | 0.64

| GHZ-4 | W-4 | $|\Psi_{max}^{\star 4}\rangle$
---|---|---|---
Concurrence | 1 | 1 | 0.39
Negativity | 0.5 | 0.43 | 0.17
Log Negativity | 1 | 0.89 | 0.43
Entanglement Entropy | 1 | 1 | 0.24
Rényi Entanglement Entropy | 1 | 1 | 0.48

Figure 2: The quantum states that are self-tested via our scheme are non-
maximally entangled. Table 2 Different measure of entanglement monotone in 2
versus 1 bi-partition for three-qubit states, $GHZ$, $W$, and
$|\Psi_{max}^{\star 3}\rangle$ respectively. $|\Psi_{max}^{\star 3}\rangle$
requires less resource than the other maximally entangled states. Table 2
Different measure of entanglement monotone in 2 versus 2 bi-partition for
four-qubit states, $GHZ$, $W$, and $|\Psi_{max}^{\star 4}\rangle$
respectively. $|\Psi_{max}^{\star 4}\rangle$ requires less resource than the
other maximally entangled states.

## Discussion.—

To sum up, we’ve introduced a paradox similar to Hardy’s paradox [42],
designed for arbitrary $N$-partite systems where each parties have the choice
of two measurements with arbitrary outcomes. Detecting a breach of our paradox
witnesses genuine multipartite non-local behavior. Additionally, we’ve proved
self-testing statements for genuinely non-local, but non-maximally
multipartite entangled states.

Genuine multipartite non-local correlations are regarded as the most potent
manifestation of genuine multipartite quantum correlations. The process of
self-testing these quantum states serves to enhance the security and
efficiency of quantum communication protocols and quantum key distribution.
Moreover, it contributes to a deeper understanding of the intricate geometry
of quantum boundaries. The inherent genuine non-locality displayed by our
self-testing state makes it a valuable asset for secure quantum key
distribution applications.

In a bipartite scenario, the Bell state is the unique maximally entangled
state. However, in a multiparty setting, this uniqueness does not hold. Take,
for instance, the tripartite scenario where two types of maximally entangled
states exist: the $GHZ$ state and the $W$ state. Interestingly, when the party
count exceeds three, there is no such well-defined characterization found in
the existing literature. It is important to note that both the $GHZ$ and $W$
states qualify as genuine entangled states. Our numerical findings, showcased
in Tables 2 and 2, reveal that the quantum state leading to maximal violation
in our inequality requires fewer resources compared to the $GHZ$ and $W$
states. It is worth emphasizing that in measuring entangled monotone, we
employ bipartition, leveraging the presence of a unique maximally entangled
state in the bipartite scenario.

In the context of robust self-testing, a typically expected outcome is that
any quantum state corresponding to $\epsilon$ sub-optimal scenario should be
approximately $\mathcal{O}(\sqrt{\epsilon})$ close to the quantum state
associated with the ideal case upto local isometries in terms of trace
distance. In this work, we have demonstrated partially robust self-testing,
specifically showcasing that the trace distance between the quantum state
linked to the ideal case and the one associated with a non-ideal scenario is
upper-bounded by $\epsilon^{\frac{1}{14}}$. It’s noteworthy that we consider
the quantum state related to the non-ideal scenario as a three-qubit state.
The numerical data that provides the substance for the aforementioned claim
has been plotted in Figure 1. Moving forward, our immediate agenda involves
developing a comprehensive robust self-testing scheme. Additionally, we plan
to delve into the exploration of the amount of randomness that can be
extracted from these correlations corresponding to various noise scales. This
exploration will be the subject of our forthcoming research efforts.

## Acknowledgement.—

Ranendu Adhikary acknowledges funding and support from Indian Statistical
Institute, Kolkata. We would like to acknowledge stimulating discussions with
Kishor Bharti, Ashutosh Rai, Tamal Guha, Subhendu B. Ghosh, and Snehasish Roy
Chowdhury.

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# Geodesically completing regular black holes by the Simpson-Visser method

Kunal Pal<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kanpur,
Kanpur 208016, India  Kuntal Pal<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Kanpur,
Kanpur 208016, India  Tapobrata Sarkar<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Kanpur,
Kanpur 208016, India

###### Abstract

Regular black holes are often geodesically incomplete when their extensions to
negative values of the radial coordinate are considered. Here, we propose to
use the Simpson-Visser method of regularising a singular spacetime, and apply
it to a regular solution that is geodesically incomplete, to construct a
geodesically complete regular solution. Our method is generic, and can be used
to cure geodesic incompleteness in any spherically symmetric static regular
solution, so that the resulting solution is symmetric in the radial
coordinate. As an example, we illustrate this procedure using a regular black
hole solution with an asymptotic Minkowski core. We study the structure of the
resulting metric, and show that it can represent a wormhole or a regular black
hole with a single or double horizon per side of the throat. Further, we
construct a source Lagrangian for which the geodesically complete spacetime is
an exact solution of the Einstein equations, and show that this consists of a
phantom scalar field and a nonlinear electromagnetic field. Finally,
gravitational lensing properties of the geodesically complete spacetime are
briefly studied.

##  I Introduction

The appearance of spacetime singularities in general relativity (GR) is a
generic feature of gravitational collapse, that leads to the formation of
black holes. Even though, in black holes the singularity is hidden from an
asymptotic observer by the event horizon, it signals the breakdown of the
theory itself in very high curvature regions, a shortcoming of GR, which is
expected to be resolved in a full quantum theory of gravity. Due to the lack
of a consistent theory of quantum gravity, one of the popular approaches in
the literature is to construct phenomenological regular spacetime metrics,
where the usual curvature invariants are finite everywhere in spacetime.
However, often the cost of these by-hand constructions is that these regular
solutions violate at least one or all of the standard energy conditions,
indicating the presence of exotic source of matter. For reviews and further
references of such constructions, see Nicolini:2008aj ; Ansoldi:2008jw ;
Maeda:2021jdc ; Sebastiani:2022wbz ; Torres:2022twv .

Typically, the presence of a curvature singularity is synonymous with the fact
that the corresponding causal geodesics are incomplete, i.e., time-like or
null geodesics cannot be well defined after a finite proper time for a test
observer Hawking:1970zqf . However, in a recent work Zhou:2022yio , it was
shown that even some of the popular regular metrics studied widely in the
literature, e.g., the Hayward metric Hayward:2005gi  and the Culetu-Ghosh-
Simpson-Visser (CGSV) regular black hole metric with asymptotically Minkowski
core Simpson:2019mud ; Culetu:2013fsa ; Ghosh:2014pba , are not geodesically
complete for negative values of the radial coordinates $r$. Consequently, the
true meaning of ‘regular black hole’ in this context needs to be addressed
more carefully. In the same work Zhou:2022yio , simple modifications of the
above two metrics were also proposed, and it was shown that it is possible to
extend the casual geodesics to all the values of proper time, thereby making
the spacetime geodesically complete.

In this paper our motivation is to explore a systematic way of deforming a
geodesically incomplete spacetime into a geodesically complete spacetime, by
using the Simpson-Visser (SV) procedure of regularising a spacetime
singularity. Consider for example the SV modification SV1 ; SV2 ; Franzin2 ;
mazza ; shaikh:MNRAS2021 ; jcch ; Bronnikov:2021uta  that can be used to
regularise the curvature singularity of the Schwarzschild and other spacetime
by introducing a single non zero parameter that assumes a constant value SV1 .
111The SV method has gained a lot of recent attention. For a non-exhaustive
list of recent works, see - Ye:2023xyv  \- Junior:2022zxo . The resultant
metric, depending on the value of this constant parameter, can interpolate
between a wormhole branch and a regular black hole branch, and in all the
cases the metric is geodesically complete for the full ranges of the radial
coordinate. However, the qualitatively different CGSV black hole
Simpson:2019mud , which represents an exponential suppression of mass to smear
out the Schwarzschild singularity at the location $r=0$ Culetu:2013fsa ;
Ghosh:2014pba ; Simpson:2019mud  does not enjoy this last property. 222To the
best of our knowledge, the phenomenological version of this metric was first
proposed in Culetu:2013fsa , along with the rotating version in Ghosh:2014pba
. However, the full significance of this metric, from carefully chosen first
principles was elucidated in Simpson:2019mud ; Simpson:2021dyo ;
Simpson:2021zfl . As explained in Zhou:2022yio , the CGSV spacetime is
strictly valid only in the region of positive values of $r$. As a result, the
effective potential encountered by a massive particle has a singularity upon
extending $r$ to negative values, in a finite proper time.

In this paper we show that if we further modify the CGSV metric by the SV
method itself, we can cure its geodesic incompleteness for $r<0$. The
resulting metric may represent a wormhole or a regular black hole with two
horizons (per side of the throat), and a single horizon (per side of the
throat). In all the cases, the curvature invariants are shown to be regular
for all values of the radial coordinates, and also the casual geodesics are
shown to be complete everywhere. We find out the possible nature of the source
of this metric for which this is an exact solution of the Einstein equations.
Furthermore, the motion of photons and the resulting effective potential
encountered by them in this geometry are discussed as well.

The method proposed here should be contrasted with the one used in
Zhou:2022yio . In that paper, a somewhat different formalism was used to make
the CGSV solution geodesically complete. With a generic static, spherically
symmetric solution of the form

$ds^{2}=-\mathcal{A}(r)\text{d}t^{2}+\mathcal{A}^{-1}(r)\text{d}r^{2}+\mathcal{B}^{2}(r)\text{d}\Omega^{2}~{},$
(1)

the CGSV spacetime is represented by
$\mathcal{A}(r)=1-(2M/r)e^{-a/r},\mathcal{B}^{2}(r)=r^{2}$, with $M$, and $a$
(not to be confused with the angular momentum of a rotating metric) are
positive, real constants. As shown in Zhou:2022yio , the modification
$\mathcal{A}(r)=1-(2M/r)e^{-a^{2}/r^{2}}$ cure the geodesic incompleteness of
the CGSV solution in regions $r<0$. In that paper, a different prescription
was put forward to make the Hayward black hole Hayward:2005gi  geodesically
complete. However, instead of a case by case resolution, here we propose that,
a more systematic way to cure geodesic incompleteness that may exist in any
regular solution of the form in Eq. (1) is to perform the replacement
$r\to\sqrt{r^{2}+r_{0}^{2}}$, with a parameter $r_{0}\geq 0$. With this
replacement, now we have the modified function
$\mathcal{A}(r)=1-(2M/\sqrt{r^{2}+r_{0}^{2}})e^{-a/\sqrt{r^{2}+r_{0}^{2}}}$,
which is a two parameter family of solutions. Here, $a$ and $r_{0}$ are real
and positive non-zero constant parameters. Indeed, the interplay between
$a,r_{0}$ and $M$ leads to interesting branches of the geodesically complete
spacetime. Furthermore, as we will argue, this modification renders any
possible geodesically incomplete regular solutions to geodesically complete
ones as well.

##  II The modified SV metric with asymptotic Minkowski core metric

Following the discussion above, we focus on the CGSV spacetime, and consider a
closely related cousin - a two-parameter extension of the Schwazschild metric
of the form

$ds^{2}=-\Big{(}1-\frac{2Me^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}}{\sqrt{r^{2}+r_{0}^{2}}}\Big{)}\text{d}t^{2}+\Big{(}1-\frac{2Me^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}}{\sqrt{r^{2}+r_{0}^{2}}}\Big{)}^{-1}\text{d}r^{2}+\Big{(}r^{2}+r_{0}^{2}\Big{)}\text{d}\Omega^{2}~{}.$
(2)

In the limit $r_{0}\rightarrow 0$, the above metric reduces to the CGSV metric
introduced in Simpson:2019mud ; Culetu:2013fsa ; Ghosh:2014pba . On the other
hand, for $a\rightarrow 0$, this metric reduces to the SV modification of the
Schwarzschild metric first introduced in SV1 . Our goal is to show that this
metric is geodesically complete, discuss its possible extension beyond $r=0$,
and find the sources for which this metric is an exact solution of the
Einstein equations. In constructing this metric, we have motivated from the SV
procedure of regularising a curvature singularity by introducing a wormhole
throat at the location of singularity SV1 , and consequently followed the SV
procedure of replacing the radial coordinate $r$ to $\sqrt{r^{2}+r_{0}^{2}}$,
without changing the $\text{d}r^{2}$ part.

###  II.1 Structure of the metric

To better understand the structure of the metric of Eq. (2), we first use a
coordinate transformation $k^{2}=r^{2}+r_{0}^{2}$ to rewrite the radial part
as

$g_{kk}=\Big{(}1-\frac{2Me^{-\frac{a}{k}}}{k}\Big{)}^{-1}\frac{k^{2}}{k^{2}-r_{0}^{2}}~{}~{}.$
(3)

The inverse component of the metric $g^{kk}$, has zeros at the location of
$\bar{\Delta}=1-\frac{2Me^{-\frac{a}{k}}}{k}=0$, as well as at $k=\pm r_{0}$,
and therefore, the final nature of the metric is determined by the interplay
of these roots. To find out the the roots of the equation

$1-\frac{2Me^{-\frac{a}{k}}}{k}=0~{},$ (4)

which we label as $k_{+}$ and $k_{-}$, we will solve this equation
numerically.333Note that, in the following we shall choose the parameter
values in such a way that this equation has only two real roots.  This
condition is the same as that of the horizon condition for the original CGSV
metric Simpson:2019mud , however in a different coordinate system from the one
used here. Note also that for Eq. (4) to have real roots, the condition
$a<\frac{2M}{e}$ must be satisfied, where, $e$ is the Euler’s number. For
$a>\frac{2M}{e}$, there is no real root of the Eq. (4), while for
$a<\frac{2M}{e}$, there exists real positive roots, which coincides with each
other for $a=\frac{2M}{e}$. Though it is also possible to write down the exact
solution of Eq. (4) in terms of the Lambert W functions, as illustrated in
Simpson:2019mud , in this paper we work only with some chosen numerical values
of the parameters to illustrate the basic properties of the spacetime in a
simple manner. The nature of the line element in Eq. (2), for different
branches of the parameter space can be classified as described below, where
for the ease of illustration we have assumed $M=1$.

Now we consider the above mentioned two cases, $a<2M/e$ and $a>2M/e$
separately. For the first case, as an illustration, we take the value of
$a=0.5<\frac{2M}{e}$. In this case we see that Eq. (4) has two real, positive
roots, with numerical values $k_{-}\sim 0.232$ and $k_{+}\sim 1.399$
respectively. The value of the other parameter $r_{0}$ is not fixed and
depending on the relative values of $r_{0}$, $k_{-}$ and $k_{+}$, several
different situations might appear.

$\bullet$ $a<\frac{2M}{e}$, $r_{0}>k_{+}$ : In this case the metric represents
a two way wormhole with a throat at $r=0~{}(k=r_{0})$, as can be seen by the
following analysis. (i) As long as the above mentioned condition is satisfied,
there is no possibility of a horizon present at spacetime. The metric can be
extended from $r=\infty$ to $r=-\infty$, through $r=0$. (ii) The analysis of
all the associated curvature scalars confirms that there is no curvature
singularity present in the spacetime.

$\bullet$ $a<\frac{2M}{e}$, $k_{-}<r_{0}<k_{+}$ : In this case the metric has
a single horizon at the location of $k_{+}$, and the throat of a wormhole at
the location $r=0~{}(k=r_{0})$. Here, the metric can be extended from
$r=\infty$ to $r=-\infty$ through $r=0$. However, due to the presence of the
event horizon, the wormhole is only one way traversable. Similarly, the
curvature scalar can be shown to be finite in this case also.

$\bullet$ $a<\frac{2M}{e}$, $r_{0}<k_{-}<k_{+}$ : For this parameter range,
the metric consists of two event horizons at the locations $k_{-}$ and
$k_{+}$. The metric also has a timelike throat at the location $r=0$, and the
range of the radial coordinate is $-\infty<r<\infty$, thus the metric is a one
way traversable wormhole.

$\bullet$ Finally, we consider the case $a>\frac{2M}{e}$ : In this case, which
is qualitatively different from the others, the metric does not have a horizon
for any values of the other parameters, and the final metric always represents
a wormhole spacetime.

It is important to note that the metric reduces to the Schwarzschild solution
for large values of the radial coordinate values $r$. However, in contrast to
that of the construction Simpson:2019mud ; Zhou:2022yio , the metric in this
case, in general, does not have a Minkowski core. The global structure of this
spacetime can be visualized by drawing the corresponding Penrose diagrams. For
different branches of the spacetime (wormhole, regular black holes with single
or double horizons per side of the throat) discussed above, the Penrose
diagrams are qualitatively similar to the black bounce and the charged black-
bounce solution discussed in SV1 ; Franzin2 , and we do not repeat them here.

###  II.2  Motion of a massive particle and geodesic completeness

To understand geodesic completeness, and hence the singularities of our
proposed line element in Eq. (2), we now consider the radial motion of a
massive test particle moving freely in this spacetime. For illustration, it is
useful to consider the general form of a static spherically symmetric line
element of Eq. (1). For our case the explicit expressions for the unknown
functions $\mathcal{A}(r)$ and $\mathcal{B}(r)$ can be read off directly.

The equation of motion of a radially moving massive free particle in this
geometry can be written as (an overdot denotes derivative with respect to the
proper time $\tau$)

$\dot{r}^{2}=E^{2}-\mathcal{A}(r)~{},$ (5)

where $E=\mathcal{A}(r)\dot{t}$ is the conserved specific energy along the
particle trajectory. From this expression, we see that the effective potential
encountered by the massive particle is $V_{eff}(r)=\mathcal{A}(r)$. Thus, for
the line element in Eq. (2), we have the effective potential to be

$V_{eff}(r)=1-\frac{2Me^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}}{\sqrt{r^{2}+r_{0}^{2}}}~{}.$
(6)

The dependence of the effective potential on the coordinate $r$ (with possible
extension to negative values of $r$) determine the behaviour of the motion of
the massive particle in this geometry, and hence whether the metric is
geodesically complete.

The first point to note is that the function $\mathcal{A}(r)$, and hence the
effective potential, is continuous across $r=0$. Specifically,
$V_{0}=V_{eff}\big{|}_{r\rightarrow 0}=1-\frac{2M}{r_{0}}e^{-a/r_{0}}~{}.$
This fact is in contrast with the CGSV metric, which is just the
$r_{0}\rightarrow 0$ limit of the metric in Eq. (2). In fact, it can be easily
checked that the effective potential of CGSV spacetime is divergent at
$r\rightarrow 0^{-}$.

Next, we find out the proper time a radially moving massive particle takes to
reach $r=0$. Assuming that the particle starts from an initial location of
$r_{i}$, the proper time it takes to reach a final location $r$ is given by
the formula

$\tau(r)=\int_{r_{i}}^{r}\frac{dr}{\sqrt{E-V_{eff}(r)}}~{}.$ (7)

Depending on the sign of $V_{0}$ it is useful to consider three different
cases.

$\bullet$ $V_{0}>0$ : Since at asymptotic infinity, the metric of Eq. (2)
reduces to the Minkowski metric, the effective potential at $r=0$ can be
positive when, for some particular choices of parameters, $a,r_{0},M$, the
function $\mathcal{A}(r)$ has no real root or has two roots. Thus in this
case, the spacetime is either horizonless or has an even number of horizons.
One such situation is illustrated in Fig. 2 by the red curve. The choice of
the parameters are made in such a way that the metric has two horizons for
positive values of $r$. We can also choose these parameters in such a way that
there are no real roots of the function $\mathcal{A}(r)$, so that in that case
as well, the effective potential at $r=0$ is positive.

To study the particle motion in this case we first consider that the energy
$E$ of the particle is less than $V_{0}$, and it starts from $r_{i}>0$. Here
the test particle cannot reach at $r=0$, and will bounce back before crossing
either of the two horizons, or after crossing both the horizons. When
$E=V_{0}$, the particle takes an infinite proper time to reach $r=0$, as seen
from Eq. (7). Finally, when $E>V_{0}$ the particle takes a finite amount of
proper time to reach at $r=0$ from the initial position. In Fig. 2 we have
plotted the proper times the particle takes to reach at $r=0$ starting from
different initial positions, by numerically integrating Eq. (7). The proper
time is finite in all cases.

Since the metric function $\mathcal{A}(r)$ is continuous across $r=0$ and the
effective potential is finite everywhere, we can extend the metric across
$r=0$ to negative values of $r$ as well. Thus a test particle can move from
$-\infty$ to $\infty$ in this spacetime, and this takes an infinite amount of
proper time.

Figure 1: Plot of the effective potential $V_{eff}$. The choices of the
parameters for three different curves shown are : $a=0.5,r_{0}=0.1,M=1$ (Red),
and $a=0.693,r_{0}=1,M=1$ (Gray), and $a=0.2,r_{0}=0.1,M=1$ (Green).

Figure 2: The plot of the proper time a radial test particle takes to reach
$r=0$ starting from different initial positions $r$. Here, the parameter
values are the same as the red curve in Fig. 2, and
$E(=1)>V_{eff}\big{|}_{r=0}$.

$\bullet$ $V_{0}<0$ : Next we consider the case when the effective potential
is negative at $r=0$ (the green curve in Fig. 2). In this case, the parameters
take values in such a way that the metric function $\mathcal{A}(r)$ has an odd
number of real zeros on the positive $r$ axis. From an analysis similar to the
previous case, we see that a test particle takes a finite amount of proper
time to reach $r=0$, and hence we can extend the spacetime beyond $r=0$.

$\bullet$ $V_{0}=0$ : Finally, when the effective potential vanishes at $r=0$,
the metric has a single horizon at that location. This case is shown in Fig. 2
with the grey curve. The metric is continuous across $r=0$, and can be
extended to negative values of $r$.

From the above analysis we see that the metric in Eq. (2), unlike the original
CGSV metric, is geodesically complete, and has an extension beyond $r=0$ to
negative values of $r$. Here, we take the extension to the negative $r$ axis
to be the same as in Eq. (2). Furthermore, unlike the geodesically complete
extension of CGSV metric proposed in Zhou:2022yio , our metric is symmetric
about $r=0$, so that an observer must cross an even number of event horizons
(if they are present) to reach $r=\infty$ to $-\infty$. It is of course
possible to make other geodesically complete extensions of our metric, which
are not symmetric about $r=0$.

###  II.3 Energy momentum tensor and the source of the geodesically complete
metric

In this subsection we compute the energy-momentum tensor (EMT) corresponding
to the line element in Eq. (2), and discuss its interpretation in terms of
possible sources of such a solution of the Einstein equations.

If we assume that our proposed metric satisfies the Einstein equation with the
energy momentum tensor $T_{\mu\nu}$, then in the general case assuming that
the spacetime under consideration has two horizon locations on the positive
$r$ axis (at $r_{+}$ and $r_{-}$ respectively), 444Since the metric extension
is symmetric about $r=0$ there are two horizons for negative values of $r$, at
$-r_{+}$ and $-r_{-}$ respectively, as well. we can obtain the following
relation

$G^{t}_{t}-G^{r}_{r}=2\mathcal{A}(r)\frac{\mathcal{B}^{\prime\prime}(r)}{\mathcal{B}(r)}=\mp(\rho+p_{r})~{},$
(8)

where $\rho$ is the energy density and $p_{r}$ is the radial pressure, and a
prime denotes a derivative with respect to the radial coordinate. Furthermore,
in the above equation the minus sign is valid when we are considering the
region outside the outer horizon $r_{+}$, or the region between $r_{-}$ and
$-{r_{-}}$ or region outside $r_{-}$ (from $r_{-}$ to $r=-\infty$). These are
precisely the regions where the metric function $\mathcal{A}(r)$ is positive.
In the other two regions, where $\mathcal{A}(r)$ is negative, the plus sign
has to be used. Hence, it is useful to write the above equation in the
following suggestive form

$2|\mathcal{A}(r)|\frac{\mathcal{B}^{\prime\prime}(r)}{\mathcal{B}(r)}=-(\rho+p_{r})~{},$
(9)

In this form, it is easy to see that though the individual components of the
EM tensor are determined by the sign of $\mathcal{A}(r)$, the sign of the sum
of radial pressure and energy density does not depend on it. Rather its sign
is fixed only by the sign of the second derivative of the areal radius. As we
shall see below, this is also related to the nature of the source of the
metric.

In our case the quantity
$\frac{\mathcal{B}^{\prime\prime}(r)}{\mathcal{B}(r)}$ is everywhere positive,
and hence the NEC is violated everywhere except the horizons (where it is
satisfied marginally) for the metric considered in this paper. Furthermore,
since for a geodesically complete black hole, the effective potential
$\mathcal{A}(r)$ encountered by a radially moving test particle must be finite
and continuous, it can be seen that the quantity $\rho+p_{r}$ is smooth
everywhere. Note that the opposite is also true, i.e., for a geodesically
incomplete black hole, like the CGSV metric, $\rho+p_{r}$ is usually
discontinuous and divergent for some values of the radial coordinate $r$. For
the CGSV metric the discontinuity is at $r=0$.

These conclusions can be verified by explicitly computing the components of
the EMT. Consider the case when the spacetime has two sets of inner and outer
horizons. Assuming that the metric satisfies the Einstein equations
$G^{\nu}_{\mu}=T^{\nu}_{\mu}$, for regions outside the outer horizon $r_{+}$,
or between $r_{-}$ and $-{r_{-}}$ or outside $-r_{-}$ (from $-r_{-}$ to
$r=-\infty$), the energy density and the radial component of the pressure are
given by

$\rho=\frac{1}{(r^{2}+r_{0}^{2})^{2}}\bigg{[}\frac{2aMr^{2}}{r^{2}+r_{0}^{2}}e^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}-r_{0}^{2}\bigg{(}1-\frac{4Me^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}}{\sqrt{r^{2}+r_{0}^{2}}}\bigg{)}\bigg{]}~{},~{}p_{r}=-\frac{r_{0}^{4}+r_{0}^{2}r^{2}+2aMr^{2}e^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}}{(r^{2}+r_{0}^{2})^{3}}~{}.$
(10)

From these we obtain

$\rho+p_{r}=-\frac{2r_{0}^{2}}{(r^{2}+r_{0}^{2})^{2}}\bigg{[}1-\frac{2Me^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}}{\sqrt{r^{2}+r_{0}^{2}}}\bigg{]}~{}.$
(11)

It is easy to check that this expression is in accord with Eq. (8).
Furthermore, using this expression, we see that, except at the locations of
the inner and outer horizons, in the regions mentioned above, the quantity
$\rho+p_{r}$ is always negative. Thus, except at the horizon locations, in the
three regions mentioned above, the NEC is everywhere violated. In the region
between two horizons, magnitudes of $\rho$ and $p_{r}$ get interchanged, and
in that case also, it can be explicitly checked that NEC is everywhere
violated.

We can now find out the source of the metric given in Eq. (2). Following,
Bronnikov:2022bud , where it was shown that a scalar and a nonlinear
electromagnetic field can act as the source of an arbitrary static spherically
symmetric metric of the general form given in Eq. (1), we consider the
following matter action

$\mathcal{S}_{m}=\int\sqrt{-g}d^{4}x\bigg{[}-\frac{1}{2}h(\phi)\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)+\mathcal{L}(\mathcal{F})\bigg{]}~{}.$
(12)

Here, $h(\phi)$ and $V(\phi)$ are functions of the scalar filed $\phi$, and
$\mathcal{L}(\mathcal{F})$ is the Lagrangian density of the nonlinear
electromagnetic field. The function $h(\phi)$ determines the nature of the
scalar field (which is self interacting and minimally coupled to gravity), and
$V(\phi)$ is the potential term in the scalar field Lagrangian density. The
scalar field is a function of the radial coordinate, and
$\mathcal{L}(\mathcal{F})$ is a function of
$\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$, $F_{\mu\nu}$ being the electromagnetic
field tensor - defined in terms of the vector potential as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.

Performing a standard analysis Bronnikov:2022bud , we obtain the following two
relations

$\displaystyle
G^{t}_{t}-G^{r}_{r}=T^{t}_{t}-T^{r}_{r}~{}\rightarrow~{}\frac{\mathcal{B^{\prime\prime}}(r)}{\mathcal{B}(r)}=-\frac{1}{4}h(r)\phi^{\prime
2}(r)~{},$ (13)
$\displaystyle~{}~{}\text{and}~{}~{}G^{t}_{t}-G^{\theta}_{\theta}=T^{t}_{t}-T^{\theta}_{\theta}=-\mathcal{F}\mathcal{L}_{\mathcal{F}}~{}.$
(14)

Utilising the parameterisation freedom of the scalar field, in the following,
we take the customary choice of the scalar field $\phi(r)=\arctan(r/c)$.
Furthermore, as the source of the nonlinear electromagnetic field, we consider
a magnetic monopole of charge $q_{m}$, so that the invariant is given by
$\mathcal{F}=2q_{m}^{2}/\mathcal{B}^{4}(r)$. Given the metric components,
$\mathcal{A}(r)$ and $\mathcal{B}(r)$, the general procedure of obtaining the
unknown function appearing in the Lagrangian has been discussed in
Bronnikov:2022bud . E.g., given functional dependence of $\phi(r)$, as above,
the function $h(r)$ obtained from Eq. (13) is given by
$h(r)=-\frac{4}{\phi^{\prime
2}(r)}\frac{\mathcal{B^{\prime\prime}}(r)}{\mathcal{B}(r)}$, while, with
$\mathcal{F}=2q_{m}^{2}/\mathcal{B}^{4}(r)$, the function
$\mathcal{L}_{\mathcal{F}}$ can be obtained by solving the differential
equation
$\mathcal{L}_{\mathcal{F}}=-\frac{1}{\mathcal{F}}(T^{t}_{t}-T^{\theta}_{\theta})$.
The remaining function $V(\phi)$ can be determined by using any of the
components of the Einstein equations. Below we discuss the nature of these
functions for the line element Eq. (2).

From Eq. (13) it is easy to see that $h(r)$ in this case is just a negative
constant.555Notice that though sign of $\rho+p_{r}$ depends on the metric
function $\mathcal{A}(r)$, once we have made the above choice of the scalar
field, the sign of the function $h(r)$ is determined only by the areal radius
$\mathcal{B}(r)$ and its second derivative. Thus, the scalar field is always
phantom in nature. On the other hand, from Eq. (14) we can obtain the NED
Lagrangian as the solution of the following differential equation

$\frac{\partial\mathcal{L}(r)}{\partial
r}=\frac{1}{(r_{0}^{2}+r^{2})^{9/2}}\Big{[}4Mre^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}\Big{(}ar^{2}\big{(}a-4\sqrt{r^{2}+r_{0}^{2}}\big{)}+c^{2}\big{(}a\sqrt{r^{2}+r_{0}^{2}}-3r^{2}\big{)}-3r_{0}^{4}\Big{)}\Big{]}~{}.$
(15)

We can write the analytical solution of this equation in terms of incomplete
Gamma functions as

$\begin{split}\mathcal{L}(r)=-4M\bigg{[}\frac{4e^{-f(r)}}{(r_{0}^{2}+r^{2})^{3/2}}+\frac{12e^{-f(r)}\Big{(}a^{2}+2a\sqrt{r_{0}^{2}+r^{2}}+2(r_{0}^{2}+r^{2})\Big{)}}{a^{3}(r_{0}^{2}+r^{2})}\\\
-\frac{1}{a^{5}}\Big{[}(a^{2}-3r_{0}^{2})\Gamma(5,f(r))+5r_{0}^{2}\Gamma(6,f(r))-r_{0}^{2}\Gamma(7,f(r))\Big{]}\bigg{]}+\mathcal{C}~{},\end{split}$
(16)

where $\mathcal{C}$ is an integration constant, and we have denoted
$f(r)=\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}$. In Fig. 4, we have plotted the
numerical solution of the differential equation Eq. (15), with the integration
constant being fixed by the condition $\mathcal{L}(r=0)=0$, for the parameter
sets corresponding to the single and double horizon (on the positive $r$ axis)
cases respectively, from Fig. 2. It is easy to see that, for both of these
sets of parameter values, the solutions for the NED field are analytic
functions of the radial coordinate and have negative values at the limit of
$r\pm\infty$. For the parameter set, where the spacetime has two horizon
locations on the positive $r$ axis, $\mathcal{L}(r)$ is positive around $r=0$,
whereas, when there is only one horizon location at the $r$ axis, the NED
Lagrangian is always negative for the boundary condition we have specified in
the numerical solution.

Figure 3: Plot of the NED Lagrangian. The parameters corresponding to the red
and green curves are same as the red and green curves in Fig. 2.

Figure 4: Plot of the NED Lagrangian $\mathcal{L}(\mathcal{F})$ as a function
of $\mathcal{F}$. Here, $q_{m}=1$, and other parameters are the same as the
red curve in Fig. 2. We also see that, here,
$\mathcal{L}(\mathcal{F}\rightarrow 0)=-245.8$.

Furthermore, it is interesting study the functional dependence of the
Lagrangian density as a function of the EM invariant $\mathcal{F}$. To obtain
this, we need to invert the relation
$\mathcal{F}=\frac{2q_{m}^{2}}{(r^{2}+r_{0}^{2})^{2}}$, and substitute in the
exact expression in Eq. (16). The resulting analytical expression for
$\mathcal{L}(\mathcal{F})$ is too complicated to provide here, rather we study
the function numerically by plotting it in Fig. 4, for parameter values such
that the metric has double horizons in the positive $r$ axis. It is easy to
see that for all physically admissible values of $\mathcal{F}$, with the
parameter values indicated in the caption, $\mathcal{L}(\mathcal{F})$ is a
regular function of $\mathcal{F}$, and takes only negative values. In
particular in the limit $\mathcal{F}\rightarrow 0$, for the numerical values
used in this plot we have $\mathcal{L}(\mathcal{F}\rightarrow 0)=-245.8$.
Notice also that for all values of the radial coordinate, with $q_{m}>0$,
$\mathcal{F}$ is always positive, and in the limit $r\rightarrow\pm\infty$, it
goes to zero. Furthermore, there is a maximum value of $\mathcal{F}$ which it
takes in the limit $r\rightarrow 0$. We see that within this range of
$\mathcal{F}$, the Lagrangian density of the NED field is an analytic
function. These conclusions are true for other branches of the spacetime under
consideration as well.

##  III Photon motion in the geodesically complete background

In this section, we discuss the motion of a photon in the background of the
metric in Eq. (2), which is essential to study the shadow structure of the
regular and geodesically complete spacetime we have introduced in this paper.
It is also important to compare and contrast the null geodesics for different
branches of spacetime, in particular, the effect of geodesic completion by
implementing a wormhole throat. Using the standard method of finding null
geodesics for a generic spherically symmetric wormhole spacetime shaikh3 , we
can write down the effective potential for an asymptotically flat metric of
the form in Eq. (1) in the units of angular momentum $L$ of the particle as

$V_{p}(r)=L^{2}\frac{\mathcal{A}(r)}{\mathcal{B}^{2}(r)}~{}.$ (17)

For the case of metric represented by Eq. (2), the expression of $V_{p}(r)$ is
given by

$V_{p}(r)=\frac{L^{2}}{r^{2}+r_{0}^{2}}\Big{(}1-\frac{2M}{\sqrt{r^{2}+r_{0}^{2}}}e^{-\frac{a}{\sqrt{r^{2}+r_{0}^{2}}}}\Big{)}~{}.$
(18)

Now depending on the range of the parameters, three different cases may
appear, which we describe below.

$\bullet$ For the wormhole branch of the full solution, we note that in
general two different types of effective potentials may appear. As depicted in
Fig. 6, the first type includes symmetrically placed single maxima per side of
the throat, and the throat is the location of the minima of the effective
potential (shown by the red curve). Here, these locations outside the throat
can act as the photon sphere for light coming from a source that is on the
same side of the throat as that of an observer. A light ray that has an impact
parameter $b(=L/E)$ greater than that of a critical impact parameter ($b_{m}$)
will turn at the photon sphere and will reach an asymptotic observer shaikh3 .
The corresponding framework for strong lensing to calculate the deflection
angle, as presented in shaikh3  will also be applicable here. On the other
hand, when the source and the observer are on different sides of the throat,
then photons with an impact parameter $b>b_{m}$ will be turned back on the
same side of the throat, and will not reach the observer who is on the other
side of the throat. However, for $b<b_{m}$, the photons will cross both the
photon sphere and the throat and reach the observer on the other side.

Further, when the maxima of the potential coincides with the location of the
throat (see the black dotted curve in Fig. 6), the throat can itself act as a
photon sphere shaikh2 . Then photons from a source that is on the same side of
the observer, with impact parameter $b$ greater than that of the critical
impact parameter for the throat $b_{th}$ will be strongly lensed to the
asymptotic observer. On the other hand, when the source and the observer are
on two different sides of the throat, then rays with $b>b_{th}$, cannot reach
the observer on the other side. However, $b<b_{th}$ does not have a turning
point and will reach the observer on the other side of the throat.

Figure 5: Effective potential $V_{p}$ in the wormhole branch. The parameters
are : $r_{0}=1.5$ (red), $r_{0}=2.5$ (black), $a=0.5,M=1$, and $L=1$.

Figure 6: Effective potential $V_{p}$ when there is a single horizon per side
of the throat. Parameters: $r_{0}=1.2$, $a=0.5,M=1$, and $L=1$.

$\bullet$ When the metric represents a black hole with a single horizon per
side of the throat, the corresponding effective potential for photon motion is
shown in Fig. 6. In this case, there are symmetrically placed locations of
maxima per side of the throat, and the effective potential is minimum at the
throat. Similarly as discussed above, these locations outside the throat will
act as photon spheres for light with an impact parameter greater than the
critical impact parameter, and will escape to the faraway observer. However,
due to the presence of a horizon outside the throat, the second case as
mentioned above (throat as a maximum of effective potential) can never appear
here.

$\bullet$ In the case of a spacetime that has two horizons per side of the
throat, there will be one photon sphere per side of the throat, see Fig. 7
inset. However, due to the presence of the event horizons on both sides of the
throat, the throat cannot act as an effective photon sphere for the motion of
null rays.

Figure 7: Effective potential when there is are two horizons per side of the
throat. Parameters: $r_{0}=0.2$, $a=0.5,M=1,L=1$. The maximum of the effective
potential outside the outer horizon is shown in the inset.

##  IV Conclusions and discussions

In this paper, we have introduced a systematic way to construct geodesically
complete metrics by following the SV method of regularising spacetime
singularities. As an example, we have illustrated the case for the CGSV metric
with an asymptotically Minkowski core. The resultant extension of the metric
to negative values of the radial coordinate is symmetric about $r=0$, and the
metric components are continuous through $r=0$. The spacetime also has rather
interesting structure in the allowed ranges of the parameter space, and can
interpolate between a wormhole, a black hole with a single horizon, and a
black hole with two horizons per side of the throat present at $r=0$. All the
three branches are complete in terms of the time-like geodesics, as we have
shown explicitly. We also find out the possible source for which the proposed
geodesically complete metric is an exact solution of the Einstein equations.
The energy momentum tensor of the metric can be interpreted in terms of a
self-interacting scalar field which minimally coupled to the gravity, and a
NED field. For all the branches of this metric, the motion of photons in this
background is discussed in details.

Here we have illustrated the use of the SV regularisation method to construct
a geodesically complete black hole using the CGSV metric as an example. It
will be interesting to consider extensions of other such “regular metrics” and
use the SV regularisation to produce geodesically complete extensions from
them. In this context, we note that the SV method of constructing geodesically
complete extensions of conventional regular black hole spacetimes used in this
paper can be used to find out geodesically complete extensions (to negative
values of the radial coordinate) of any spherically symmetric static metric of
the general form of Eq. (1). Moreover, these extensions are symmetric with
respect to the radial coordinate, which ensures that the same form of the
metric functions can be used for smooth matching through the negative values
of the radial coordinate as well. We therefore expect the method put forward
here to be useful in a broad context.

## Acknowledgements

We thank Bidyut Dey and Rajibul Shaikh for discussions and help, and Tian Zhou
for a useful correspondence.

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