# Maximal Secret Reconstruction, Teleportation and Bell’s Inequality

Pratishtha Abrol  Centre for Quantum Science and Technology, International
Institute of Information Technology, Hyderabad, Hyderabad – 500032, Telangana,
India.
Center for Security, Theory and Algorithmic Research, International Institute
of Information Technology, Hyderabad, Hyderabad – 500032, Telangana, India.
Pahulpreet Singh  Centre for Quantum Science and Technology, International
Institute of Information Technology, Hyderabad, Hyderabad – 500032, Telangana,
India.
Center for Computational Natural Science and Bioinformatics, International
Institute of Information Technology, Hyderabad, Hyderabad – 500032, Telangana,
India  Indranil Chakrabarty  Centre for Quantum Science and Technology,
International Institute of Information Technology, Hyderabad, Hyderabad –
500032, Telangana, India.
Center for Security, Theory and Algorithmic Research, International Institute
of Information Technology, Hyderabad, Hyderabad – 500032, Telangana, India.
<EMAIL_ADDRESS>pahulpreet.singh<EMAIL_ADDRESS>
###### Abstract

A tripartite state is said to be a potential resource for secret sharing if
the state imposes restrictions on the teleportation fidelity of the bipartite
dealer–reconstructor and dealer–assistant channels in addition of being useful
for the state reconstruction singh2024controlled . Given a secret shareable
state in a pure three-qubit system acin2001three ; acin2001classification , we
are able to characterize the set of states with maximum possible
reconstruction fidelity (abbreviated as MSR states) for a fixed value of the
maximum teleportation fidelity that can be obtained out of both the
dealer–receiver channels. Similarly for a value giving the maximum of Bell-
CHSH value of both dealer–reconstructor and dealer–assistant channels, we are
able to find the maximum achievable reconstruction fidelity. Interestingly, we
find that all secret shareable states satisfy Bell’s inequality in both
dealer–reconstructor and dealer–assistant partitions. This brings out a new
mutual exclusivity between secret shareable state and Bell’s inequality
violations. Our result paves the way in identifying the best candidate among
the secret sharing resource states in achieving the maximum reconstruction
fidelity thus by setting the practical information transfer limit in a
possible resource theoretic extension of secret sharing. It also brings out a
new kind of mutual exclusiveness between the bipartite correlation and in the
ability of secret sharing in a tripartite setting.

##  I Introduction

In quantum information theory, entanglement schrodinger1935discussion ;
horodecki2009quantum  plays a significant role in various communication
protocols. These involve sending of qubits (known or unknown)
horodecki1996teleportation ; bennett1993teleporting ; pati2023teleportation ;
chakrabarty2010teleportation ; adhikari2008quantum ; bennett2005remote  or
classical bits bennett1992communication ; srivastava2019one ;
das2015distributed ; patro2017non ; vempati2021witnessing ;
roy2018deterministic ; wehner2018quantum  from one location to another without
violating the no-cloning theorem wootters1982single . Entanglement also
enables the transfer of the coherence of quantum states using fewer resources
pati2023teleportation . All of these help us to develop resource theory of
entanglement shahandeh2019resource . Understanding its significance as a
resource, entanglement in qubit-qubit datta2005entanglement ; li2009generation
, qutrit-qutrit dajka2008origination ; chkecinska2007separability  and qubit-
qudit vinjanampathy2012quantum ; mundra2019broadcasting  systems are
broadcasted along with other quantifiers of resources. It can be further used
to build up a quantum networkdas2018robust .
Entanglement, as a form of nonlocal correlation plays a major role as a
resource in sharing of classical secrets and also in the controlled
reconstruction of quantum states. Multiparty entanglement beyond bipartite
entanglement acts as a resource in secure quantum communication tasks like
controlled state reconstruction and secret sharing. In a general sense, secret
sharing is a process where a dealer shares a secret with $n$ share holders
such that only $k(k<n)$ or more share holders can reveal the secret but any
number of share holders fewer than $k$ cannot reveal the secret. Such a secret
sharing scheme is known as $(k,n)$ threshold scheme. This idea is further
explored in quantum information theory where we are able to share classical
bits as secrets using entangled resources. Interestingly, when we use a
quantum state then the secret sharing protocol will have certain nuances. In
the original article on quantum secret sharing by Hillery et. al.
hillery1999quantum  this was referred as quantum information splitting. In
consequent studies, the protocol has also been referred to as controlled
quantum teleportation (CQT) garg2023estimation  and controlled state
reconstruction (CSR) and sometimes loosely as quantum secret sharing (QSS)
hillery1999quantum ; cleve1999share ; karlsson1999quantum ;
bandyopadhyay2000teleportation ; markham2008graph ; li2010semiquantum ;
adhikari2010probabilistic ; ray2016sequential ; sazim2015retrieving ;
tittel2001experimental ; schmid2005experimental ; schmid2006experimental ;
bogdanski2008experimental . In a recent work singh2024controlled  a new
distinction was made in between the controlled state reconstruction (CSR) and
quantum secret sharing (QSS). Based on that we are able to identify the
conditions for a state to be useful for secret sharing within the set of
states which are useful for controlled state reconstruction.
Following Bell’s demonstration bell1964einstein , in 1964, that quantum
mechanics cannot be reconciled with the concept of local realism, there have
been several advancements into investigating nonlocality in quantum systems.
Bell inequalities, for example, establish an upper limit on the correlations
observed in measurement statistics of multi party systems clauser1969proposed
; svetlichny1987distinguishing . Exceeding this upper limit implies the
presence of a nonlocal correlation among the parties involved. For bipartite
systems there is a more convincing way to quantify the Bell violation in the
form of a closed formula horodecki1996teleportation . This highlights the
relation between the entanglement, bell’s inequality and teleportation
fidelity of states in a more explicit way with examples from either sides.
Bipartite bell violation also plays a significant role in context of
correlation and information processing abilities in multiparty situations
pandya2016complementarity .
In this work, we begin with a three party framework where without any loss of
generality we assume Alice is the dealer, Bob and Charlie are the assistant
and reconstructor respectively. A three-qubit state will be called as a secret
sharing resource if the state reconstruction fidelity at the Charlie’s side is
greater than $\frac{2}{3}$ in addition to the assumption that the fidelities
of Alice-Charlie (Dealer–Reconstructor) and Alice-Bob (Dealer–Assistant)
channels are less than $\frac{2}{3}$ singh2024controlled . This comes from the
idea that the individual information gain by the share holders should not be
more than the classical limit. Here, in this work we study how the degree of
faithfulness in transmission of a secret through a secret sharing protocol
relates with the subsystem’s individual ability to transmit a state through a
teleportation protocol. We characterize those states which will have maximum
reconstruction fidelity, given the maximum value of the bipartite
teleportation fidelities of Alice-Charlie (Dealer–Reconstructor) and Alice-Bob
(Dealer–Assistant) channels. We also give a single parameter canonical
representation of this maximal secret reconstructed resources (labeled as MSR
state) from the set of all secret shareable resources in the pure tripartite
state regime. We explore how the Bell-CHSH values of the bipartite
dealer–reconstructor, dealer–assistant subsystems correlate with the
maximization of reconstruction fidelities. In particular we have shown that
all secret shareable states satisfy Bell’s inequality. Equivalently, we can
say that three-qubit pure states with bipartite Bell violation is not suitable
for quantum secret sharing. This clearly brings out a new kind of mutually
exclusiveness of bipartite Bell violation and the secret sharing capacity of
the tripartite resource. In a larger sense, it brings out new kind of mutual
exclusivity indicating that for processes like secret sharing we probably need
tripartite states with minimum bipartite correlations. This entire study is
carried out for the three-qubit pure states (analytical proofs are for three-
qubit GHZ class of states with real coefficients and numerical results for the
larger class of three qubit GHZ states which encompasses all possible three-
qubit vectors acin2001three ; acin2001classification ).
The organization of the paper is explained as follows. In section II we give
brief overview of the concepts which form the basis of our work. Section III
introduces the theorem for maximal Secret reconstruction for given
teleportation fidelities of the bipartite channels connecting the dealer. We
also give the canonical representation of the states for which the maximal
reconstruction is possible. In Section IV we discuss the correlation between
the Bell’s CHSH value of the same bipartite channels and the reconstruction
fidelity of the state, In addition to that we establish a mutual exclusiveness
between the bipartite Bell violation and the secret shareable states. Section
V is devoted to conclusions and scope for future discussions.

##  II Related concepts

In this section, we provide an introduction to the various concepts that are
employed in our study. These include teleportation fidelity, controlled state
reconstruction fidelity and, the concept of bell’s inequality violation.

###  II.1 Teleportation Fidelity

Quantum teleportation is the process of transferring unknown quantum
information from one location to other. Teleportation fidelity is a measure of
how well a quantum state can be faithfully transmitted using an entangled
quantum channel aided with classical communication. The fidelity is a measure
of the similarity between the teleported state and the original state. Let us
consider a scenario involving Alice being the sender and Bob the receiver. To
set up the protocol, Alice and Bob share an entangled resource. Suppose Alice
wants to send the quantum state $\ket{\Psi}$. Alice would then perform a joint
measurement on her particle of the shared entangled state and $\ket{\psi}$,
based on which she sends classical information to Bob. Bob can then apply a
unitary operation to retrieve the teleported state $\ket{\psi^{\prime}}$. The
teleportation fidelity can be given by the overlap between the original state
and the teleported state:

$F^{T}=|\braket{\psi}{\psi^{\prime}}|^{2}$ (1)

Consecutively, the fidelity ranges from 0 to 1, 1 indicating perfect
teleportation, as in the case of the Bell state
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. We can transfer an
unknown state perfectly hillery1999quantum , considering no interference from
external sources. In other words, Bell state can be considered as an optimal
resource for transferring quantum information. It was also shown that
coherence of a quantum state can also be teleported with much lesser resource
pati2023teleportation . A fidelity less than 1 indicates some deviation or
error in the teleportation protocol.

It can be shown that one can still carry out teleportation without having an
entangled resource, classically to achieve a fidelity $\frac{2}{3}$, if not
$1$. Any quantum resource giving a fidelity greater than $\frac{2}{3}$, is in
general considered as a resource as there is quantum advantage associated in
beating the classical limit. As a resource, we consider a two-qubit mixed
state horodecki1996teleportation ,

$\rho_{AB}=\frac{1}{4}(I\otimes I+\sum\limits_{i}r_{i}.\sigma_{i}\otimes
I+\sum\limits_{i}s_{i}.I\otimes\sigma_{i}+\sum\limits_{ij}t_{ij}\sigma_{i}\otimes\sigma_{j}),$
(2)

where $\sigma_{i}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli matrices;
$r_{i}=(r_{1},r_{2},r_{3})$, $s_{i}=(s_{1},s_{2},s_{3})$ are the local Bloch
vectors and $t_{ij}=Tr(\rho(\sigma_{i}\otimes\sigma_{j}))$ are the elements of
the correlation matrix T = $[t_{ij}]_{3\times 3}$. The maximum teleportation
fidelity that can be achieved for such a state $\rho_{AB}$ is given by,

$F_{AB}=\frac{1}{2}(1+\frac{1}{3}\vartheta_{2}(\rho)),$ (3)

where $\vartheta_{2}(\rho)=Tr(\sqrt{T^{\dagger}T})$. The above formula
characterizes the teleportation fidelity in terms of the parameters of the
resource state. For a teleportation protocol to be successful, the
teleportation fidelity should be greater than $\frac{2}{3}$. This does not
mean that there do not exist entangled states for which the teleportation
fidelity is less than $\frac{2}{3}$ horodecki1996teleportation ;
adhikari2008quantum ; chakrabarty2011deletion ; chakrabarty2010teleportation .
In other words, not all entangled states guarantee a successful teleportation.

###  II.2 Controlled State Reconstruction and Secret Sharing

A controlled state reconstruction process involves reconstructing a quantum
state shared by the dealer at the share holder’s location. This requires
cooperation of other shareholder in the process of reconstruction. In the most
simplistic scenario when there is one dealer (Alice) and two share holders
(Bob and Charlie), we consider a general three-qubit state,

$\displaystyle\rho_{ABC}=\frac{1}{8}[I^{\otimes
3}+\sum_{i=1}^{3}a_{i}.\sigma_{i}\otimes I^{\otimes 2}+{}$
$\displaystyle\sum_{j=1}^{3}I\otimes b_{j}.\sigma_{j}\otimes
I+\sum_{k=1}^{3}I^{\otimes 2}\otimes c_{k}.\sigma_{k}+{}$
$\displaystyle\sum_{i,j=1}^{3}q_{ij}\sigma_{i}\otimes\sigma_{j}\otimes
I+\sum_{i,k=1}^{3}r_{ik}\sigma_{i}\otimes I\otimes\sigma_{k}+{}$
$\displaystyle\sum_{j,k=1}^{3}s_{jk}I\otimes\sigma_{j}\otimes\sigma_{k}+\sum_{i,j,k=1}^{3}t_{ijk}\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}],$
(4)

as the resource state. Here $a_{i},b_{j},c_{k}$ are local Bloch vectors and
the correlation matrices are given by,
$Q=\\{q_{ij}\\}=Tr(\rho_{ABC}(\sigma_{i}\otimes\sigma_{j}\otimes I))$,
$R=\\{r_{ik}\\}=Tr(\rho_{ABC}(\sigma_{i}\otimes I\otimes\sigma_{k}))$ and
$S=\\{s_{jk}\\}=Tr(\rho_{ABC}(I\otimes\sigma_{j}\otimes\sigma_{k}))$. These
correlation matrices are of order $3\times 3$. Here
$\tau=t_{ijk}=Tr(\rho_{ABC}(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}))$ is
the correlation tensor. The expression for maximum reconstruction fidelity for
a three-qubit state $\rho_{ABC}$ when Alice is the dealer, Bob is the
assistant and Charlie is the reconstructor, is given by

$\mathcal{F}_{ABC}^{CSR}=\frac{1}{2}+\frac{1}{6}\vartheta_{3}(\rho)$ (5)

where $\vartheta_{3}(\rho)$ is a function of the parameters of the state,
given by

$\vartheta_{3}(\rho)=\frac{\|R+T\|_{1}+\|R-T\|_{1}}{2}.$ (6)

If the reconstruction fidelity $\mathcal{F}^{CSR}>\frac{2}{3}$ , then we have
a quantum advantage as mentioned in singh2024controlled . This measure helps
us to identify the states that are useful for controlled state reconstruction.

Note that the secret sharing process is not exactly equivalent to a
reconstruction process. That is, along with successful reconstruction with a
fidelity greater than $\frac{2}{3}$, we have to ensure that the share holders
will not be able to reconstruct the secret on their own. This independent
reconstruction can be taken equivalent to a teleportation protocol and only be
possible if the bipartite channel achieve teleportation fidelity more than
$\frac{2}{3}$. So to identify a three-qubit state $\rho_{ABC}$ which is useful
for the secret sharing protocol we need additional constraints that inhibit
teleportation across bipartitions. Since, for a channel to be useful for
teleportation, we have $F>\frac{2}{3}$, we have the conditions:

$F_{AB}\leq\frac{2}{3},F_{AC}\leq\frac{2}{3}$ (7)

along with the condition $\mathcal{F}^{CSR}>\frac{2}{3}$ as shown in
singh2024controlled  for a dealer ($A$) and share holders ($B$ and $C$)
setting. These conditions together ensure faithful transport of the secret,
while also accounting for dishonest recipients.

Note: 1\. In the later section we will call the maximum achievable
teleportation fidelity of a channel as simple teleportation fidelity to avoid
confusion.
2\. From now on, for the purpose of this paper, we will be referring to
teleportation fidelity in the notation $F$ and reconstruction fidelity as
$\mathcal{F}$.

###  II.3 Bell’s Inequality Violation

Bell’s inequality violation is a phenomenon in quantum mechanics that
demonstrates the non classical nature of correlations between entangled
particles. The violation of the inequality implies the presence of nonlocal
correlations that cannot be explained by classical realism. According to local
realism, physical properties of a system have well defined values, independent
of measurement settings on distant particles. The Bell-CHSH inequality, for a
pair of entangled pparticles with measurement settings $a,b$ and outcomes
$A,B$ is given by:

$|\mathcal{S}|\leq 2$ (8)

where $\mathcal{S}$ is the CHSH correlation given by
$\mathcal{S}=E(a,b)+E(a^{\prime},b)+E(a,b^{\prime})-E(a^{\prime},b^{\prime})$
and $E(a,b)$ is the correlation between the outcomes $A$ and $B$ for
measurement settings $a$ and $b$. All bipartite pure entangled states violate
the Bell inequality yu2012all . For a two-qubit mixed state to violate the
Bell-CHSH inequality, it must hold the condition that the Bell-CHSH value
$\mathcal{S}$ is greater than 2, to imply the presence of nonlocal correlation
that cannot be described by the assumptions of local realism. The Bell-CHSH
observable quantity for any arbitrary bipartite state $\rho$ (2) can be
defined as

$\mathcal{S}(\rho)=2\sqrt{M(\rho)}$ (9)

where $M(\rho)=m_{1}+m_{2}$. $m_{1}$ and $m_{2}$ are the two greatest
eigenvalues of $T^{T}_{\rho}T_{\rho}$, the correlation matrix for the two-
qubit state being $T_{\rho}$ defined as
$(T_{\rho})_{ij}=Tr[\rho(\sigma_{i}\otimes\sigma_{j})]$ where $\sigma_{i}$
represent the respective Pauli Matrices. Drawing from the condition for Bell
Inequality Violation, it is clear that, $M(\rho)\leq 1$ is equivalent to the
Bell inequality not being violated.

It is interesting to note, that in the context of teleportation fidelity, it
is an established fact that $\vartheta_{2}(\rho)\geq M(\rho)$
horodecki1996teleportation , for an arbitrary mixed spin-$\frac{1}{2}$ state
to be useful for teleportation. This inequality results from the definition of
the quantity $\vartheta_{2}(\rho)=\Sigma_{i=1}^{3}\sqrt{m_{i}}$ with $m_{i}$
being the eigenvalues of the matrix $T^{T}T$. Since $M(\rho)>1$ for a state
violating Bell’s inequality, we can estimate

$\displaystyle F_{max}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}(1+\frac{1}{3}M(\rho))>\frac{2}{3}$ (10)

Note that the absence of Bell’s inequality violation does not rule out the
presence of entanglement or nonlocality.

##  III  Reconstruction Fidelity Versus Teleportation Fidelity in Secret
Sharing

In this section we obtain the relation between the reconstruction fidelity of
the shared pure tripartite resource state and the maximum teleportation
fidelity between the two bipartite channels involving the dealer. In
particular, we consider those states that are useful for secret sharing in the
class of the generalized GHZ (Greenberger-Horne-Zeilinger) states
acin2001three ; acin2001classification  representing all possible pure states
in the tripartite scenario. We show that there exists a limit to the maximal
reconstruction fidelity such a tripartite state can achieve, given the maximal
achievable teleportation fidelity that can be obtained from the bipartite
channels (dealer–reconstructor/dealer–assistant). Furthermore, we are also
able to identify the family of states that will achieve maximum reconstruction
fidelity by giving its canonical form.
Let us choose a state from the generalized GHZ (Greenberger-Horne-Zeilinger)
class of states with real coefficients shared between three parties – Alice
($A$) as the dealer, Bob ($B$) as the assistant and Charlie ($C$) as the
reconstructor. It is important to mention that these set of states that can be
converted into the GHZ state $\frac{1}{\sqrt{2}}(\ket{000}+\ket{111})$ using
Stochastic Local Operation and Classical Communication (SLOCC) with non zero
probability acin2001three ; acin2001classification . These states can be
expressed as :

$\displaystyle\ket{\psi(GHZ^{R})}_{ABC}=$
$\displaystyle\lambda_{0}\ket{000}+\lambda_{1}\ket{100}+\lambda_{2}\ket{101}{}$
(11) $\displaystyle+\lambda_{3}\ket{110}+\lambda_{4}\ket{111},$

where $\lambda_{i}\geq 0,\Sigma_{i}\lambda_{i}^{2}=1$ This state is a
derivative of the Acin state acin2001classification  obtained by putting the
complex part $\phi$ associated with $\lambda_{1}$ as $e^{i\phi}\lambda_{2}$ to
$0$. The reconstruction fidelity at Charlie’s ($C$) location, for such a state
in terms of $\lambda_{i}$, is given by,

$\displaystyle\mathcal{F(GHZ^{R})}_{ABC}=\frac{1}{2}+\frac{1}{6}\vartheta_{3}(\rho);{}$
$\displaystyle\vartheta_{3}(\rho)=4\lambda_{0}\text{max}\\{\lambda_{2},\lambda_{4}\\}+1$
(12)

Here in this setting we consider the maximum ($F_{max}$) of the teleportation
fidelities $F_{AB}$ and $F_{AC}$ of both dealer–assistant ($A-B$) and
dealer–reconstructor ($A-C$) channels of the tripartite state
$\ket{\psi(GHZ^{R})}$. In other words, $F_{max}=max(F_{AB},F_{AC})$. In the
same spirit we can write $\vartheta_{2}(\rho)$ of $F_{max}$ as
$\vartheta_{2}(\rho)=max(\vartheta_{2}(\rho_{AB}),\vartheta_{2}(\rho_{AC}))$.
It turns out that $\vartheta_{2}(\rho)$ depends upon two correlation matrices
$R_{AC}$ and $Q_{AB}$ of the channels $AC$ and $AB$ respectively:

$\vartheta_{2}(\rho)=\text{max}\\{||R||_{AC},||Q||_{AB}\\},$ (13)

where

$\displaystyle||R||=2\lambda_{0}\lambda_{2}+2\sqrt{\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2})^{2}}$
(14)
$\displaystyle||Q||=2\lambda_{0}\lambda_{3}+2\sqrt{\lambda_{0}^{2}\lambda_{3}^{2}+(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2}}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{3}^{2}-\lambda_{2}^{2})^{2}}$
(15)

It can be seen from the equations (III) and (III) that the value of
$\vartheta_{2}(\rho)$ depends on the value of $\lambda_{2}$ and $\lambda_{3}$.
If $\lambda_{2}>\lambda_{3}$ we have $\vartheta_{2}(\rho)=||R||_{AB}$ and if
$\lambda_{3}>\lambda_{2},\vartheta_{2}(\rho)=||Q||$. At
$\lambda_{2}=\lambda_{3}$ the values of $||R||=||Q||$.

###### Theorem 1

Given a three-qubit pure state $\ket{\psi(GHZ^{R})}_{ABC}$, which is achieving
a quantum advantage for secret sharing, implying that $F_{AB}\leq\frac{2}{3}$
and $F_{AC}\leq\frac{2}{3}$ ( equivalently we can say that
$F_{max}\leq\frac{2}{3}$ or $\vartheta_{2}(\rho)\leq 1$) and
$\mathcal{F}_{ABC}>\frac{2}{3}$ (or equivalently,$\vartheta_{3}(\rho)>1$), for
a teleportation fidelity of $F_{max}$, the reconstruction fidelity is bounded
by the following inequality:

$\mathcal{F_{ABC}}\leq 2F_{max}-\frac{1}{3}$ (16)

which can also be expressed as

$\vartheta_{3}(\rho)\leq 1+2\vartheta_{2}(\rho)$ (17)

Proof. Let us define 2 states, an arbitrary state
$\rho^{a}\equiv\\{\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$
and a boundary state that achieves equality in the above theorem,
$\rho^{b}\equiv\\{\Lambda_{0},\Lambda_{1},\Lambda_{2},\Lambda_{3},\Lambda_{4}\\}$.
We can divide this proof into three parts as

$\vartheta_{3}(\rho^{a})\overset{(1)}{\leq}\vartheta_{3}(\rho^{b})\overset{(2)}{=}2\vartheta_{2}(\rho^{b})+1\overset{(3)}{\leq}2\vartheta_{2}(\rho^{a})+1$
(18)

with

  1. 1.

$\forall\rho^{a}\exists\rho^{b}:\vartheta_{3}(\rho^{a})\leq\vartheta_{3}(\rho^{b})$,
given $\vartheta_{2}(\rho^{a})=\vartheta_{2}(\rho^{b})$

  2. 2.

$\forall\rho^{b};\vartheta_{3}(\rho^{b})=2\vartheta_{2}(\rho^{b})+1$

  3. 3.

$\forall\rho^{a}\exists\rho^{b}:\vartheta_{2}(\rho^{a})\geq\vartheta_{2}(\rho^{b})$

From the definition of teleportation fidelity (13) it is evident that, keeping
$\lambda_{0},\lambda_{1},\lambda_{4}$ constant, increasing $\lambda_{2}$
decreases $\lambda_{3}$ and hence, increases $||R||$ and decreases $||Q||$ and
vice versa. To minimize $\vartheta_{2}(\rho)$ we require
$\lambda_{2}=\lambda_{3}=0$. Hence we can define
$\rho^{b}\equiv\\{\Lambda_{0},\Lambda_{1},0,0,\Lambda_{4}\\}$, at which point

$\vartheta_{2}(\rho^{b})=\sqrt{1+4\lambda_{4}^{4}+4\lambda_{4}^{2}\lambda_{0}^{2}-4\lambda_{4}^{2}}\leq\vartheta_{2}(\rho^{a})$
(19)

Thus, part 3 is true. Now, for $\rho^{b}$ we have
$\vartheta_{3}(\rho^{b})=4\lambda_{0}\lambda_{4}+1$. Replacing these values in
the theorem, we can achieve equality at the point
$\lambda_{4}=\frac{1}{\sqrt{2}}$. Since this establishes part 2 and the
existence of a boundary, we can define this state as
$\rho^{b}\equiv\\{\frac{1}{\sqrt{2}}\cos\theta,\frac{1}{\sqrt{2}}\sin\theta,0,0,\frac{1}{\sqrt{2}}\\}$.
We can now calculate the values for $\vartheta_{2}(\rho^{b})$ and
$\vartheta_{3}(\rho^{b})$ in terms of $\theta$ as
$\vartheta_{2}(\rho^{b})=\cos\theta$ and
$\vartheta_{3}(\rho^{b})=2\cos\theta+1$.
Now,, assert that
$\vartheta_{2}(\rho^{a})=\vartheta_{2}(\rho^{b})=\cos\theta$.

CASE 1: If $\lambda_{2}>\lambda_{3}:$

$\vartheta_{2}(\rho^{a})=||R_{a}||=\cos\theta=\vartheta_{2}(\rho^{b})$ (20)

  1. 1.

if
$\lambda_{2}\geq\lambda_{4}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{2}+1$
Replace the value of $\cos\theta$ in $\vartheta_{3}(\rho^{b})$

$\displaystyle\vartheta_{3}(\rho^{b})$
$\displaystyle=4\lambda_{0}\lambda_{2}+4\sqrt{\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}}$
$\displaystyle+2\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2})^{2}}+1$
$\displaystyle>4\lambda_{0}\lambda_{2}+1=\vartheta_{3}(\rho^{a})$

  2. 2.

if
$\lambda_{4}>\lambda_{2}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{4}+1$
Simplifying the last term it is obvious that,

$\displaystyle\vartheta_{3}(\rho^{b})$
$\displaystyle>4\lambda_{0}\lambda_{2}+4\sqrt{\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}}$
$\displaystyle+2\sqrt{4\lambda_{0}^{2}\lambda_{4}^{2}}+1$
$\displaystyle>4\lambda_{0}\lambda_{4}+1=\vartheta_{3}(\rho^{a})$

Thus, if $\lambda_{2}>\lambda_{3}$ and
$\vartheta_{2}(\rho^{a})=\vartheta_{2}(\rho^{b})$ then
$\vartheta_{3}(\rho^{b})>\vartheta_{3}(\rho^{a})$.

CASE 2: If $\lambda_{2}<\lambda_{3}:$

$\vartheta_{2}(\rho^{a})=||Q_{a}||=\cos\theta=\vartheta_{2}(\rho^{b})$ (21)

  1. 1.

if
$\lambda_{3}>\lambda_{2}\geq\lambda_{4}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{2}+1$
Replace the value of $\cos\theta$ in $\vartheta_{3}(\rho^{b})$

$\displaystyle\vartheta_{3}(\rho^{b})$
$\displaystyle>4\lambda_{0}\lambda_{3}+1$
$\displaystyle>4\lambda_{0}\lambda_{2}+1=\vartheta_{3}(\rho^{a})$

  2. 2.

if
$\lambda_{4}>\lambda_{2}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{4}+1$
Simplify the last term to the form:

$\displaystyle\vartheta_{3}(\rho^{b})$
$\displaystyle>4\lambda_{0}\lambda_{3}+4\sqrt{\lambda_{0}^{2}\lambda_{3}^{2}+(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2}}$
$\displaystyle+2\sqrt{4\lambda_{0}^{2}\lambda_{4}^{2}}+1$
$\displaystyle>4\lambda_{0}\lambda_{4}+1=\vartheta_{3}(\rho^{a})$

Thus, if $\lambda_{3}>\lambda_{2}$ and $\hskip
5.69046pt\vartheta_{2}(\rho^{a})=\vartheta_{2}(\rho^{b})$ then
$\vartheta_{3}(\rho^{b})>\vartheta_{3}(\rho^{a})$. Thus, for all arbitrary
states belonging to the GHZ class as defined above, the theorem is true,
achieving equality only if $\lambda_{2}=\lambda_{3}=0$. $\blacksquare$

Figure 1: Fidelity of Reconstruction vs Teleportation fidelity: The
reconstruction fidelity of the tripartite pure state acin2001three ;
acin2001classification  is plotted against the maximum of the teleportation
fidelities of the (dealer–reconstructor) and (dealer–assistant) channels. The
red coloured line indicates the MSR states with the standard GHZ state shown
as the green point is considered as an example to this class.

Thus, we can establish the fact that there exists a limit to the maximal
reconstruction fidelity $\ket{\psi(GHZ^{R})}_{ABC}$ can achieve, given the
teleportation fidelity of the bipartite reduced states of $AB$ and $AC$
channels. Interestingly, in the the Figure (1), we show numerically that aour
result holds for Generalized GHZ class of states (sometimes referred as Acin
states) :
$\ket{\psi(GHZ^{R})}_{ABC}=\lambda_{0}\ket{000}+\lambda_{1}e^{i\phi}\ket{100}+\lambda_{2}\ket{101}+\lambda_{3}\ket{110}+\lambda_{4}\ket{111}$
cannonically representing any three-qubit vector in general.

Maximal Secret Reconstructed State (MSR) : In this article we are able to
characterize those states for which maximum secret reconstruction is possible
among the secret sharing resources. We call these states to be Maximal Secret
Reconstructed State (MSR). For a given value of the maximum teleportation
fidelity $F_{max}$, these states give the maximum reconstruction fidelity.
This class of states are obtained from the equality condition in Theorem 1, by
putting the values $\lambda_{2}=\lambda_{3}=0$ and
$\lambda_{4}=\frac{1}{\sqrt{2}}$ in the state $\ket{\psi(GHZ^{R})}$. This can
also be identified in the Figure (1 by the red line over the ash triangle.
States belonging to this family can be represented as:

$\ket{\psi}_{GHZ^{R}}=\lambda_{0}\ket{000}+\lambda_{1}\ket{100}+\frac{1}{\sqrt{2}}\ket{111}$
(22)

where, $\lambda_{i}\in\mathbf{R},\mu_{i}\geq
0,\Sigma_{i}\lambda_{i}^{2}=\frac{1}{2}$. Canonically, it can also have a
single parameter representation as:

$\ket{\psi}_{GHZ^{R}}=\frac{1}{\sqrt{2}}(\cos{\theta}\ket{000}+\sin{\theta}\ket{100})+\frac{1}{\sqrt{2}}\ket{111}$
(23)

with $\theta\in[0,\frac{\pi}{2}]$.

Example 1. The GHZ state:  A classic example of a state belonging to this
class is the GHZ state ($\theta=\frac{\pi}{2}$):
$\ket{GHZ}=\frac{1}{\sqrt{2}}\ket{000}+\frac{1}{\sqrt{2}}\ket{111}$, where
$\lambda_{0}=\frac{1}{\sqrt{2}}$ and $\lambda_{1}=0$. For this state, we find
that the maximum teleportation fidelity of the bipartite channels
$F_{max}(GHZ)=\frac{2}{3}$ and the reconstruction fidelity
$\mathcal{F(GHZ)}=1$ which satisfies the equation 16. This clearly indicates
that the GHZ state falls into the class of Maximum secret reconstructed class
of state (MSR) by achieving maximum possible reconstruction fidelity in
addition of being classified as a secret sharing resource (labeled as the
green dot in the Figure (1).

Example 2. $\theta=\frac{\pi}{4}$:  Such a state would take the form
$\ket{\psi}=\frac{1}{2}\ket{000}+\frac{1}{2}\ket{100}+\frac{1}{\sqrt{2}}\ket{111}$.
Knowing the coefficients, we can substitute them in the definitions for
bipartite teleportation fidelity and reconstruction fidelity. For
teleportation fidelity, from (13), $\vartheta_{2}(\rho)=\frac{1}{\sqrt{2}}$,
and hence $F_{max}=\frac{6+\sqrt{2}}{12}$ $<\frac{2}{3}$ (3). From definition
of reconstruction fidelity, $\vartheta_{3}(\rho)=\sqrt{2}+1$, and from (5)
$\mathcal{F}=\frac{4+\sqrt{2}}{6}$ $>\frac{2}{3}$. Clearly,
$\mathcal{F}-2F=-\frac{1}{3}$ which achieves equality on Theorem 1. Clearly
the state $|\psi\rangle$ falls into the MSR class of states.

##  IV Reconstruction fidelity versus Bell’s Inequality Violation

In this section, we explore the relationship between the Bell’s Bell-CHSH
value of the dealer-reconstructor and dealer-assistant channels with the
reconstruction fidelity of the tripartite GHZ class state.
Considering the generalized GHZ class of states defined in equation (11), we
find the reduced density matrices $\rho_{AC}$ and $\rho_{AB}$ for the
dealer–reconstructor and dealer–assistant bipartitions respectively. It can be
calculated that the correlation matrix for each bipartition, represented as
$T_{\rho}=Tr(\sigma_{i}\otimes\sigma_{j}\rho)$ which takes the value of $R$
for the subsystem AC and $Q$ for subsystem AB. Similar to the teleportation
fidelity, the eigenvalues of the matrix $T_{\rho}^{T}T_{\rho}$ will differ by
the factor of $\lambda_{2}$ and $\lambda_{3}$ for the two bipartitions. Note
that all three of the eigenvalues from each bipartition will remain non-
negative.

###### Theorem 2

(Mutual Exclusivity Theorem: ) All states that are useful for secret sharing
satisfies Bell’s inequality ($\mathcal{S}_{max}\leq 2$). Equivalently, the
maximum bipartite Bell’s Inequality Violation ($\mathcal{S}_{max}>2$) would
imply that the state is not useful for faithful secret sharing.

Proof. We already know that for a state to be useful for secure secret
sharing, we have the 2 necessary conditions; one for the bipartitions, (
$F_{AB}\leq\frac{2}{3}$ and $F_{AC}\leq\frac{2}{3}$ or equivalently
$F_{max}\leq\frac{2}{3}$) and for the entire three-qubit state
$\mathcal{F}\geq\frac{2}{3}$. We can use this condition to establish a similar
limit on Bell-CHSH value
($\mathcal{S}_{max}=max\\{\mathcal{S}_{AC},\mathcal{S}_{AB}\\}$ or
equivalently $M_{max}=\max\\{M(\rho_{AC}),M(\rho_{AB})\\}$) through equation
(10).

$\displaystyle
F_{max}\leq\frac{2}{3}\Rightarrow\frac{2}{3}\geq\frac{1}{2}(1+\frac{1}{3}M_{max}){}$
$\displaystyle\Rightarrow 1\geq M_{max}\Rightarrow\mathcal{S}_{max}\leq 2$
(24)

This clearly indicates that those states for which $F_{max}\leq\frac{2}{3}$
are not nonlocal in the dealer–reconstructor and dealer–assistant bipartite
regime as it is independent of whether $\mathcal{F}\geq\frac{2}{3}$. So
eventually for secret shareable states being a subset of that will satisfy the
same conclusion. This makes sense from the perspective that if there are more
correlation in the bipartite channels more information individual can access
without involving others. This creates a mutual exclusiveness of the secret
sharing capacity and the bipartite nonlocality. $\blacksquare$

###### Theorem 3

Given a three-qubit pure state $\ket{\psi(GHZ^{R})}$ which is useable for
secret sharing under the below mentioned conditions:

$\displaystyle F_{max}\leq\frac{2}{3}\Rightarrow\vartheta_{2}(\rho)\leq 1$
$\displaystyle\mathcal{F}_{ABC}\geq\frac{2}{3}\Rightarrow\vartheta_{3}(\rho)\geq
1$

the relation between Bell’s CHSH value
$\mathcal{S}_{max}=max(\mathcal{S}_{AB},\mathcal{S}_{AC})$ (equivalently
$M_{max}=max(M_{AB},M_{AC})$) and reconstruction fidelity $\mathcal{F}$ can be
presented as:

$6\mathcal{F}_{ABC}\leq\mathcal{S}_{max}+4$ (25)

or, in terms of their factors as:

$\vartheta_{3}(\rho)\leq 2\sqrt{M_{max}}+1$ (26)

Proof. As before, we define 2 states, an arbitrary state
$\rho^{a}\equiv\\{\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$
and a boundary state,
$\rho^{b}\equiv\\{\Lambda_{0},\Lambda_{1},\Lambda_{2},\Lambda_{3},\Lambda_{4}\\}$.
We can divide this proof into three parts as

$\vartheta_{3}(\rho^{a})\overset{(1)}{\leq}\vartheta_{3}(\rho^{b})\overset{(2)}{=}2\sqrt{M_{\max}(\rho^{b})}+1\overset{(3)}{\leq}2\sqrt{M_{\max}(\rho^{a})}+1$
(27)

with

  1. 1.

$\forall\rho^{a}\exists\rho^{b}:\vartheta_{3}(\rho^{a})\leq\vartheta_{3}(\rho^{b})$,
given $M(\rho^{a})=M(\rho^{b})$

  2. 2.

$\forall\rho^{b};\vartheta_{3}(\rho^{b})=2\sqrt{M(\rho^{b})}+1$

  3. 3.

$\forall\rho^{a}\exists\rho^{b}:M(\rho^{a})\geq\vartheta_{2}(\rho^{b})$

We have already seen that $M(\rho)$ is minimized at
$\lambda_{2}=\lambda_{3}=0$, thus part $3$ is true at
$\rho^{b}\equiv\\{\Lambda_{0},\Lambda_{1},0,0,\Lambda_{4}\\}$. Following
similar steps as in proof for Theorem 1, we can obtain the equality at
$\lambda_{4}=\frac{1}{\sqrt{2}}$ thus defining the family of states lying on
the boundary in the form
$\rho^{b}\equiv\\{\frac{1}{\sqrt{2}}\cos\theta,\frac{1}{\sqrt{2}}\sin\theta,0,0,\frac{1}{\sqrt{2}}\\}$.
Again, we have, in terms of $\theta$, $M(\rho^{b})=\cos^{2}\theta$ and
$\vartheta_{3}(\rho^{b})=2\cos\theta+1$.

CASE 1: If $\lambda_{2}>\lambda_{3}:$

$\displaystyle M(\rho^{a})$
$\displaystyle=1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{2}^{2}-\lambda_{3}^{2}){}$
(28) $\displaystyle=\cos^{2}\theta=M(\rho^{b})$

  1. 1.

if
$\lambda_{2}\geq\lambda_{4}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{2}+1$
Replace the value of $\cos\theta$ in $\vartheta_{3}(\rho^{b})$

$\displaystyle\vartheta_{3}(\rho^{b})=2\cos\theta+1$
$\displaystyle=1+2\sqrt{1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{2}^{2}-\lambda_{3}^{2})}$
$\displaystyle>1+2\sqrt{4\lambda_{0}^{2}\lambda_{3}^{2}}>1+4\lambda_{0}\lambda_{2}=\vartheta_{3}(\rho^{a})$

  2. 2.

if
$\lambda_{4}>\lambda_{2}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{4}+1$
Similarly, simplify the last term to the form:

$\displaystyle\vartheta_{3}(\rho^{b})=2\cos\theta+1$
$\displaystyle=1+2\sqrt{1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{2}^{2}-\lambda_{3}^{2})}$
$\displaystyle>1+2\sqrt{4\lambda_{0}^{2}\lambda_{4}^{2}}=1+4\lambda_{0}\lambda_{4}=\vartheta_{3}(\rho^{a})$

Thus, if $\lambda_{2}>\lambda_{3}$ and $M(\rho^{a})=M(\rho^{b})$ then
$\vartheta_{3}(\rho^{b})>\vartheta_{3}(\rho^{a})$.

CASE 2: If $\lambda_{3}>\lambda_{2}:$

$\displaystyle M(\rho^{a})$
$\displaystyle=1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{3}^{2}-\lambda_{2}^{2}){}$
(29) $\displaystyle=\cos^{2}\theta=M(\rho^{b})$

  1. 1.

if
$\lambda_{3}>\lambda_{2}\geq\lambda_{4}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{2}+1$
Replace the value of $\cos\theta$ in $\vartheta_{3}(\rho^{b})$

$\displaystyle\vartheta_{3}(\rho^{b})=2\cos\theta+1$
$\displaystyle=1+2\sqrt{1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{3}^{2}-\lambda_{2}^{2})}$
$\displaystyle>1+2\sqrt{4\lambda_{0}^{2}\lambda_{3}^{2}}>1+4\lambda_{0}\lambda_{2}=\vartheta_{3}(\rho^{a})$

  2. 2.

if
$\lambda_{4}>\lambda_{2}\Rightarrow\vartheta_{3}(\rho^{a})=4\lambda_{0}\lambda_{4}+1$
Similarly, simplify the last term to the form:

$\displaystyle\vartheta_{3}(\rho^{b})=2\cos\theta+1$
$\displaystyle=1+2\sqrt{1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{3}^{2}-\lambda_{2}^{2})}$
$\displaystyle>1+2\sqrt{4\lambda_{0}^{2}\lambda_{4}^{2}}=1+4\lambda_{0}\lambda_{4}=\vartheta_{3}(\rho^{a})$

Thus, if $\lambda_{3}>\lambda_{2}$ and $M(\rho^{a})=M(\rho^{b})$ then
$\vartheta_{3}(\rho^{b})>\vartheta_{3}(\rho^{a})$. Thus, for all arbitrary
states belonging to the GHZ class as defined above, the theorem is true,
achieving equality only if $\lambda_{2}=\lambda_{3}=0$. $\blacksquare$

This is also an important result, proving that the maximum value of $M$
doesn’t exceed $1$. In other words, the maximum of Bell’s CHSH values of both
the dealer–reconstructor and dealer–assistant channels is less than equal to
$2$ (i.e $\mathcal{S}(\rho)\leq 2)$ in all cases of a secret shareable state.
This means that no relevant bipartitions of a state agreeing to the
aforementioned conditions to qualify for faithful secret sharing violates
Bell’s inequality. In the Figure (2) we numerically show that this is true for
much larger class of states like Generalized GHZ class (Acin states
acin2001classification ; acin2001three ) with boundary being given by the red
colour line. The standard GHZ state is one example in the boundary state,
represented by the green point in the Figure (2).

Note: Refer to the Appendix for complete calculations.

Figure 2: Reconstruction fidelity vs max of Bell’s CHSH values: The
reconstruction fidelity of three-qubit pure states acin2001three ;
acin2001classification  is plotted against the maximum of Bell’s CHSH values
for both dealer–reconstructor and dealer–assistant channels. The red line
represents the states taking the maximum value for the reconstruction
fidelity. The green dot being the standard GHZ state is depicted as an example
in the figure

##  V Conclusion

In conclusion, this work identifies those secret sharing resources for which
maximum reconstruction of the secret is possible. Our result is based on the
set of Acin states representing the class of all tripartite pure states. We
prove the analytical results for Acin states with vanishing complex
coefficient. We call this class as Maximal Secret Reconstructed (MSR) class of
states. This class of states are obtained as a result of the bound that we
establish on the reconstruction fidelity for a given value of the maximum of
bipartite teleportation fidelities of the dealer–assistant and
dealer–reconstructor channels. Additionally, our study also evaluates the
suitability of a state for reliable secret sharing in light of the Bell-CHSH
operator values. Remarkably, we find that all secret shareable states does not
violate Bell’s inequality when observed across both the dealer–reconstructor
and dealer–assistant partitions bringing out a new mutual exclusivity. In the
same spirit, we also find the the maximum reconstruction fidelity that can be
achieved if the Bell-CHSH values in the dealer–reconstructor and
dealer–assistant partitions are known. Our study has been fundamental in the
spirit of realizing the capacity of secret sharing states in terms of
reconstructing the secret bringing out the limits of probable resource theory
based on quantum secret sharing. It also brings out the mutually exclusiveness
of both bipartite correlations with tripartite capacity of a quantum states in
a general sense conforming to such previous insights in different context
pandya2016complementarity .

## References

  * [1] Pahulpreet Singh and Indranil Chakrabarty.  Controlled state reconstruction and quantum secret sharing.  Physical Review A, 109(3):032406, 2024.
  * [2] Antonio Acín, A Andrianov, E Jané, and Rolf Tarrach.  Three-qubit pure-state canonical forms.  Journal of Physics A: Mathematical and General, 34(35):6725, 2001\.
  * [3] Antonio Acín, Dagmar Bruß, Maciej Lewenstein, and Anna Sanpera.  Classification of mixed three-qubit states.  Physical Review Letters, 87(4):040401, 2001.
  * [4] Erwin Schrödinger.  Discussion of probability relations between separated systems.  In Mathematical Proceedings of the Cambridge Philosophical Society, volume 31, pages 555–563. Cambridge University Press, 1935.
  * [5] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki.  Quantum entanglement.  Reviews of modern physics, 81(2):865, 2009.
  * [6] Ryszard Horodecki, Michał Horodecki, and Paweł Horodecki.  Teleportation, bell’s inequalities and inseparability.  Physics Letters A, 222(1-2):21–25, 1996.
  * [7] Charles H Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K Wootters.  Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels.  Physical review letters, 70(13):1895, 1993.
  * [8] Arun K Pati, Vijeth Aradhya, Indranil Chakrabarty, Subhasree Patro, et al.  Teleportation of quantum coherence.  Physical Review A, 108(4):042620, 2023.
  * [9] Indranil Chakrabarty.  Teleportation via a mixture of a two qubit subsystem of a n-qubit w and ghz state.  The European Physical Journal D, 57(2):265–269, 2010.
  * [10] Satyabrata Adhikari, N Ganguly, Indranil Chakrabarty, and BS Choudhury.  Quantum cloning, bell’s inequality and teleportation.  Journal of Physics A: Mathematical and Theoretical, 41(41):415302, 2008.
  * [11] Charles H Bennett, Patrick Hayden, Debbie W Leung, Peter W Shor, and Andreas Winter.  Remote preparation of quantum states.  IEEE Transactions on Information Theory, 51(1):56–74, 2005.
  * [12] Charles H Bennett and Stephen J Wiesner.  Communication via one-and two-particle operators on einstein-podolsky-rosen states.  Physical review letters, 69(20):2881, 1992.
  * [13] Chirag Srivastava, Anindita Bera, Aditi Sen, Ujjwal Sen, et al.  One-shot conclusive multiport quantum dense coding capacities.  Physical Review A, 100(5):052304, 2019.
  * [14] Tamoghna Das, R Prabhu, Aditi Sen, Ujjwal Sen, et al.  Distributed quantum dense coding with two receivers in noisy environments.  Physical Review A, 92(5):052330, 2015.
  * [15] Subhasree Patro, Indranil Chakrabarty, and Nirman Ganguly.  Non-negativity of conditional von neumann entropy and global unitary operations.  Physical Review A, 96(6):062102, 2017.
  * [16] Mahathi Vempati, Nirman Ganguly, Indranil Chakrabarty, and Arun K Pati.  Witnessing negative conditional entropy.  Physical Review A, 104(1):012417, 2021.
  * [17] Saptarshi Roy, Titas Chanda, Tamoghna Das, Aditi Sen De, and Ujjwal Sen.  Deterministic quantum dense coding networks.  Physics Letters A, 382(26):1709–1715, 2018.
  * [18] Stephanie Wehner, David Elkouss, and Ronald Hanson.  Quantum internet: A vision for the road ahead.  Science, 362(6412):eaam9288, 2018.
  * [19] William K Wootters and Wojciech H Zurek.  A single quantum cannot be cloned.  Nature, 299(5886):802–803, 1982.
  * [20] Farid Shahandeh and Farid Shahandeh.  The resource theory of entanglement.  Quantum Correlations: A Modern Augmentation, pages 61–109, 2019\.
  * [21] Animesh Datta, Steven T Flammia, and Carlton M Caves.  Entanglement and the power of one qubit.  Physical Review A, 72(4):042316, 2005.
  * [22] Jian Li and GS Paraoanu.  Generation and propagation of entanglement in driven coupled-qubit systems.  New Journal of Physics, 11(11):113020, 2009.
  * [23] Jerzy Dajka and Jerzy Łuczka.  Origination and survival of qudit-qudit entanglement in open systems.  Physical Review A, 77(6):062303, 2008.
  * [24] Agata Chęcińska and Krzysztof Wodkiewicz.  Separability of entangled qutrits in noisy channels.  Physical Review A, 76(5):052306, 2007.
  * [25] Sai Vinjanampathy and ARP Rau.  Quantum discord for qubit–qudit systems.  Journal of Physics A: Mathematical and Theoretical, 45(9):095303, 2012.
  * [26] Rounak Mundra, Dhrumil Patel, Indranil Chakrabarty, Nirman Ganguly, and Sourav Chatterjee.  Broadcasting of quantum correlations in qubit-qudit systems.  Physical Review A, 100(4):042319, 2019.
  * [27] Siddhartha Das, Sumeet Khatri, and Jonathan P Dowling.  Robust quantum network architectures and topologies for entanglement distribution.  Physical Review A, 97(1):012335, 2018.
  * [28] Mark Hillery, Vladimír Bužek, and André Berthiaume.  Quantum secret sharing.  Physical Review A, 59(3):1829, 1999.
  * [29] Anuma Garg and Satyabrata Adhikari.  Estimation of power in the controlled quantum teleportation through the witness operator.  arXiv preprint arXiv:2307.16574, 2023.
  * [30] Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo.  How to share a quantum secret.  Physical review letters, 83(3):648, 1999.
  * [31] Anders Karlsson, Masato Koashi, and Nobuyuki Imoto.  Quantum entanglement for secret sharing and secret splitting.  Physical Review A, 59(1):162, 1999.
  * [32] Somshubhro Bandyopadhyay.  Teleportation and secret sharing with pure entangled states.  Physical Review A, 62(1):012308, 2000.
  * [33] Damian Markham and Barry C Sanders.  Graph states for quantum secret sharing.  Physical Review A, 78(4):042309, 2008.
  * [34] Qin Li, Wai Hong Chan, and Dong-Yang Long.  Semiquantum secret sharing using entangled states.  Physical Review A, 82(2):022303, 2010.
  * [35] Satyabrata Adhikari, Indranil Chakrabarty, and Pankaj Agrawal.  Probabilistic secret sharing through noisy quantum channels.  arXiv preprint arXiv:1012.5570, 2010.
  * [36] Maharshi Ray, Sourav Chatterjee, and Indranil Chakrabarty.  Sequential quantum secret sharing in a noisy environment aided with weak measurements.  The European Physical Journal D, 70:1–11, 2016.
  * [37] Sk Sazim, V Chiranjeevi, Indranil Chakrabarty, and Kannan Srinathan.  Retrieving and routing quantum information in a quantum network.  Quantum Information Processing, 14(12):4651–4664, 2015.
  * [38] Wolfgang Tittel, Hugo Zbinden, and Nicolas Gisin.  Experimental demonstration of quantum secret sharing.  Physical Review A, 63(4):042301, 2001.
  * [39] Christian Schmid, Pavel Trojek, Mohamed Bourennane, Christian Kurtsiefer, Marek Żukowski, and Harald Weinfurter.  Experimental single qubit quantum secret sharing.  Physical review letters, 95(23):230505, 2005.
  * [40] Ch Schmid, Pavel Trojek, Sascha Gaertner, Mohamed Bourennane, Ch Kurtsiefer, Marek Zukowski, and Harald Weinfurter.  Experimental quantum secret sharing.  Fortschritte der Physik: Progress of Physics, 54(8-10):831–839, 2006.
  * [41] Jan Bogdanski, Nima Rafiei, and Mohamed Bourennane.  Experimental quantum secret sharing using telecommunication fiber.  Physical Review A, 78(6):062307, 2008.
  * [42] John S Bell.  On the einstein podolsky rosen paradox.  Physics Physique Fizika, 1(3):195, 1964.
  * [43] John F Clauser, Michael A Horne, Abner Shimony, and Richard A Holt.  Proposed experiment to test local hidden-variable theories.  Physical review letters, 23(15):880, 1969.
  * [44] George Svetlichny.  Distinguishing three-body from two-body nonseparability by a bell-type inequality.  Physical Review D, 35(10):3066, 1987.
  * [45] Palash Pandya, Avijit Misra, and Indranil Chakrabarty.  Complementarity between tripartite quantum correlation and bipartite bell-inequality violation in three-qubit states.  Physical Review A, 94(5):052126, 2016.
  * [46] Indranil Chakrabarty, Nirman Ganguly, and Binayak S Choudhury.  Deletion, bell’s inequality, teleportation.  Quantum Information Processing, 10(1):27–32, 2011.
  * [47] Sixia Yu, Qing Chen, Chengjie Zhang, Choy Heng Lai, and Choo Hiap Oh.  All entangled pure states violate a single bell’s inequality.  Physical review letters, 109(12):120402, 2012.

##  Appendix A Teleportation Fidelity Detailed Analysis

Given the correlation matrices $R$ and $Q$ for the subsystems AC and AB, we
can calculate the value of $\vartheta_{2}(\rho)=\max\\{||R||,||Q||\\}$. where,

$\displaystyle||R||=2\lambda_{0}\lambda_{2}+2\sqrt{\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2})^{2}}{}$
$\displaystyle||Q||=2\lambda_{0}\lambda_{3}+2\sqrt{\lambda_{0}^{2}\lambda_{3}^{2}+(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2}}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{3}^{2}-\lambda_{2}^{2})^{2}}$
(30)

Analysing these values in the cases we need to compare with reconstruction
fidelity, we can rewrite the values.

CASE 1: if $\lambda_{2}>\lambda_{3}:$

$\displaystyle\vartheta_{2}(\rho)=2\lambda_{0}\lambda_{2}+2\sqrt{\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2})^{2}}{}$
$\displaystyle=2\lambda_{0}\lambda_{2}+2\sqrt{2\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{4}^{2}+4\lambda_{0}^{2}(\lambda_{2}^{2}-\lambda_{3}^{2})+(\lambda_{2}^{2}+\lambda_{4}^{2}-\lambda_{0}^{2}-\lambda_{1}^{2}-\lambda_{3}^{2})^{2}}{}$

CASE 2: if $\lambda_{3}>\lambda_{2}:$

$\displaystyle\vartheta_{2}(\rho)=2\lambda_{0}\lambda_{3}+2\sqrt{\lambda_{0}^{2}\lambda_{3}^{2}+(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2}}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{1}^{2}+(\lambda_{0}^{2}-\lambda_{1}^{2}+\lambda_{4}^{2}+\lambda_{3}^{2}-\lambda_{2}^{2})^{2}}{}$
$\displaystyle=2\lambda_{0}\lambda_{2}+2\sqrt{2\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})}{}$
$\displaystyle+\sqrt{4\lambda_{0}^{2}\lambda_{4}^{2}+4\lambda_{0}^{2}(\lambda_{3}^{2}-\lambda_{2}^{2})+(\lambda_{0}^{2}+\lambda_{1}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2}-\lambda_{4}^{2})^{2}}{}$

Using these simplifications, it is easy to see that

$\vartheta_{2}(\rho)>2\lambda_{0}\lambda_{2}+2\lambda_{0}\lambda_{4}$ (33)

##  Appendix B Bell Violation Detailed Analysis

For subsystems AC and AB, we have the correlation matrices $R$ and $Q$. The
eigenvalues of $R^{T}R$ and $Q^{T}Q$ can be calculated, respectively as:

$\displaystyle[\hskip 5.69046pt$ $\displaystyle
4\lambda_{0}^{2}\lambda_{2}^{2},$ $\displaystyle
4\lambda_{0}^{2}\lambda_{2}^{2}+4(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2},$
$\displaystyle
4\lambda_{0}^{2}\lambda_{2}^{2}+4\lambda_{0}^{2}(\lambda_{4}^{2}-\lambda_{3}^{2})+(\lambda_{0}^{2}+\lambda_{1}^{2}+\lambda_{3}^{2}-\lambda_{2}^{2}-\lambda_{4}^{2})^{2}$
$\displaystyle]$

and

$\displaystyle[\hskip 5.69046pt$ $\displaystyle
4\lambda_{0}^{2}\lambda_{3}^{2},$ $\displaystyle
4\lambda_{0}^{2}\lambda_{3}^{2}+4(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2},$
$\displaystyle
4\lambda_{0}^{2}\lambda_{3}^{2}+4\lambda_{0}^{2}(\lambda_{4}^{2}-\lambda_{2}^{2})+(\lambda_{0}^{2}+\lambda_{1}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2}-\lambda_{4}^{2})^{2}$
$\displaystyle]$

Consequently, it is obvious that the largest two values are the last two in
each case.

$M=1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}|\lambda_{2}^{2}-\lambda_{3}^{2}|$

We can simplify $M$ in multiple ways, to compare with reconstruction fidelity
in each case.

CASE 1: if $\lambda_{2}>\lambda_{3}:$

$\displaystyle M$
$\displaystyle=1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{2}^{2}-\lambda_{3}^{2}){}$
(34)
$\displaystyle=4\lambda_{0}^{2}\lambda_{2}^{2}+(\lambda_{0}^{2}+\lambda_{1}^{2}-\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2})^{2}{}$
$\displaystyle+4\lambda_{0}^{2}(\lambda_{2}^{2}-\lambda_{3}^{2})+4(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}+4\lambda_{0}^{2}\lambda_{4}^{2}{}$
$\displaystyle=4\lambda_{0}^{2}\lambda_{4}^{2}+(\lambda_{0}^{2}+\lambda_{1}^{2}-\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2})^{2}{}$
$\displaystyle+4\lambda_{0}^{2}(2\lambda_{2}^{2}-\lambda_{3}^{2})+4(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{4})^{2}$

CASE 2: if $\lambda_{3}>\lambda_{2}:$

$\displaystyle M$
$\displaystyle=1-4(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3})^{2}+4\lambda_{0}^{2}(\lambda_{3}^{2}-\lambda_{2}^{2}){}$
(35)
$\displaystyle=4\lambda_{0}^{2}\lambda_{3}^{2}+(\lambda_{0}^{2}+\lambda_{1}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2}-\lambda_{4}^{2})^{2}{}$
$\displaystyle+4\lambda_{0}^{2}(\lambda_{3}^{2}-\lambda_{2}^{2})+4(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2}+4\lambda_{0}^{2}\lambda_{4}^{2}{}$
$\displaystyle=4\lambda_{0}^{2}\lambda_{4}^{2}+(\lambda_{0}^{2}+\lambda_{1}^{2}+\lambda_{2}^{2}-\lambda_{3}^{2}-\lambda_{4}^{2})^{2}{}$
$\displaystyle+4\lambda_{0}^{2}(2\lambda_{3}^{2}-\lambda_{2}^{2})+4(\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{4})^{2}$

It is now obvious that

$M>4\lambda_{0}^{2}\lambda_{2}^{2}+4\lambda_{0}^{2}\lambda_{4}^{2}$ (36)