#  Resource theoretic efficacy of the single copy of a two-qubit entangled
state in a sequential network

Arun Kumar Das<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Debarshi
Das<EMAIL_ADDRESS>S. N. Bose National Centre for Basic Sciences,
Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Shiladitya Mal
<EMAIL_ADDRESS>Department of Physics and Center for Quantum Frontiers
of Research and Technology (QFort), National Cheng Kung University, Tainan
701, Taiwan  Dipankar Home<EMAIL_ADDRESS>Centre for Astroparticle
Physics and Space Science, Bose Institute, Block EN, Sector V, Salt Lake,
Kolkata 700 091, India  A. S. Majumdar<EMAIL_ADDRESS>S. N. Bose National
Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106,
India

###### Abstract

How best one can recycle a given quantum resource, mitigating the various
difficulties involved in its preparation and preservation, is of considerable
importance for ensuring efficient applications in quantum technology. Here we
demonstrate quantitatively the resource theoretic advantage of reusing the
single copy of a two-qubit entangled state towards information processing. To
this end, we consider a scenario of sequential detection of the given
entangled state by multiple independent observers on each of the two spatially
separated wings. In particular, we consider equal numbers of sequential
observers on the two wings. We first determine the upper bound on the number
of observers who can detect entanglement employing suitable entanglement
witness operators. In terms of the parameters characterizing the entanglement
consumed and the robustness of measurements, we then compare the above
scenario with the corresponding scenario involving the sharing of multiple
copies of identical two-qubit states among the two wings. This reveals a clear
resource theoretic advantage of recycling the single copy of a two-qubit
entangled state.

##  I Introduction

One of the most counterintuitive features of quantum mechanics is quantum
entanglement ent1 ; ent2 ; ent3 , which leads to nonclassical phenomena like
Bell nonlocality belln1 ; belln2  and Einstein-Podolsky-Rosen steering eprs1 ;
eprs2 ; eprs3 . Apart from the foundational significance of probing the
incompatibility between quantum mechanical predictions and the local realist
descriptions of nature, quantum entanglement serves as a resource for various
information processing and communication tasks. To name a few, some such well-
established tasks include quantum teleportation teleport , quantum dense
coding dense_code , quantum key distribution qkd , certification of genuine
randomness rand , and quantum random access codes rac1 .

In a real laboratory set-up, preparation of any quantum resource always faces
different types of complications prep1 ; prep2 . Moreover, it is an extremely
difficult task to prepare quantum resources having a high degree of isolation
from environmental interactions env1 ; env2 ; env3 ; env4 . Therefore, the
efficient use of resources is one of the primary challenges in the backdrop of
current endeavour of building quantum technology. To this end, the possibility
of recycling the same resource several times is of great advantage. A
particular network scenario suitable for this purpose comprises a single copy
of a bipartite entangled state with multiple pairs of independent sequential
observers in the two spatially separated wings, where each of these observers
makes unsharp measurement and delivers the accessed particle to the next
observer silva_prl .

In silva_prl  the authors showed that at most two independent observers at one
wing can violate the Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality CHSH
with an observer on the other wing by sharing a pair of entangled spin-$1/2$
particles. Note that all except the last one in a sequence of multiple
observers cannot perform sharp or projective measurements, since it is desired
that some amount of entanglement must survive in the post-measured state in
order to be utilized by the subsequent observer. It was shown in
mathematics_mal , that the upper bound of two observers on one wing who can
share nonlocality of a two-qubit state in the scenario of unbiased measurement
settings, is based on the optimality of the unsharp measurement framework um1
; unsharp_measurement  with respect to the trade-off between information gain
and disturbance in a quantum measurement igd1 ; igd2 . It is thus important to
employ such measurements in order to obtain optimal performance in the above
sequential network scenario.

The issue of sequential detection of different quantum correlations by
multiple observers has been investigated both theoretically and experimentally
in several subsequent works using the unsharp measurement formalism. Such
works include, for example, steering a single system multiple times
sasmal_steering ; shenoy ; choi_steering ; shashank_steering ; steernew21 ;
estwo ; estwo2 , exploring Bell-type nonlocality in various settings npj_2018
; Qu2017 ; debarshi_facets ; rennew ; Saha19 ; foletto_entanglement ; expnew ;
zhang21 ; cglmp ; prl_brown , witnessing entanglement of bipartite and
tripartite states bera_entanglement ; ananda_entanglement ;
srivastava_entanglement , sharing of nonlocal advantage of quantum coherence
coherence_sounak , and quantum contextuality akpan . Applications of
sequential detection of quantum correlations in different information
processing tasks have also been reported rantt ; cctt ; appln1tt ; ractt ;
appln4tt ; appln5tt ; sroytt ; rsp_shounak . Recently, the technique of
choosing different sharpness parameters for the different measurement settings
of each observer has been proposed for obtaining the possibility of unbounded
number of observers sharing Bell-nonlocality prl_brown ; bntwo ; cabello , and
this type of result has also been probed towards random access code generation
rac_debarshi .

The above works have stimulated wide interest in recycling various types of
quantum correlations for their use in multi-observer networks. A natural
question emerges in this context as to if any quantitative advantage can be
gained from the resource theoretic perspective by the reuse of correlations in
a single copy of a quantum state. In the present work we answer this question
in the affirmative by computing the consumed resource (entanglement in the
present context), as well as the resource theoretic cost of measurement
rom_prl  required for witnessing entanglement ewo ; oew1 . For this purpose,
we consider a single copy of a bipartite two-qubit entangled state that is
shared between multiple observers on both the wings, who sequentially access
and perform measurements on the state. We first determine the essential figure
of merit in this scenario, that is given by the maximum number of observers
who can successfully detect entanglement contained in the bipartite two-qubit
state. We next compute the entanglement consumed in the detection process, as
well as the robustness of the required measurements, and compare these
quantities with those obtained by using several identical copies of the
initial state shared by the same pairs of observers on both the wings.

Benchmarking of a sequential network from a resource theoretic perspective has
remained hitherto unexplored in the literature. By the formalism used here, we
are able to quantitatively demonstrate the resource theoretic advantage of a
sequential network compared to the case when multiple copies of entangled
states are used as resource.

In particular, we consider a scenario involving equal number, say, $n$ numbers
of Alices and Bobs (who represent multiple observers here and all Alices are
spatially separated from all Bobs) at both wings. First, the pair Alice1-Bob1
performs a measurement to detect entanglement of the initially shared state
employing a suitable entanglement witness operator ewo , and then, Alice1
passes her particle to Alice2, whereas, Bob1 passes his particle to Bob2.
Next, the pair Alice2-Bob2 repeats the same procedure of witnessing
entanglement, and passes the particles to the next pair in the sequence, i.e.,
Alice3-Bob3, and the process continues. An extension of the above scenario may
also be considered for an asymmetric case involving an unequal number of
Alices and Bobs. In the latter case the process is similar to the first
scenario, except that Alice1 can be paired not only with Bob1 but also with
Bob2, Bob3 and so on. Similarly, o ther Alices can be paired with multiple
Bobs to share and witness entanglement. It is to be noted that in both the
scenarios each observer measures independently on her/his subsystem, being
ignorant about the measurement choices and outcomes of others. Additionally,
we restrict ourselves to protocols without bias111For a very recent approach
on studying recycling of entanglement, see hallnew ., where each observer
employs the same value of the sharpness parameter corresponding to all the
settings of her/his non-projective measurement.

We next focus towards addressing the main aim of the present work, viz., a
resource theoretic comparison of performance of sequential network schemes
based on single copy entangled state, with that of schemes based on multi-copy
entangled states shared by the same number of observers. In order to compare
the sequential and the non-sequential scenarios we utilize the detectability
(or visibility) in terms of the expectation value of the entanglement witness
operator oew1 , as well as the information extraction capability of the
involved measurement, defined quantitatively via the robustness of measurement
rom_prl . In terms of the above two parameters, we show that the sequential
measurement protocol provides advantages in witnessing entanglement by
multiple pair of observers in terms of the resources consumed.

We arrange the rest of the paper in the following way. In Sec. II, we provide
a brief overview of some basic tools employed in our analysis, such as
entanglement witness operators, unsharp measurement and robustness of
measurements. Next, in Sec. III, we provide details of our symmetric network
scenario. The main results regarding the bounds on the number of observers,
and the resource theoretic comparisons pertaining to the first scenario are
discussed in Sec. IV. Sec. V contains an analysis of the asymmetric extension
of the above scenario. Finally, in Sec. VI we summarize our results with some
concluding discussions.

##  II Basic tools

In this section, we present the basic ideas of entanglement witness operators,
unsharp measurements and robustness of measurements. These concepts will be
used later for presenting the main results of this paper.

###  II.1 Entanglement Witness Operators

A Hermitian operator $W$ is called an entanglement witness operator if there
exists at least one entangled state $\rho_{e}\notin\mathcal{S}$ such that
$\text{Tr}(W\rho_{e})<0$ and $\text{Tr}(W\rho)\geq 0$ for all
$\rho\in\mathcal{S}$ with $\mathcal{S}$ being the set of all separable states
ewo . One can find out an entanglement witness operator for each entangled
state. However, finding out the optimal entanglement witness operator for a
given entangled state is not always easy oew1 . For detecting entanglement,
one is usually interested in decomposing an entanglement witness operator in
terms of local quantum measurements. This enables performing the detection
process using local quantum measurements ewo .

Consider the state
$|\psi^{+}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$. The optimal
entanglement witness operator for this state is given by ewo ,

$W=\frac{1}{4}\Big{(}\mathbb{I}\otimes\mathbb{I}+\sigma_{z}\otimes\sigma_{z}-\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}\Big{)}.$
(1)

The advantage of this entanglement witness operator is that it can be
implemented in the laboratory by performing a three correlated local quantum
measurements in the bases associated with the Pauli operators
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$.

If in the preparation process of the state $|\psi^{+}\rangle$ some random
noise acts, then the resultant state may turn out to be

$\rho=p|\psi^{+}\rangle\langle\psi^{+}|+(1-p)\frac{\mathbb{I}}{2}\otimes\frac{\mathbb{I}}{2},$
(2)

where $0<p\leq 1$; $\frac{\mathbb{I}}{2}\otimes\frac{\mathbb{I}}{2}$ denotes
white noise and $(1-p)$ is the strength of the noisy process. The entanglement
witness operator $W$ remains optimal for the state $\rho$ as well ewo .

###  II.2 Optimality of Unsharp Measurements

A fundamental feature of quantum theory is that no information about a system
can be obtained without perturbing its state heisenberg . Projective
measurements, also known as strong measurements, are the most informative at
the cost of maximally disturbing the initial state. On the other hand, weak
measurements are characterised by broad pointer states, and are less
informative, but affect lesser the initial state silva_prl , thereby
reflecting a nontrivial trade off between information gain and disturbance
igd1 ; igd2 . In our sequential networks, it is important to employ such weak
measurements which optimize the information gain-disturbance trade-off to
achieve best performance, rather than some random choice of measurement. In
mathematics_mal , it was shown that unsharp measurement which is an one-
parameter Positive Operator-Valued Measure (POVM) satisfies the optimality
criteria.

Generalized quantum measurement or POVM um1 ; unsharp_measurement  is defined
by a set of positive operators that add to identity, i.e.,
$E\equiv\\{E_{i}|\sum E_{i}=\mathbb{I},0\leq E_{i}\leq\mathbb{I}\\}$. Consider
the dichotomic observable $\vec{\sigma}\cdot\hat{n}$, which is the spin
component observable for qubits along the direction $\hat{n}$. Here
$\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is a vector composed of
three Pauli operators and $\hat{n}$ is a unit vector in $\mathbb{R}^{3}$.
Given the observable $\vec{\sigma}\cdot\hat{n}$, one can define the dichotomic
unsharp observable
$E^{\lambda}_{\hat{n}}=E^{\lambda}_{+|\hat{n}}-E^{\lambda}_{-|\hat{n}}$ uno2
associated with the sharpness parameter $\lambda\in(0,1]$, where

$E^{\lambda}_{\pm|\hat{n}}=\lambda
P_{\pm|\hat{n}}+\frac{1-\lambda}{2}\mathbb{I}$ (3)

are the effect operators and satisfy
$E^{\lambda}_{+}+E^{\lambda}_{-}=\mathbb{I},\,0\leq
E^{\lambda}_{\pm}\leq\mathbb{I}$.

The probabilities of getting the outcomes $+1$ and $-1$, when the above
unsharp measurement is performed on the state $\rho$, are given by
$\text{Tr}[\rho E^{\lambda}_{+|\hat{n}}]$ and $\text{Tr}[\rho
E^{\lambda}_{-|\hat{n}}]$, respectively. The expectation value of
$E^{\lambda}_{\hat{n}}$ for a given $\rho$ is defined as,

$\displaystyle\langle E^{\lambda}_{\hat{n}}\rangle=\text{Tr}[\rho
E^{\lambda}_{+|\hat{n}}]-\text{Tr}[\rho
E^{\lambda}_{-|\hat{n}}]=\lambda\langle\vec{\sigma}\cdot\hat{n}\rangle,$ (4)

where $\langle\vec{\sigma}\cdot\hat{n}\rangle$ $=$
$\text{Tr}[\rho(P_{+|\hat{n}}-P_{-|\hat{n}})]$ denotes the expectation value
of the observable $\vec{\sigma}\cdot\hat{n}$ under projective measurement. The
post-measurement state can be determined using the generalized von Neumann-
Lüders transformation rule um1 ; unsharp_measurement  as follows,

$\rho\rightarrow\frac{\sqrt{E^{\lambda}_{\pm|\hat{n}}}\,\rho\,\sqrt{E^{\lambda}_{\pm|\hat{n}}}}{\text{Tr}(\rho
E^{\lambda}_{\pm|\hat{n}})}.$ (5)

In the weak measurement formalism with broad pointer states, the optimal
information gain-disturbance trade-off is characterised by a condition
involving two parameters called quality factor (F) and precision (G). An
optimal pointer satisfies $F^{2}+G^{2}=1$, which implies maximal information
gain for a given amount of disturbance silva_prl . In the unsharp measurement
formalism described above, it turns out that
$G=\lambda,F=\sqrt{1-\lambda^{2}}$ mathematics_mal , thereby satisfying the
optimality condition.

###  II.3 Robustness of Measurements

The concept of “Robustness of Measurements” (RoM) has been formulated recently
to quantify the informativeness of a measurement rom_prl . Given a particular
measurement $E$, RoM of $E$, denoted by $R(E)$, quantifies to what extent $E$
is a resourceful measurement. When a measurement returns an arbitrary outcome
$i$ with probability $q(i)$ independent of the quantum state measured, the
measurement is called a trivial measurement rom_prl . Such a measurement has
POVM elements $E_{i}$ with $E_{i}=q(i)\mathbb{I}$ for all $i$. It is evident
that trivial measurements are not informative or resourceful at all.

RoM is defined as the minimal amount of noise that needs to be added to the
measurement such that the measurement becomes a trivial one rom_prl . Suppose,
instead of always performing the measurement $E=\\{E_{i}\\}$, one performs a
different measurement $F=\\{F_{i}\\}$ sometimes. The informativeness of the
measurement $E$ can be captured by the minimal probability of this other
measurement $F$ that makes the overall measurement trivial. Hence, the RoM can
be formally defined as,

$\displaystyle R(E)=\min_{F,q}r$ $\displaystyle\text{such that}\hskip
11.38092pt\frac{E_{i}+rF_{i}}{1+r}=q(i)\,\mathbb{I}\hskip
11.38092pt\forall\,\,i,$ $\displaystyle\hskip 14.22636ptF_{i}\geq
0\,\,\forall\,i\,\,\,\text{and}\,\,\,\sum_{i}F_{i}=\mathbb{I}.$ (6)

Here, the minimization is taken over all noise measurements $F=\\{F_{i}\\}$
and all probability distributions $q=\\{q(i)\\}$.

It was shown in rom_prl  that RoM can be written as,

$R(E)=\sum_{i}||E_{i}||_{\infty}-1,$ (7)

where $||E_{i}||_{\infty}$ is the operator norm of $E_{i}$. Note that
$||E_{i}||_{\infty}$ is equal to the maximum eigen value of
$\sqrt{E_{i}^{\dagger}E_{i}}$ hayashi_book . Hence, for the unsharp
measurement
$E^{\lambda}_{\hat{n}}\equiv\\{E^{\lambda}_{+|\hat{n}},E^{\lambda}_{-|\hat{n}}\\}$
defined in Eq.(3), we have

$R(E^{\lambda}_{\hat{n}})=\lambda,$ (8)

i.e., the sharpness parameter. Using the resource theory of measurement
informativeness rom_prl , any trivial measurement can be considered as a free
measurement and any measurement which is not trivial is a resourceful
measurement. Hence, RoM characterises the amount of resource in a measurement
i.e., its information extraction capacity.

##  III Setting up the scenario

In the present study, we will consider the particular scenario as described
below (see Fig. 1).

######  Scenario 1.

We consider $n$ number of sequential Alices (Alice1, Alice2, Alice3, $\cdots$,
Alicen) and $n$ number of sequential Bobs (Bob1, Bob2, Bob3, $\cdots$, Bobn),
where $n$ is a-priori arbitrarily large. At first, the pair Alice1-Bob1
detects the entanglement of the initial state $\rho_{i}$ of two
spin-$\frac{1}{2}$ particles using entanglement witness operator. Alice1 then
passes her particle to Alice2 and Bob1 passes his particle to Bob2.
Alice2-Bob2 then detects entanglement of the two-qubit state received after
the measurements by Alice1 and Bob1. Consequently, Alice2 and Bob2 pass their
particles to Alice3 and Bob3 respectively, and so on. The process is
terminated when the Alicen-Bobn pair is unable to detect entanglement.

Figure 1: An entangled state $\rho_{i}$ of two spin-$\frac{1}{2}$ particles is
initially shared between Alice1 and Bob1, and subsequently, between multiple
Alices (Alice1, Alice2, Alice3, $\cdots$) on one wing and multiple Bobs (Bob1,
Bob2, Bob3, $\cdots$) on the other wing, sequentially. Multiple Alices perform
measurements on one particle sequentially and multiple Bobs perform
measurements on the other particle sequentially.

In the above scenario we consider the following assumptions:

1) Each Alice (Bob) performs measurements independent of the measurement
settings and outcomes of the previous Alices (Bobs).

2) All possible measurement settings of each Alice (Bob) are equally probable,
i.e., we are considering unbiased input scenario for each Alice and each Bob.

3) In a given run of the experiment, each Alice (Bob) applies the same value
of the sharpness parameter for all of her (his) measurement settings.

In the case of sharp projective measurement, one obtains the maximum amount of
information at the cost of maximum disturbance to the state. In the scenarios
considered by us, Alicei (Bobi) passes on the respective particle to Alicei+1
(Bobi+1) after performing suitable measurement. Hence, in this case, Alicei
(Bobi) needs to perform measurement for detecting entanglement by disturbing
the state minimally such that some entanglement remains in the post-
measurement state to be detected by Alicei+1 (Bobi+1). This can be achieved in
the unsharp measurement formalism as the disturbance is minimized for any
fixed amount of information gain in this formalism for qubits mathematics_mal
; sasmal_steering .

###  III.1 Modified entanglement witness operator in unsharp measurement
formalism

Note that the entanglement witness operator (1) can be implemented in the
laboratory by performing projective quantum measurements. Since, in our
scenarios, all Alices (Bobs), except the last Alice (Bob) in a sequence,
perform unsharp measurements, the entanglement witness operator (1) needs to
be modified accordingly. In order to modify the entanglement witness operator
(1), we follow the process described in bera_entanglement ;
ananda_entanglement .

Suppose, Alicei and Bobj perform unsharp measurement of spin component
observables $\vec{\sigma}\cdot\hat{n}_{i}$ and $\vec{\sigma}\cdot\hat{m}_{j}$
respectively. The sharpness parameters associated with the measurements by
Alicei and Bobj are denoted by $\xi_{i}$ and $\lambda_{j}$ respectively with
$\xi_{i},\lambda_{j}$ $\in$ $(0,1]$. We will follow this notation throughout
the paper.

The joint probability of obtaining the outcomes $a_{i}$, $b_{j}$ (with
$a_{i},b_{j}\in\\{+1,-1\\}$), when Alicei and Bobj perform the above unsharp
measurements, can be evaluated using the expression,

$\mbox{Tr}\Big{[}\rho\Big{(}E^{\xi_{i}}_{a_{i}|\hat{n}_{i}}\otimes
E^{\lambda_{j}}_{b_{j}|\hat{m}_{j}}\Big{)}\Big{]},$

where $\rho$ is state shared by Alicei and Bobj; and the expressions of
$E^{\xi_{i}}_{a_{i}|\hat{n}_{i}}$ and $E^{\lambda_{j}}_{b_{j}|\hat{m}_{j}}$
are defined following Eq.(3). The expectation value of the the above joint
measurement in the state $\rho$ is given by,

$\displaystyle\langle E^{\xi_{i}}_{\hat{n}_{i}}\otimes
E^{\lambda_{i}}_{\hat{m}_{j}}\rangle$
$\displaystyle=\mbox{Tr}\Big{[}\Big{\\{}\Big{(}E^{\xi_{i}}_{+|\hat{n}_{i}}-E^{\xi_{i}}_{-|\hat{n}_{i}}\Big{)}\otimes\Big{(}E^{\lambda_{j}}_{+|\hat{m}_{j}}-E^{\lambda_{j}}_{-|\hat{m}_{j}}\Big{)}\Big{\\}}\rho\Big{]}$
$\displaystyle=\xi_{i}\,\lambda_{j}\,\mbox{Tr}\Big{[}\Big{\\{}\Big{(}P_{+|\hat{n}_{i}}-P_{-|\hat{n}_{i}}\Big{)}\otimes\Big{(}P_{+|\hat{m}_{j}}-P_{-|\hat{m}_{j}}\Big{)}\Big{\\}}\rho\Big{]}$
$\displaystyle=\xi_{i}\,\lambda_{j}\,\langle\vec{\sigma}\cdot\hat{n}_{i}\otimes\vec{\sigma}\cdot\hat{m}_{j}\rangle,$
(9)

where
$\langle\vec{\sigma}\cdot\hat{n}_{i}\otimes\vec{\sigma}\cdot\hat{m}_{j}\rangle$
is the expectation value under projective measurements. We can hence use the
substitution
$\langle\vec{\sigma}\cdot\hat{n}_{i}\otimes\vec{\sigma}\cdot\hat{m}_{j}\rangle\rightarrow\xi_{i}\,\lambda_{j}\,\langle\vec{\sigma}\cdot\hat{n}_{i}\otimes\vec{\sigma}\cdot\hat{m}_{j}\rangle$
in order to obtain the modified entanglement witness operator for the case of
unsharp measurements. For any $\xi_{i},\lambda_{j}$ $\in$ $(0,1]$, the
modified entanglement witness operator for the state
$|\psi^{+}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$ has the following
form,

$\displaystyle W^{(\xi_{i},\lambda_{j})}$
$\displaystyle=\frac{1}{4}\Big{(}\mathbb{I}\otimes\mathbb{I}+\xi_{i}\sigma_{z}\otimes\lambda_{j}\sigma_{z}-\xi_{i}\sigma_{x}\otimes\lambda_{j}\sigma_{x}-\xi_{i}\sigma_{y}\otimes\lambda_{j}\sigma_{y}\Big{)}$
$\displaystyle=\frac{1}{4}\Big{[}\mathbb{I}\otimes\mathbb{I}+\xi_{i}\lambda_{j}\Big{(}\sigma_{z}\otimes\sigma_{z}-\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}\Big{)}\Big{]}.$
(10)

Now, for any separable state $\rho_{s}\in\mathcal{S}$, we have

$\displaystyle\text{Tr}\Big{(}W^{(\xi_{i},\lambda_{j})}\rho_{s}\Big{)}$
$\displaystyle=\text{Tr}\Big{[}\big{(}\xi_{i}\,\lambda_{j}\,W+\frac{1}{4}(1-\xi_{i}\,\lambda_{j})\mathbb{I}\otimes\mathbb{I}\big{)}\rho_{s}\Big{]}$
$\displaystyle=\xi_{i}\,\lambda_{j}\text{Tr}\Big{(}W\rho_{s}\Big{)}+\frac{1}{4}(1-\xi_{i}\,\lambda_{j})$
$\displaystyle\geq
0\,\,\,\,\,\,\,\,\forall\,\rho_{s}\in\mathcal{S}\,\,\,\,\,\,\text{as}\,\,0<\xi_{i},\lambda_{j}\leq
1,$ (11)

showing that $W^{(\xi_{i},\lambda_{j})}$ is a valid entanglement witness
operator.

##  IV Witnessing entanglement by sequential observers

We now focus on the task of entanglement detection in our sequential network
scenario. We take three different types of the initially shared states
$\rho_{i}$.

###  IV.1 Initially shared maximally entangled two-qubit state

Let us consider that Alice1 and Bob1 initially share the Bell state given by,
$|\psi^{+}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$. Different pairs
of Alice and Bob (i.e., Alice1-Bob1, Alice2-Bob2, Alice3-Bob3, $\cdots$,
Alicen-Bobn) try to detect entanglement sequentially. At first, we will
address the following question: how many such pairs of Alice and Bob can
sequentially detect entanglement.

Let Alicei and Bobi ($i\in\\{1,2,\cdots,n\\}$) perform unsharp measurements
with sharpness parameters $\xi_{i}$ and $\lambda_{i}$ respectively. The pair
Alice1-Bob1 can detect entanglement if the following condition is satisfied:

$\text{Tr}\Big{[}W^{(\xi_{1},\lambda_{1})}|\psi^{+}\rangle\langle\psi^{+}|\Big{]}<0,$
(12)

where the operator $W^{(\xi_{1},\lambda_{1})}$ is given by Eq.(10). After
simplification, we get the following condition from (12),

$\xi_{1}\,\lambda_{1}>\frac{1}{3}$ (13)

Next, let us find out the post measurement state received by the pair
Alice2-Bob2 from Alice1-Bob1. As Alice2 (Bob2) acts independent of the
measurement setting and outcome of Alice1 (Bob1) in each experimental run, we
take average over the measurement settings and outcomes by Alice1 and Bob1.
Hence, the state received, on average, by Alice2-Bob2 from Alice1-Bob1 is
given by,

$\displaystyle\rho_{A_{2}B_{2}}=\frac{1}{9}\sum_{n_{1},m_{1},a_{1},b_{1}}\Big{(}\sqrt{E^{\xi_{1}}_{a_{1}|\hat{n}_{1}}}\otimes\sqrt{E^{\lambda_{1}}_{b_{1}|\hat{m}_{1}}}\Big{)}\,|\psi^{+}\rangle\langle\psi^{+}|\,\Big{(}\sqrt{E^{\xi_{1}}_{a_{1}|\hat{n}_{1}}}\otimes\sqrt{E^{\lambda_{1}}_{b_{1}|\hat{m}_{1}}}\Big{)},$
(14)

with $\hat{n}_{1},\hat{m}_{1}\in\\{\hat{x},\hat{y},\hat{z}\\}$ and
$a_{1},b_{1}\in\\{+1,-1\\}$. Here, we have used the fact that each of Alice1
and Bob1 performs any of the three local unsharp measurements associated with
the observables $\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$ in each experimental
run in order to implement the entanglement witness operator (10). We have also
used here the assumtion that all possible measurement settings of Alice1 and
that of Bob1 are equally probable. After simplification, we get from Eq.(14),

$\displaystyle\rho_{A_{2}B_{2}}$
$\displaystyle=p|\psi^{+}\rangle\langle\psi^{+}|+(1-p)\frac{\mathbb{I}}{2}\otimes\frac{\mathbb{I}}{2}$
$\displaystyle\text{with}\,\,\,p=\frac{1}{9}\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}.$
(15)

Since, the state (15) has the form given by Eq.(2), the Alice2-Bob2 pair again
uses the same entanglement witness operator given by Eq.(10) to detect
entanglement. Hence, Alice2-Bob2 can detect entanglement if the following
condition is satisfied,

$\text{Tr}\Big{[}W^{(\xi_{2},\lambda_{2})}\,\rho_{A_{2}B_{2}}\Big{]}<0,$ (16)

which implies the condition,

$\xi_{2}\,\lambda_{2}>\frac{3}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}}.$
(17)

Proceeding in a similar way, it can be shown that the state
$\rho_{A_{3}B_{3}}$ received, on average, by Alice3-Bob3 from Alice2-Bob2 has
the similar form of Werner state (2) and the pair Alice3-Bob3 can detect
entanglement if

$\text{Tr}\Big{[}W^{(\xi_{3},\lambda_{3})}\,\rho_{A_{3}B_{3}}\Big{]}<0,$ (18)

i.e., when

$\xi_{3}\,\lambda_{3}>\frac{27}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}}.$
(19)

Repeating the above steps, it can be shown that Alice4-Bob4 can detect
entanglement if the following condition is satisfied,

$\xi_{4}\,\lambda_{4}>\frac{243}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{3}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{3}^{2}}\Big{)}}.$
(20)

Similar conditions can be found out that ensure entanglement detection by the
other pairs, i.e., Alicen-Bobn.

Now, our purpose is to investigate what is the maximum number of the
sequential pairs succeed in witnessing the entanglement of the shared two-
qubit state. Combining Eqs.(13), (17), (19) and (20) and performing some
analytical calculations (see Appendix A for details), we get that Alice1-Bob1,
Alice2-Bob2 and Alice3-Bob3 can detect entanglement if the following
conditions are satisfied simultaneously,

$\displaystyle\xi_{1}=\lambda_{1}=0.58+\delta_{1}\hskip
7.11317pt\text{with}\hskip 7.11317pt0\leq\delta_{1}<<1,$ (21)
$\displaystyle\xi_{2}=\lambda_{2}=0.66+\delta_{2}\hskip
7.11317pt\text{with}\hskip 7.11317pt0\leq\delta_{2}<<1,$ (22)
$\displaystyle\xi_{3}=\lambda_{3}=0.79+\delta_{3}\hskip
7.11317pt\text{with}\hskip 7.11317pt0\leq\delta_{3}<<1.$ (23)

Here the numerical digits appearing in the the above conditions are rounded to
two decimal places.

If Alice1-Bob1, Alice2-Bob2 and Alice3-Bob3 perform measurements with
sharpness parameters satisfying Eqs.(21) (22), (23) , then it can be shown
that Alice4-Bob4 cannot witness the entanglement even if they perform
projective measurements, i.e., with $\xi_{4}=\lambda_{4}=1$ (see Appendix A
for details). Thus, at most three sequential pairs of Alice and Bob can
witness the entanglement in this case.

###  IV.2 Initially shared two-qubit Werner state

Let Alice1 and Bob1 initially share the two-qubit Werner state given by,

$\rho_{i}=p|\psi^{+}\rangle\langle\psi^{+}|+(1-p)\frac{\mathbb{I}}{2}\otimes\frac{\mathbb{I}}{2},$
(24)

where $0<p\leq 1$. For this state, the optimal entanglement witness operator
remains the same as before, i.e., it is $W$ given by Eq.(1) ewo .

In this case at most three sequential pairs of Alice and Bob (for example,
Alice1-Bob1, Alice2-Bob2 and Alice3-Bob3) can witness the entanglement using
the entanglement witness operator given by Eq.(10). Furthermore, we get the
following results,

(1) When $0.80<p\leq 1$, each of the pairs Alice1-Bob1, Alice2-Bob2 and
Alice3-Bob3 can detect the entanglement. Other pairs Alicei-Bobi with
$i\in\\{4,5,6,\cdots\\}$ cannot detect entanglement.

(2) When $0.57<p\leq 0.80$, each of the two pairs Alice1-Bob1 and Alice2-Bob2
can detect the entanglement. Other pairs Alicei-Bobi with
$i\in\\{3,4,5,\cdots\\}$ cannot detect entanglement.

(3) When $0.33<p\leq 0.57$, only the pair Alice1-Bob1 can detect the
entanglement. Other pairs Alicei-Bobi with $i\in\\{2,3,4,\cdots\\}$ cannot
detect entanglement.

###  IV.3 Initially shared non-maximally entangled two-qubit pure state

Suppose that Alice1 and Bob1 initially share a non-maximally entangled two-
qubit pure state given by,

$|\Psi\rangle=\cos\theta|01\rangle+\sin\theta|10\rangle,$ (25)

with $0<\theta<\frac{\pi}{4}$. For this state also, the optimal entanglement
witness operator is given by Eq.(1) ewo .

In this case too, at most three pairs of Alice and Bob (e.g., Alice1-Bob1,
Alice2-Bob2, Alice3-Bob3) can witness entanglement sequentially through the
entanglement witness operator (10). Further, the following results are
obtained.

(1) When $\frac{\pi}{8}\leq\theta<\frac{\pi}{4}$, each of the pairs
Alice1-Bob1, Alice2-Bob2 and Alice3-Bob3 can detect the entanglement. Other
pairs Alicei-Bobi with $i\in\\{4,5,6,\cdots\\}$ cannot detect entanglement.

(2) When $\frac{\pi}{17}\leq\theta<\frac{\pi}{8}$, each of the two pairs
Alice1-Bob1 and Alice2-Bob2 can detect the entanglement. Other pairs Alicei-
Bobi with $i\in\\{3,4,5,\cdots\\}$ cannot detect entanglement.

(3) When $0<\theta<\frac{\pi}{17}$, only the pair Alice1-Bob1 can detect the
entanglement. Other pairs Alicei-Bobi with $i\in\\{2,3,4,\cdots\\}$ cannot
detect entanglement.

###  IV.4 Advantage of the sequential measurement scenario

For sequential detection of entanglement by multiple pairs of observers, it is
important to ensure that the expectation values of the witness operator for
different pairs become as much negative as possible, for feasibility of actual
detection. In our case, as we discussed earlier, maximum three pairs of Alice
and Bob can detect entanglement. Hence, for our purpose, we define
‘Detectability’ ($D$) as the sum of the expectation values of the entanglement
witness operators for all the three pairs.

Mathematically, detectability, is defined as

$\displaystyle D=\sum_{i}D_{ii}$
$\displaystyle=\sum_{i}\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{A_{i}B_{i}}\Big{]}$
$\displaystyle\hskip 8.5359pt\text{with}\hskip
8.5359pt\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{A_{i}B_{i}}\Big{]}<0\,\,\forall
i,$ (26)

where $\rho_{A_{i}B_{i}}$ is the state shared by the pair Alicei-Bobi.

We now demonstrate the advantage of sequential scenario when Alice1 and Bob1
initially share the Bell state
$\rho_{A_{1}B_{1}}=|\psi^{+}\rangle\langle\psi^{+}|$ with
$|\psi^{+}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$. In this case,
maximum three pairs can detect entanglement. Hence, $D$ in this case is given
by,

$\displaystyle D=$ $\displaystyle D_{11}+D_{22}+D_{33}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{A_{i}B_{i}}\Big{]},$
(27)

where $\rho_{A_{i}B_{i}}$ is the state shared, on average, by the pair Alicei-
Bobi. Based on the analysis described in Sec. IV.1, it can be shown that

$\displaystyle D=$
$\displaystyle\frac{1}{4}(1-3\xi_{1}\lambda_{1})+\frac{1}{12}\Big{[}3-\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\xi_{2}\lambda_{2}\Big{]}$
(28)
$\displaystyle+\frac{1}{108}\Big{[}27-\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}\xi_{3}\lambda_{3}\Big{]}.$

Now, maximum detectability $D_{\text{max}}$ is defined as the maximum negative
value, i.e., the minimum value of $D$ under the constraint that each of the
three pairs detect entanglement. The maximum detectability $D_{\text{max}}$
serves as a tool to quantify the the ability of the three pairs to detect
entanglement in an experiment. $D_{\text{max}}$ can be evaluated as follows,

$\displaystyle
D_{\text{max}}=\min_{\xi_{1},\lambda_{1},\xi_{2},\lambda_{2},\xi_{3},\lambda_{3}}D$
$\displaystyle\text{such that}\hskip
11.38092ptD_{ii}=\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{A_{i}B_{i}}\Big{]}<0\,\,\forall
i\in\\{1,2,3\\}.$ (29)

We get $D_{\text{max}}=-0.20$, which is obtained for
$\xi_{1}=\lambda_{1}=0.73$, $\xi_{2}=\lambda_{2}=0.80$, and
$\xi_{3}=\lambda_{3}=1$.

Next, let us evaluate the total RoM that is needed to achieve the above-
mentioned $D_{\text{max}}$. As mentioned earlier, for the unsharp measurement
$E^{\lambda}_{\hat{n}}\equiv\\{E^{\lambda}_{+|\hat{n}},E^{\lambda}_{-|\hat{n}}\\}$
defined by Eq.(3), we have $R(M)=\lambda$. Hence, the total RoM, denoted by
$R_{\text{total}}(M)$, needed to achieve the above-mentioned $D_{\text{max}}$
is given by,

$\displaystyle R_{\text{total}}(M)$
$\displaystyle=\sum_{i=1}^{3}(\xi_{i}+\lambda_{i})\,\,\,\text{such
that}\,\,\,D=D_{\text{max}}=-0.20$ $\displaystyle=2(0.73+0.80+1)$
$\displaystyle=5.06$ (30)

$R_{\text{total}}(M)$ quantifies the total amount of resource consumed while
performing the unsharp measurements necessary for achieving the maximum
detectability.

Next, we will compare the resource requirement in terms of the entanglement
consumed and total RoM between the sequential measurement scenario (1) and the
corresponding non-sequential measurement scenario. The non-sequential
measurement scenario involves three pairs of observers- Alice1-Bob1,
Alice2-Bob2 and Alice3-Bob3, where each of these three pairs share one copy of
the two-qubit state. Hence, this non-sequential measurement scenario involves
total three copies of a two-qubit state. Each pair detects entanglement of
their shared state using the witness operator (10). For the purpose of
meaningful comparison of the sequential and the non-sequential scenarios, we
need to ensure that the detectability $D$ remains the same for both the
scenarios. This can be accomplished by taking different types of initial
states (three copies each) for the non-sequential scenario, as discussed
below.

####  IV.4.1 Comparison with non-sequential scenario with three copies of
Werner state

Let us first consider that three identical Werner states,

$\rho_{\text{W}}=p|\psi^{+}\rangle\langle\psi^{+}|+(1-p)\frac{\mathbb{I}}{2}\otimes\frac{\mathbb{I}}{2}$
(31)

with $0<p\leq 1$ are shared between the three pairs of parties. Now, we would
like to evaluate the total amount of entanglement (denoted by $\eta$) that is
necessary in the non-sequential measurement scenario for the detectability in
the non-sequential measurement scenario given by,
$D^{\text{NS}}=\sum_{i=1}^{3}\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{\text{W}}\Big{]}$
being equal to the maximum detectability in the sequential measurement
scenario denoted by $D^{\text{S}}_{\text{max}}$, when the RoM for each party
(and, hence, the total RoM) in the sequential measurement case is equal to
that in the non-sequential measurement case. Since, $D^{\text{S}}$ =
$D^{\text{S}}_{\text{max}}$ in the sequential measurement scenario is achieved
for $\xi_{1}=\lambda_{1}=0.73$, $\xi_{2}=\lambda_{2}=0.80$, and
$\xi_{3}=\lambda_{3}=1$, we consider that the different parties in the non-
sequential measurement scenario perform unsharp measurements with respective
sharpness parameters given by, $\xi_{1}=\lambda_{1}=0.73$,
$\xi_{2}=\lambda_{2}=0.80$, and $\xi_{3}=\lambda_{3}=1$. With these values of
the sharpness parameter, the condition,
$D^{\text{NS}}=D^{\text{S}}_{\text{max}}$ implies that $p=0.58$. For $p=0.58$,
the concurrence of the Werner state $\rho_{\text{W}}$ is given by
$C(\rho_{\text{W}})=\dfrac{1}{2}(3p-1)=0.37$. Since, a total of three copies
of $\rho_{\text{W}}$ is involved in the non-sequential measurement scenario,
the total amount of entanglement $\eta$ necessary in the non-sequential case
is given by $\eta=3C(\rho_{\text{W}})=1.12$. On the other hand, only $1$ ebit
is sufficient in the sequential measurement scenario for achieving
$D^{\text{S}}_{\text{max}}$ as only one copy of the maximally entangled state
is involved. Hence, for achieving the same amount of detectability in the
sequential and the non-sequential measurement scenarios with the
resourcefulness of the measurements by any observer being equal in both the
scenarios (i.e., when the RoM of any observer is equal in b oth scenarios),
the non-sequential measurement scenario needs greater amount of initial
entanglement compared to the sequential measurement scenario. This result is
summarized in Table 1. This demonstrates a resource theoretic advantage of the
sequential measurement scenario over the non-sequential measurement scenario.

| Sequential | Non-sequential
---|---|---
| measurement | measurement
| scenario | scenario
Detectability $D$ | $-0.20$ | $-0.20$
Total robustness of |  |
measurement $R_{\text{total}}(M)$ | $5.06$ | $5.06$
Total | for $\rho_{\text{W}}$ |  $1$ ebit |  $1.12$ ebits
entanglement | for $\rho_{p}$ |  $1$ ebit |  $1.14$ ebits
consumed $\eta$ | for $|\Psi\rangle$ |  $1$ ebit |  $1.11$ ebits
Table 1: The total amount of required initial entanglement $\eta$ when each
party performs equally resourceful measurements in both scenarios. Here, the
non-sequential measurement scenario involves either three copies of the same
Werner state $\rho_{W}$, or three copies of the same state with colored noise
$\rho_{p}$, or three copies of the same non-maximally pure entangled state
$|\Psi\rangle$. All numerical values presented in this table are rounded to
two decimal places.

It is possible to demonstrate further the advantage of the sequential
measurement scenario from another perspective, as well. Let us now take the
total entanglement consumed $\eta$ in the two scenarios to be equal. The
detectability $D^{\text{NS}}$ in the non-sequential measurement scenario is
taken to be equal to the maximum detectability $D^{\text{S}}_{\text{max}}$ in
the sequential measurement scenario. Under these conditions, we compare the
total RoM, i.e., $R_{\text{total}}(M)$ in the two scenarios. Total
entanglement consumption in the sequential measurement scenario starting with
a maximally entangled initial state is $\eta$ = $1$ ebit. Hence, for the non-
sequential measurement scenario we prepare three identical Werner states with
the total amount of entanglement being equal to that in the sequential
measurement scenario, i.e., $3\times C(\rho_{\text{W}})=1$. Thus, each of the
three copies of the Werner states has concurrence $\frac{1}{3}$. This can be
achieved with $p=0.56$ for each Werner state $\rho_{\text{W}}$. Next, the
condition
$D^{\text{NS}}=\sum_{i=1}^{3}\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{\text{W}}\Big{]}=D^{\text{S}}_{\text{max}}=-0.20$
with $p=0.56$ gives the following condition,

$\xi_{1}\lambda_{1}+\xi_{2}\lambda_{2}+\xi_{3}\lambda_{3}=2.28.$ (32)

For simplicity, let us assume that $\xi_{i}=\lambda_{i}$ for all
$i\in\\{1,2,3\\}$ in the non-sequential measurement scenario. Note that this
condition is necessary for achieving the the maximum detectability in the
sequential measurement scenario. With these, the above condition becomes

$\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=2.28.$ (33)

Moreover, we have to ensure that each pair detects entanglement in the non-
sequential measurement scenario, i.e.,
$\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{\text{W}}\Big{]}<0$ for all
$i\in\\{1,2,3\\}$, where $\rho_{\text{W}}$ is the Werner state with the state
parameter $p=0.56$. Now, the condition
$\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,\rho_{\text{W}}\Big{]}<0$ for all
$i\in\\{1,2,3\\}$ with $p=0.56$ and $\xi_{i}=\lambda_{i}$ for all
$i\in\\{1,2,3\\}$ implies that $\lambda_{i}>0.77$ for all $i\in\\{1,2,3\\}$.
Our task is to determine the minimum amount of total RoM in the non-sequential
measurement scenario when $\lambda_{i}>0.77$ for all $i\in\\{1,2,3\\}$ and
when the condition (33) is satisfied. The total RoM in the non-sequential
measurement scenario with $\xi_{i}=\lambda_{i}\hskip 14.22636pt\forall
i\in\\{1,2,3\\}$ is given by
$R_{\text{total}}(M)=2(\lambda_{1}+\lambda_{2}+\lambda_{3})$. Minimizing over
the sharpness parameters,

$\displaystyle
R^{\text{min}}_{\text{total}}(M)=\min_{\lambda_{1},\lambda_{2},\lambda_{3}}R_{\text{total}}(M)=\min_{\lambda_{1},\lambda_{2},\lambda_{3}}2(\lambda_{1}+\lambda_{2}+\lambda_{3})$
$\displaystyle\hskip 14.22636pt\text{such that}\hskip
11.38092pt\lambda_{i}>0.77\,\,\forall\,i\in\\{1,2,3\\},$ $\displaystyle\hskip
14.22636pt\text{and}\hskip
11.38092pt\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=2.28.$ (34)

We obtain, $R^{\text{min}}_{\text{total}}(M)=5.20$. On the other hand, the
total RoM in the sequential measurement scenario is given by $5.06$, as
mentioned earlier. Hence, for achieving the same amount of detectability using
the same amount of total initial entanglement in the sequential and the non-
sequential measurement scenarios, the total resourcefulness of the
measurements (in terms of total RoM) necessary in the non-sequential
measurement scenario turns out to be greater than that in the sequential
measurement scenario. Thus a resource theoretic advantage of the sequential
measurement scenario over the non-sequential one is clearly implied. These
results are depicted in Table 2.

| Sequential | Non-sequential
---|---|---
| measurement | measurement
| scenario | scenario
Detectability $D$ | $-0.20$ | $-0.20$
Total entaglement |  |
consumed $\eta$ |  $1$ ebit |  $1$ ebit
Total robustness | for $\rho_{\text{W}}$ | $5.06$ | $\geq 5.20$
of measurement | for $\rho_{p}$ | $5.06$ | $\geq 5.18$
$R_{\text{total}}(M)$ | for $|\Psi\rangle$ | $5.06$ | $\geq 5.20$
Table 2: The required total robustness of measurement $R_{\text{total}}(M)$
when the same amount of total entanglement is consumed in the both scenarios.
The non-sequential measurement scenario involves either three copies of the
same Werner state $\rho_{W}$, or three copies of the same state with colored
noise $\rho_{p}$, or three copies of the same non-maximally pure entangled
state $|\Psi\rangle$. All numerical values presented in the table are rounded
to two decimal places.

####  IV.4.2 Comparison with non-sequential scenario with three copies of
mixed state with colored noise

Now, we consider that the non-sequential measurement scenario involves three
copies of the following state,

$\rho_{p}=p|\phi^{+}\rangle\langle\phi^{+}|+\frac{1-p}{2}(|01\rangle\langle
01|+|10\rangle\langle 10|),$ (35)

with $0<p\leq 1$.

Before proceeding, we will construct the optimal entanglement witness operator
of the above state using the method described in ewo ; mlew1 ; mlew2 . First,
we compute the eigen vector corresponding to the negative eigen value of
$\rho_{p}^{T_{B}}$, where $\rho_{p}^{T_{B}}$ denotes the partial transposition
of $\rho_{p}$. Then the entanglement witness operator is given by the
partially transposed projector onto this eigen vector. Following this
calculation, we obtain that the optimal entanglement witness operator for this
state is given by,

$\widetilde{W}=\frac{1}{4}\Big{(}\mathbb{I}\otimes\mathbb{I}-\sigma_{x}\otimes\sigma_{x}+\sigma_{y}\otimes\sigma_{y}-\sigma_{z}\otimes\sigma_{z}\Big{)}.$
(36)

Now, suppose Alicei performs unsharp measurements with sharpness parameter
$\xi_{i}$ $\in$ $(0,1]$ and Bobj performs unsharp measurements with sharpness
parameter $\lambda_{j}$ $\in$ $(0,1]$. Following the calculations mentioned in
Sec. III.1, it can be shown that the modified entanglement witness operator of
the state (35) for Alicei and Bobj is given by,

$\widetilde{W}^{(\xi_{i},\lambda_{j})}=\frac{1}{4}\Big{[}\mathbb{I}\otimes\mathbb{I}-\xi_{i}\,\lambda_{j}\Big{(}\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}\Big{)}\Big{]}.$
(37)

Following the method mentioned in Sec. III.1, it can be easily shown that for
any separable state $\rho_{s}\in\mathcal{S}$,
$\text{Tr}\Big{(}\widetilde{W}^{(\xi_{i},\lambda_{j})}\rho_{s}\Big{)}\geq 0$.

Next, we calculate the total amount of entanglement $\eta$ that is necessary
in the non-sequential measurement scenario involving three copies of
$\rho_{p}$ given by (35) for the detectability in the non-sequential
measurement scenario:
$D^{\text{NS}}=\sum_{i=1}^{3}\text{Tr}\Big{[}\widetilde{W}^{(\xi_{i},\lambda_{i})}\,\rho_{p}\Big{]}$
being equal to the maximum detectability $D^{\text{S}}_{\text{max}}=-0.20$ in
the sequential measurement scenario when the RoM for each party (and, hence,
the total RoM) in sequential case is equal to that in the non-sequential case.

In the sequential measurement scenario, $D^{\text{S}}_{\text{max}}$ is
achieved for $\xi_{1}=\lambda_{1}=0.73$, $\xi_{2}=\lambda_{2}=0.80$, and
$\xi_{3}=\lambda_{3}=1$. With these values of the sharpness parameter, the
condition, $D^{\text{NS}}=D^{\text{S}}_{\text{max}}$ implies that $p=0.69$.
With $p=0.69$, the concurrence of the state $\rho_{p}$ given by (35) becomes
$C(\rho_{p})=2p-1=0.38$. Since, a total of three copies of $\rho_{p}$ is
involved in the non-sequential measurement scenario, the total amount of
entanglement $\eta$ necessary in the non-sequential case is given by
$\eta=3C(\rho_{p})=1.14$. On the other hand, only $1$ ebit is sufficient in
the sequential measurement scenario for achieving $D^{\text{S}}_{\text{max}}$.
This result is summarized in Table 1. We observe that similar to the previous
case, here also the sequential measurement scenario is advantageous in terms
of the total amount of entanglement necessary for achievi ng a fixed amount of
detectability using a fixed RoM by each party.

Next, we take the total entanglement consumed $\eta$ in the two scenarios to
be equal (= 1 ebit) and the detectability $D^{\text{NS}}$ in the non-
sequential measurement scenario to be equal to the maximum detectability
$D^{\text{S}}_{\text{max}}=-0.20$ in the sequential measurement scenario.
Under these conditions, we compare the total RoM, i.e., $R_{\text{total}}(M)$
in the two scenarios. Here also, we assume that $\xi_{i}=\lambda_{i}$ for all
$i\in\\{1,2,3\\}$ in the non-sequential measurement scenario for simplicity.
Since the total entanglement in the non-sequential case should be $1$ ebit, we
can write $3\times C(\rho_{p})=1$, which gives $p=0.67$. Next, the condition
$D^{\text{NS}}=\sum_{i=1}^{3}\text{Tr}\Big{[}\widetilde{W}^{(\xi_{i},\lambda_{i})}\,\rho_{p}\Big{]}=-0.20$
with the assumption $\xi_{i}=\lambda_{i}$ for all $i\in\\{1,2,3\\}$ in the
non-sequential measurement scenario implies that

$\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=2.26.$ (38)

Further, we ensure that each pair detects entanglement in the non-sequential
measurement scenario, i.e.,
$\text{Tr}\Big{[}\widetilde{W}^{(\xi_{i},\lambda_{i})}\,\rho_{p}\Big{]}<0$ for
all $i\in\\{1,2,3\\}$, where $\rho_{p}$ is the state (35) with the state
parameter $p=0.67$. This gives $\lambda_{i}>0.77$ for all $i\in\\{1,2,3\\}$.
Under these conditions, we get that the total RoM in the non-sequential
measurement scenario is given by,
$R_{\text{total}}(M)=2(\lambda_{1}+\lambda_{2}+\lambda_{3})\geq 5.18$.

As mentioned earlier, the total RoM in the sequential measurement scenario is
$5.06$. Hence, for achieving the same amount of detectability using the same
amount of total entanglement in the sequential and the non-sequential
measurement scenarios, the total resourcefulness of the measurements (in terms
of total RoM) necessary in the non-sequential measurement scenario is greater
than that in the sequential measurement scenario. These results are depicted
in Table 2. This again signifies a resource theoretic advantage of the
sequential measurement scenario over the non-sequential one.

####  IV.4.3 Comparison with non-sequential scenario with three copies of non-
maximally entangled pure state

Next, we perform a similar comparison between the sequential and the non-
sequential measurement scenarios, when the non-sequential measurement scenario
involves three copies of the identical non-maximally pure entangled state
given by,

$|\Psi\rangle=\cos\theta|01\rangle+\sin\theta|10\rangle,$ (39)

with $0<\theta<\frac{\pi}{4}$. In the non-sequential measurement scenario,
each pair of Alice and Bob detects entanglement of the above pure state
through the entanglement witness operator given by Eq.(10).

Similar to the earlier cases, at first, we calculate the total amount of
entanglement $\eta$ that is necessary in the non-sequential measurement
scenario involving three copies of $|\Psi\rangle$ for the detectability in the
non-sequential measurement scenario:
$D^{\text{NS}}=\sum_{i=1}^{3}\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,|\Psi\rangle\langle\Psi|\Big{]}$
being equal to the maximum detectability $D^{\text{S}}_{\text{max}}=-0.20$ in
the sequential measurement scenario when the RoM for each party (and, hence,
the total RoM) in sequential case is equal to that in the non-sequential case.
The results in this case can be obtained following the similar steps as
adopted in the previous two cases. These results are summarized in Table 1.
Here too, it is evident that the sequential measurement scenario is
advantageous in terms of the total amount of entanglement necessary for
achieving a fixed amount of detectability using a fixed RoM by each party.

Now, let us consider that the total entanglement consumed $\eta$ in the non-
sequential measurement scenario is equal to that in the sequential measurement
scenario (= 1 ebit). Also, consider that the detectability
$D^{\text{NS}}=\sum_{i=1}^{3}\text{Tr}\Big{[}W^{(\xi_{i},\lambda_{i})}\,|\Psi\rangle\langle\Psi|\Big{]}$
in the non-sequential measurement scenario is equal to the maximum
detectability $D^{\text{S}}_{\text{max}}=-0.20$ in the sequential measurement
scenario. Under these conditions, we compare the total RoM, i.e.,
$R_{\text{total}}(M)$ in the two scenarios. Similar to the earlier cases, here
also, we assume $\xi_{i}=\lambda_{i}$ for all $i\in\\{1,2,3\\}$ in the non-
sequential measurement scenario for simplicity. The results obtained in this
case are presented in Table 2. Hence, for achieving the same amount of
detectability using the same amount of total initial entanglement in the
sequential and the non-sequential measurement scenarios, the total
resourcefulness of the measurements (in terms of total RoM) necessary in the
non-sequential measurement scenario turns out to be greater than that in the
sequential measurement scenario. Thus, the advantage of the sequential
measurement scenario over the non-sequential one is reinforced.

##  V More number of observers in an asymmetric scenario

In the scenario considered so far in this work, we assume that the number of
sequential Alices on one side is equal to the number of sequential Bobs on the
other. We call this scenario as a “Symmetric Scenario”. At this stage it may
be pertinent to ask the question as to what happens if we relax this condition
of symmetry. To this end, let us now consider sequential detection of
entanglement in the case when the number of Alices is not equal in general to
the number of Bobs. We consider that the Bell state given by,
$|\psi^{+}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$ is initially
shared.

It may be noted here that an asymmetric scenario involving a single Alice and
multiple Bobs (Bob1, Bob2, Bob3, $\cdots$, Bobm) was discussed in
bera_entanglement . In this case, it was shown that at most twelve Bobs can
detect entanglement with the single Alice bera_entanglement .

Next, consider the scenario involving two Alices (Alice1 and Alice2) and
multiple Bobs (Bob1, Bob2, Bob3, $\cdots$, Bobm). At first, the pair
Alice1-Bob1 detects entanglement using the entanglement witness operator (10).
As mentioned in Sec. IV.1, this pair can detect entanglement if the condition
(13) is satisfied. The state $\rho_{A_{2}B_{2}}$ received, on average, by
Alice2-Bob2 from Alice1-Bob1 is given by Eq.(15). Since, there are only two
Alices in the sequence, Alice2 performs projective measurements. On the other
hand, all subsequent Bobs (i.e., Bob2, Bob3, $\cdots$, Bobm) perform unsharp
measurements. The sharpness parameter associated with the measurement by Bobi
(with $i\in\\{2,3,\cdots,m\\}$) is denoted by $\lambda_{i}$. Hence, the
modified entanglement witness operator used by the pair Alice2-Bobi is given
by,

$\displaystyle
W^{(\lambda_{i})}=\frac{1}{4}\Big{[}\mathbb{I}\otimes\mathbb{I}+\lambda_{i}\Big{(}\sigma_{z}\otimes\sigma_{z}-\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}\Big{)}\Big{]}.$
(40)

This can be obtained from the entanglement witness operator given by Eq.(1)
and following the analysis mentioned in Sec. III.1. It can be shown that for
any separable state $\rho_{s}\in\mathcal{S}$, we have
$\text{Tr}\Big{(}W^{(\lambda_{j})}\rho_{s}\Big{)}\geq 0$.

Alice2 and Bob2 can witness entanglement if

$\text{Tr}\Big{[}W^{(\lambda_{2})}\,\rho_{A_{2}B_{2}}\Big{]}<0,$ (41)

which implies the following condition,

$\lambda_{2}>\frac{3}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}}.$
(42)

The average state shared between Alice2 and Bob3 is given by,

$\displaystyle\rho_{A_{2}B_{3}}=\frac{1}{3}\sum_{m_{2},b_{2}}\Big{(}\mathbb{I}\otimes\sqrt{E^{\lambda_{2}}_{b_{2}|\hat{m}_{2}}}\Big{)}\,\rho_{A_{2}B_{2}}\,\Big{(}\mathbb{I}\otimes\sqrt{E^{\lambda_{2}}_{b_{2}|\hat{m}_{2}}}\Big{)},$
(43)

with $\hat{m}_{2}\in\\{\hat{x},\hat{y},\hat{z}\\}$ and $b_{2}\in\\{+1,-1\\}$.
Here also, we use the assumption that all possible measurement settings of
Bob2 are equally probable.

Similarly, Alice2 and Bob3 can witness entanglement if the following condition
is satisfied,

$\text{Tr}\Big{[}W^{(\lambda_{3})}\,\rho_{A_{2}B_{3}}\Big{]}<0.$ (44)

The above condition implies the following,

$\lambda_{3}>\frac{9}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}}.$
(45)

Repeating the above calculations for other subsequent Bobs (i.e., Bob4, Bob5,
$\cdots$, Bobm), we can derive the conditions on $\lambda_{4}$, $\lambda_{5}$,
$\cdots$.

Next, we determine what is the maximum number of Bobs succeed in witnessing
the entanglement of the shared state with Alice2. Combining the conditions
(13), (42), (45), other similar conditions on $\lambda_{4}$, $\lambda_{5}$,
$\cdots$, and performing analytical calculations as described in Appendix A,
we get that Bob2, Bob3, $\cdots$, Bob8 can detect entanglement with Alice2, if
the following conditions are satisfied simultaneously,

$\displaystyle\xi_{1}\,=\lambda_{1}=0.58+\tilde{\delta_{1}}\,\,\text{with}\,\,0\leq\tilde{\delta_{1}}<<1,$
$\displaystyle\lambda_{2}=0.44+\tilde{\delta_{2}}\,\,\text{with}\,\,0\leq\tilde{\delta_{2}}<<1,$
$\displaystyle\lambda_{3}=0.47+\tilde{\delta_{3}}\,\,\text{with}\,\,0\leq\tilde{\delta_{3}}<<1,$
$\displaystyle\lambda_{4}=0.51+\tilde{\delta_{4}}\,\,\text{with}\,\,0\leq\tilde{\delta_{4}}<<1,$
$\displaystyle\lambda_{5}=0.56+\tilde{\delta_{5}}\,\,\text{with}\,\,0\leq\tilde{\delta_{5}}<<1,$
$\displaystyle\lambda_{6}=0.63+\tilde{\delta_{6}}\,\,\text{with}\,\,0\leq\tilde{\delta_{6}}<<1,$
$\displaystyle\lambda_{7}=0.74+\tilde{\delta_{7}}\,\,\text{with}\,\,0\leq\tilde{\delta_{7}}<<1,$
$\displaystyle\lambda_{8}=0.95+\tilde{\delta_{8}}\,\,\text{with}\,\,0\leq\tilde{\delta_{8}}<<1.$
(46)

Here the numerical values appearing in the above conditions are rounded to two
decimal places.

If Alice1, Alice2, Bob1, Bob2, Bob3, $\cdots$, Bob8 perform particular
measurements with sharpness parameters satisfying the conditions mentioned in
(46), then it can be shown that Bob9 cannot witness entanglement with Alice2
even if Bob9 performs projective measurements, i.e., with $\lambda_{9}=1$.
Hence, at most eight sequential Bobs can detect entanglement in this scenario
with two Alices.

Next, let us consider the scenario involving three Alices (Alice1, Alice2 and
Alice3) and multiple Bobs (Bob1, Bob2, Bob3, $\cdots$, Bobm). At first, the
pair Alice1-Bob1 detects entanglement using the entanglement witness operator
(10). Then Alice1 and Bob1 pass their particles to Alice2 and Bob2,
respectively. The pair Alice2-Bob2 detects entanglement using the same
entanglement witness operator (10) and passes the respective particles to the
pair Alice3-Bob3 who performs measurements to detect entanglement. Now, Bob3
passes his particle to Bob4 so that the pair Alice3-Bob4 can detect
entanglement. Subsequently, Bob4 passes the particle to Bob5, and so on. In
this scenario, following the aforementioned calculations, it can be shown that
Bob3, Bob4 and Bob5 can detect entanglement with Alice3. No additional Bob can
detect entanglement with Alice3. Hence, at most five Bobs can detect
entanglement.

Finally, if we consider that there are four Alices (Alice1, Alice2, Alice3 and
Alice4) and multiple Bobs (Bob1, Bob2, Bob3, $\cdots$, Bobm), then the result
derived in Sec. IV.1 for the symmetric scenario tells us that it is not
possible for four pairs (i.e., Alice1-Bob1, Alice2-Bob2, Alice3-Bob3 and
Alice4-Bob4) to detect entanglement sequentially. Entanglement detection is
possible only up to the third pair.

Before concluding, it may be noted that a resource theoretic comparison of the
above asymmetric scenarios can be performed with the corresponding non-
sequential scenarios involving multiple copies of mixed or non-maximally pure
entangled initial states. As expected, similar to the case of the symmetric
scenario, here too it is possible to observe advantages of the sequential
scenario in terms of the entanglement consumed and robustness of measurement.

##  VI Concluding Discussions

To summarize, our analysis presented in this paper clearly shows a hitherto
unexplored resource theoretic advantage of recycling the single copy of a two-
qubit entangled state towards performing information processing tasks by
multiple observers. Specifically, we have analyzed in detail a scenario
involving multiple independent observers acting sequentially on each of the
two spatially separated wings that initially share the resource of a single
copy of a two-qubit entangled state. The number of observers on the two wings
may be equal or unequal, depending upn the type of network considered. We have
estimated the maximum number of observers that can detect entanglement
sequentially using only one pair of qubits. Our results demonstrate that one
can reduce the number of physical qubits needed in the context of performing
different entanglement-assisted quantum tasks multiple times in network
scenarios.

Furthermore, we have performed a quantitative analysis of the advantage of the
sequential measurement scenario involving multiple sequential observers on
both wings from the perspective of resource requirements. In particular, we
have shown that the above scenario can help in reducing the necessary
requirement of the total initial entanglement, as well as the resource cost of
the measurements necessary for detecting entanglement by multiple sequential
observers. These advantages demonstrate the benefits of recycling single-shot
entanglement in various entanglement-assisted quantum information processing
and communication tasks in practical contexts, in comparison with various
standard schemes employing either multiple copies of two-qubit entangled
states, or multipartite entangled states for performing such tasks.

The analysis of this paper may be extended to certain interesting directions.
The scheme of unsharp measurement that we have adotpted in this work employs
the same value of the sharpness parameter associated with measurement in any
direction by an individual observer in a given run of the experiment.
Considering different sharpness parameters for different measurement input
settings by an observer can lead to a significant increase in the number of
allowed observers prl_brown ; bntwo ; cabello . The resource theoretic
efficacy of such a scheme would then be worthwhile to study. A fertile
direction of study could also be to generalize the scheme formulated in the
present paper towards studying resource theoretic efficacy of sequential
sharing of single-shot entanglement in the context of multi-qubit and two-
qudit higher dimensional entangled states. Finally, categorizing various
information processing tasks involving sequential measurements rantt ; cctt ;
appln1tt ; ractt ; appln4tt ; appln5tt ; sroytt ; rsp_shounak ; rac_debarshi
in terms of their resource theoretic advantages is another potentially
attractive direction of future study.

## Acknowledgement

AKD, DH, ASM acknowledge support from the project no. DST/ICPS/QuEST/2018/98
of the Department of Science and Technology, Government of India. AKD
acknowledges Shashank Gupta for useful discussions. DD acknowledges Science
and Engineering Research Board (SERB), Government of India for financial
support through a National Post Doctoral Fellowship (File No.:
PDF/2020/001358). SM acknowledges the Ministry of Science and Technology in
Taiwan (Grant no. 110-2811-M-006 -501). DH acknowledges support from the NASI
Senior Scientist Fellowship.

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## Appendix A

The condition for Alice1 and Bob1 to witness entanglement is given by,

$\xi_{1}\,\lambda_{1}>\frac{1}{3}$ (47)

Similarly, the condition for Alice2 and Bob2 to witness entanglement is given
by,

$\xi_{2}\,\lambda_{2}>\frac{3}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}}.$
(48)

Now, in order to ensure maximum number of sequential observers witnessing
entanglement, Alice1 and Bob1 should choose the sharpness parameters of their
measurements in such a way that these can detect entanglement causing minimal
disturbance to the state. In any unsharp measurement of the form (3) on
qubits, the disturbance can be reduced by reducing the associated sharpness
parameter mathematics_mal ; sasmal_steering . Hence, Alice1 and Bob1 should
choose sharpness parameters satisfying the following relation,

$\xi_{1}\,\lambda_{1}=\frac{1}{3}+\epsilon_{1}\,\,\,\text{with}\,\,\,0<\epsilon_{1}<<1.$
(49)

Next, the sharpness parameters of the measurements by Alice2 and Bob2 also
should ensure detection of entanglement causing minimal disturbance to the
state. For this, we have to minimize the right hand side of (48) under the
constraint given by Eq.(49). In other words, we have to perform the following
optimization problem,

$\displaystyle\max_{\xi_{1},\lambda_{1}}f_{1}(\xi_{1},\lambda_{1})$
$\displaystyle\text{such that}\hskip
11.38092pt\xi_{1}\,\lambda_{1}=\frac{1}{3}+\epsilon_{1}\hskip
11.38092pt\text{and}\hskip 11.38092pt0<\xi_{1},\lambda_{1}\leq 1$
$\displaystyle\text{where}\,\,f_{1}(\xi_{1},\lambda_{1})=\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}.$
(50)

Now, taking $\epsilon_{1}=10^{-2}$, we obtain $\xi_{1}=\lambda_{1}=0.58$.
Hence, the condition (48) becomes

$\xi_{2}\,\lambda_{2}>0.43.$ (51)

(Note that all numerical values appearing in this appendix are rounded to two
decimal places.)

Next, we find out the conditions under which Alice1-Bob1, Alice2-Bob2 and
Alice3-Bob3 can witness entanglement in such a way that the measurement by
each observer causes minimum possible disturbance to the state. Alice3-Bob3
can detect entanglement if

$\xi_{3}\,\lambda_{3}>\frac{27}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}}.$
(52)

In order to ensure minimum possible disturbance by Alice3-Bob3 while winessing
entanglement, we minimize the right hand side of (52) performing the following
optimization problem,

$\displaystyle\max_{\xi_{2},\lambda_{2}}f_{2}(\xi_{2},\lambda_{2})$
$\displaystyle\text{such that}\hskip
11.38092pt\xi_{2}\,\lambda_{2}=0.43+\epsilon_{2},\,\,\text{with}\,\,\,0<\epsilon_{2}<<1,\hskip
11.38092pt0<\xi_{2},\lambda_{2}\leq 1$
$\displaystyle\text{where}\,\,f_{2}(\xi_{2},\lambda_{2})=\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}\hskip
11.38092pt\text{with}\hskip 11.38092pt\xi_{1}=\lambda_{1}=0.58.$ (53)

Taking $\epsilon_{2}=10^{-2}$, we obtain $\xi_{2}=\lambda_{2}=0.66$. With
these values, the condition (52) becomes

$\xi_{3}\,\lambda_{3}>0.62.$ (54)

Proceeding in a similar way, we check the conditions on the parameters under
which Alice1-Bob1, Alice2-Bob2, Alice3-Bob3 and Alice4-Bob4 can witness
entanglement, given by

$\xi_{4}\,\lambda_{4}>\frac{243}{\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{3}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{3}^{2}}\Big{)}}.$
(55)

Here the corresponding optimization problem is,

$\displaystyle\max_{\xi_{3},\lambda_{3}}f_{3}(\xi_{3},\lambda_{3})$
$\displaystyle\text{such that}\hskip
11.38092pt\xi_{3}\,\lambda_{3}=0.62+\epsilon_{3},\,\,\text{with}\,\,\,0<\epsilon_{3}<<1,\hskip
11.38092pt0<\xi_{3},\lambda_{3}\leq 1$
$\displaystyle\text{where}\,\,f_{3}(\xi_{3},\lambda_{3})=\Big{(}1+2\sqrt{1-\xi_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{1}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{2}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\xi_{3}^{2}}\Big{)}\Big{(}1+2\sqrt{1-\lambda_{3}^{2}}\Big{)}$
$\displaystyle\hskip 270.30118pt\text{with}\hskip
11.38092pt\xi_{1}=\lambda_{1}=0.58,\,\,\xi_{2}=\lambda_{2}=0.66.$ (56)

Taking $\epsilon_{3}=10^{-2}$, we get $\xi_{3}=\lambda_{3}=0.79$. However,
with these values, Eq.(55) becomes

$\xi_{4}\,\lambda_{4}>1.13.$ (57)

Since $\xi_{4},\lambda_{4}\in(0,1]$, the above condition cannot be satisfied.
Therefore, at most three pairs (Alice1-Bob1, Alice2-Bob2, Alice3-Bob3) can
detect entanglement.