# Ability of unbounded pairs of observers to achieve quantum advantage in
random access codes with a single pair of qubits

Debarshi Das<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Arkaprabha
Ghosal<EMAIL_ADDRESS>Centre for Astroparticle Physics and Space
Science (CAPSS), Bose Institute, Block EN, Sector V, Salt Lake, Kolkata 700
091, India  Ananda G. Maity<EMAIL_ADDRESS>S. N. Bose National
Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106,
India  Som Kanjilal<EMAIL_ADDRESS>Harish-Chandra Research
Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019, India  Arup
Roy<EMAIL_ADDRESS>Department of Physics, A B N Seal College, Cooch
Behar, West Bengal 736 101, India

###### Abstract

Complications in preparing and preserving quantum correlations stimulate
recycling of a single quantum resource in information processing and
communication tasks multiple times. Here, we consider a scenario involving
multiple independent pairs of observers acting with unbiased inputs on a
single pair of spatially separated qubits sequentially. In this scenario, we
address whether more than one pair of observers can demonstrate quantum
advantage in some specific $2\rightarrow 1$ and $3\rightarrow 1$ random access
codes. Interestingly, we not only address these in the affirmative, but also
illustrate that unbounded pairs can exhibit quantum advantage. Furthermore,
these results remain valid even when all observers perform suitable projective
measurements and an appropriate separable state is initially shared.

##  I Introduction

Random access code (RAC) [1, 2, 3] is one of the fundamental communication
protocols which, when assisted with quantum resources, manifests the
astonishing potential of quantum systems in the context of information
processing. In a $n\rightarrow m$ RAC, $n$ is the number of bits ($x_{0}$,
$x_{1}$, $\cdots$, $x_{n-1}$) accessed by the sender, say, Alice. On the other
hand, $m$ is the number of bits that Alice is allowed to send the receiver,
say, Bob with $m<n$. In each run, Bob chooses the number $y$ randomly (where
$y\in\\{0,1,\cdots,n-1\\}$) and tries to guess the bit $x_{y}$ accessed by
Alice, but unknown to Bob. The efficacy of RAC is limited when only classical
strategies are employed. However, one can surpass the best classical
strategies using quantum resources, e.g., by using either quantum
communication [3] or classical bit communications assisted with a shared
bipartite quantum state [4, 5].

RAC assisted with quantum resources was initially introduced [1, 2, 3] in
order to demonstrate the immense capabilities of quantum systems in
information processing tasks. The state of a $m$-qubit system can be
represented by a unit vector in a $2^{m}$ dimensional complex Hilbert space,
which opens up the possibility of encoding and transmitting classical
information with exponentially fewer qubits, for example, Alice encoding $n$
bits into a $m$-qubit system (where $n>>m$) and sending it to Bob. However,
due to the Holevo bound [6], $m$ qubits cannot transmit more than $m$
classical bits of information faithfully. Hence, it can be inferred that
exponentially many degrees of freedom of a quantum system remain inaccessible.
Nevertheless, the situation becomes interesting when Bob does not need to know
all the $n$ bits of information together and chooses which bit of classical
information he would like to extract out of the encoding. In order to extract
different bits of information, Bob performs different measurements and these
measurements are in general not commuting. Thus, by choosing a particular
measurement, Bob inevitably disturbs the state and destroys some or all the
information that would have been revealed by other possible measurements. This
leads to the idea of RAC assisted with quantum resources. RAC has served as a
powerful quantum communication task with various applications ranging from
quantum finite automata [2, 3, 7], communication complexity [8, 9, 10, 11,
12], non-local games [13], network coding [14, 15], locally decodable codes
[16, 17, 18], dimension witnessing [19, 20, 21, 22], quantum state learning
[23], self-testing [24, 25, 26, 27], quantum randomness certification [28],
quantum key distribution [29], studies of no-signaling resources [30] to
characterising quantum mechanics from information-theoretic principles [31].
Experimental demonstrations of RAC protocols have also been reported [32, 33].

In the present study, we consider RAC using classical communications assisted
with shared quantum correlations. In reality, it is experimentally difficult
to create any quantum correlation. Moreover, environmental interactions
unavoidably degrade the efficacy of any quantum correlation. To cope with
these, one can recycle a single copy of any quantum resource multiple times.
Furthermore, this also indicates how much quantumness in a correlation is
preserved even after few cycles of local operations. Historically, this issue
was first addressed by Silva et al. [34], where two spatially separated
spin-$\frac{1}{2}$ particles were assumed to be shared between a single Alice
and multiple independent Bobs. In this scenario, the maximum number of Bobs
was deduced [34, 35, 36, 37, 38, 39] that can demonstrate Bell nonlocality
[40]. This idea of sharing quantum correlations by multiple sequential
observers has been extended in different contexts as well [41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]. The applications of sequential
sharing of quantum correlations in different information processing tasks have
also been demonstrated [27, 57, 58, 59, 60, 61, 62, 63]. In all these studies,
multiple observers performing sequential measurements on only one qubit have
been considered, whereas the present study contemplates multiple observers
performing sequential measurements on each of the two qubits. This is a more
general and practical scenario for re-utilizing quantum correlations in
commercial quantum technologies.

In particular, we focus on recycling a single quantum resource in sequentially
carrying out RAC tasks multiple times. Here, we consider the scenario where a
two-qubit state is shared between two spatially separated wings. Multiple
independent Alices (say, Alice1, Alice2, Alice3, $\cdots$) and multiple
independent Bobs (say, Bob1, Bob2, Bob3, $\cdots$) act sequentially on the
first and second qubit respectively with unbiased inputs. At first,
Alice1-Bob1 executes the RAC task with the initially shared two-qubit state.
Afterwards, Alice1 passes her qubit to Alice2 and Bob1 passes his qubit to
Bob2. Next, Alice2-Bob2 also passes the two qubits to Alice3-Bob3 after
performing the RAC task and so on.

In the above scenario, we show that unbounded pairs of Alice-Bob (i.e.,
Alice1-Bob1, Alice2-Bob2, $\cdots$) can gain quantum advantage in executing
RAC tasks. Specifically, we demonstrate that the above result holds 1) when
all pairs always perform some particular $2\rightarrow 1$ RAC, 2) when all
pairs always perform some particular $3\rightarrow 1$ RAC task, 3) when each
of the pairs always performs either a $2\rightarrow 1$ RAC or a $3\rightarrow
1$ RAC independent of other pairs, 4) when each pair performs a $2\rightarrow
1$ RAC and a $3\rightarrow 1$ RAC with different probabilities independent of
other pairs. While comparing the classical and quantum strategies to
demonstrate quantum advantage, we restrict the amount of shared classical bits
to be equal to the amount of shared quantum bits. This constraint is quite
natural in the sense that classical bits, similar to qubits, are expensive
resources [5, 64, 65, 66]. Since, the aforementioned scenario involves two
qubits, quantum strategies are compared with the classical ones assisted with
two bits from a common source.

The rest of the paper is arranged as follows. In Section II we review the
$2\rightarrow 1$ and $3\rightarrow 1$ RAC protocols assisted with classical
communication and a two-qubit state. The scenario considered by us and the
main results are presented in Section III. Finally, we conclude with a short
discussion in Section IV.

##  II $2\rightarrow 1$ and $3\rightarrow 1$ RAC protocols assisted with
classical communication and a two-qubit state

Let us now describe the $n\rightarrow 1$ (with $n\in\\{2,3\\}$) RAC protocol
using limited classical communication and shared two-qubit state. At first,
Alice is given a string of $n$ bits $x=(x_{0},x_{1},\cdots,x_{n-1})$ chosen
randomly from a uniform distribution with $x_{i}\in\\{0,1\\}$ for all
$i\in\\{0,1,\cdots,n-1\\}$. Next, depending on the input bit string, Alice
performs one of the $2^{n}$ dichotomic measurements denoted by
$A_{x_{0}x_{1}\cdots x_{n-1}}$ on her qubit. The outcome of the measurement
$A_{x_{0}x_{1}\cdots x_{n-1}}$ is denoted by $a_{x_{0}x_{1}\cdots
x_{n-1}}\in\\{0,1\\}$. Alice then communicates the outcome of her measurement
to Bob with one bit of information. Next, Bob tries to guess one of the $n$
bits $x_{y}$ (with $y\in\\{0,1,\cdots,n-1\\}$) given to Alice (in each run $y$
is chosen randomly). For this purpose, Bob performs one of the $n$ dichotomic
measurements denoted by $B_{y}$ on his qubit. The outcome of the measurement
$B_{y}$ is denoted by $b_{y}\in\\{0,1\\}$. Finally, Bob’s guess is given by
$a_{x_{0}x_{1}\cdots x_{n-1}}\oplus b_{y}$. Hence, the RAC task will be
successful, i.e., Bob’s guess will be correct if and only if
$a_{x_{0}x_{1}\cdots x_{n-1}}\oplus b_{y}=x_{y}$.

In the present study, we will quantify the efficacy of the RAC protocol by
minimum success probability defined as,

$\displaystyle P_{\texttt{Min}}^{n\rightarrow 1}$
$\displaystyle=\min\limits_{x_{0},x_{1},\cdots,x_{n-1},y}\,\,P(a_{x_{0}x_{1}\cdots
x_{n-1}}\oplus b_{y}=x_{y}).$ (1)

##  III Results

We consider a scenario involving multiple independent Alices and multiple
independent Bobs as described in Fig. 1. Alice1-Bob1 initially shares one pair
of qubit in the singlet state,
$|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$. This first pair
performs the aforementioned RAC task and then, Alice1, Bob1 pass their
particles to Alice2, Bob2 respectively. Alice2, Bob2 also pass their particles
to Alice3, Bob3 respectively after executing the RAC. In this way, the process
continues. Note that each of the observers act with unbiased inputs. Here we
want to find out how many pairs of Alice and Bob can exhibit quantum
advantage. If any pair performs projective measurements, it will disturb the
state maximally and the next pair may not get any quantum advantage. Hence, in
order to continue the above sequential RAC task with multiple pairs of Alice-
Bob, we consider weak measurements by all pairs. We should choose the weak
measurement formalism in such a way that the disturbance due to this
measurement is minimized for any given amount of information gain [34]. One
such example is unsharp measurement (a particular class of Positive Operator-
Valued Measure or POVM) [67] with generalized von Neumann-Lüders state-
transformation rule [35, 42].

Figure 1: Scenario for performing $n\rightarrow 1$ RAC task with multiple
pairs of observers sequentially.

In the present paper, we consider two particular RAC tasks. The first one is
the $2\rightarrow 1$ RAC task assisted with two (quantum or classical) bits,
shared from a common source and having maximally mixed marginal at the
receiver’s end. In the classical strategy, a source produces two correlated
bits which are shared by Alice and Bob. The two binary values $0$ and $1$ of
Bob’s bit are equiprobable. Consequently, Alice’s encoding and Bob’s decoding
strategies are now assisted with these bits. The minimum success probability
of such a classical RAC task is always less than or equal to $\frac{1}{2}$
[5]. In case of quantum strategy, two-qubit states with maximally mixed
marginal at Bob’s end can only be shared in the context of this task and
$P^{2\rightarrow 1}_{\texttt{Min}}>\frac{1}{2}$ implies quantum advantage.

Another RAC task that we consider is the $3\rightarrow 1$ RAC task assisted
with two (quantum or classical) bits shared from a common source. There is no
restriction on the marginals of the shared bits in this case. For classical
strategies, the minimum success probability is always less than or equal to
$\frac{1}{2}$ [5]. Hence, $P^{3\rightarrow 1}_{\texttt{Min}}>\frac{1}{2}$
ensures quantum advantage.

Suppose the pair Alicek-Bobk for arbitrary $k\in\\{1,2,\cdots\\}$ performs the
above RAC using the shared Bell-diagonal state,

$\rho^{k}_{AB}=\frac{1}{4}\left(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k}_{ii}\,\sigma_{i}\otimes\sigma_{i}\right),$
(2)

where $(t^{k}_{uu})^{2}\geq(t^{k}_{vv})^{2}\geq(t^{k}_{ww})^{2}$ for an
arbitrary choice of $u\neq v\neq w\in\\{1,2,3\\}$; $\sigma_{i}$ with $i=1,2,3$
are the three Pauli matrices.

Next, let us present the encoding-decoding strategies adopted by the pair
Alicek-Bobk. In case of the $2\rightarrow 1$ RAC, Alicek performs one of the
four POVMs denoted by
$A^{k}_{x_{0}x_{1}}\equiv\\{E^{k,0}_{x_{0}x_{1}},E^{k,1}_{x_{0}x_{1}}\\}$ with
$(x_{0},x_{1})\in\\{(00),(01),(10),(11)\\}$, where

$E^{k,\,a^{k}_{x_{0}x_{1}}}_{x_{0}x_{1}}=\frac{1}{2}\left[\mathbf{I}_{2}+\lambda^{k}\,(-1)^{a^{k}_{x_{0}x_{1}}}\,\,\left(\hat{u}^{k}_{x_{0}x_{1}}\cdot\vec{\sigma}\right)\right].$
(3)

Bobk performs one of the two POVMs denoted by
$B^{k}_{y}\equiv\\{E^{k,0}_{y},E^{k,1}_{y}\\}$ with $y\in\\{0,1\\}$, where

$E^{k,\,b^{k}_{y}}_{y}=\frac{1}{2}\left[\mathbf{I}_{2}+\eta^{k}\,(-1)^{b^{k}_{y}}\,\left(\hat{v}^{k}_{y}\cdot\vec{\sigma}\right)\right].$
(4)

Here $\lambda^{k}$, $\eta^{k}$ $\in$ $(0,1]$ are the sharpness parameters;
$\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$;
$a^{k}_{x_{0}x_{1}},\,b^{k}_{y}\in\\{0,1\\}$ denote the outcomes of the POVMs
$A^{k}_{x_{0}x_{1}}$ performed by Alicek and $B^{k}_{y}$ performed by Bobk
respectively. The unit vectors $\hat{u}^{k}_{x_{0}x_{1}}$ and
$\hat{v}^{k}_{y}$ are given by,

$\displaystyle\hat{u}^{k}_{x_{0}x_{1}}=\left(\dfrac{(-1)^{x_{0}}t^{k}_{11}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}}},\dfrac{(-1)^{x_{1}}t^{k}_{22}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}}},0\right),$
(5)
$\displaystyle\hat{v}^{k}_{0}=\left(1,0,0\right),\quad\quad\quad\quad\quad\quad\quad\quad\quad\hat{v}^{k}_{1}=\left(0,1,0\right).$
(6)

On the other hand, for executing the aforementioned $3\rightarrow 1$ RAC,
Alicek performs one of the eight possible POVMs denoted by
$A^{k}_{x_{0}x_{1}x_{2}}\equiv\\{E^{k,0}_{x_{0}x_{1}x_{2}},E^{k,1}_{x_{0}x_{1}x_{2}}\\}$
with $x_{i}\in\\{0,1\\}$ for all $i\in\\{0,1,2\\}$, where

$E^{k,\,a^{k}_{x_{0}x_{1}x_{2}}}_{x_{0}x_{1}x_{2}}=\frac{1}{2}\left[\mathbf{I}_{2}+\lambda^{k}\,(-1)^{a^{k}_{x_{0}x_{1}x_{2}}}\,\,\left(\hat{u}^{k}_{x_{0}x_{1}x_{2}}\cdot\vec{\sigma}\right)\right].$
(7)

Bobk performs one of the three POVMs denoted by
$B^{k}_{y}\equiv\\{E^{k,0}_{y},E^{k,1}_{y}\\}$ with $y\in\\{0,1,2\\}$, where

$E^{k,\,b^{k}_{y}}_{y}=\frac{1}{2}\left[\mathbf{I}_{2}+\eta^{k}\,(-1)^{b^{k}_{y}}\,\left(\hat{v}^{k}_{y}\cdot\vec{\sigma}\right)\right].$
(8)

We choose the unit vectors
$\hat{u}^{k}_{x_{0}x_{1}x_{2}}=\dfrac{\vec{u}^{k}_{x_{0}x_{1}x_{2}}}{|\vec{u}^{k}_{x_{0}x_{1}x_{2}}|}$
and $\hat{v}^{k}_{y}$ as follows,

$\displaystyle\vec{u}^{k}_{x_{0}x_{1}x_{2}}=\Big{(}(-1)^{x_{0}}t^{k}_{11},\,(-1)^{x_{1}}t^{k}_{22},\,(-1)^{x_{2}}t^{k}_{33}\Big{)},$
(9)
$\displaystyle\hat{v}^{k}_{0}=\left(1,0,0\right),\quad\hat{v}^{k}_{1}=\left(0,1,0\right),\quad\hat{v}^{k}_{2}=\left(0,0,1\right).$
(10)

With these, we can present the following lemma (for proof, see Appendix A),
which will be useful for probing the main result,

######  Lemma 1.

Let Alicek-Bobk performs the $n\rightarrow 1$ RAC task (where $n=2$ or $n=3$)
with a two-qubit Bell-diagonal state (2) using the above unsharp measurements.
Then the pair achieves minimum success probability strictly greater than
$\frac{1}{2}$ if $\min\limits_{i\leq n}\left[(t^{k}_{ii})^{2}\right]\neq 0$.

Next, we want to find out the post-measurement state $\rho^{k+1}_{AB}$
received, on average, by Alicek+1-Bobk+1 from Alicek-Bobk. When Alicek-Bobk
performs the $2\rightarrow 1$ RAC, following the generalized von Neumann-
Lüder’s transformation rule, we have (see Appendix B)

$\displaystyle\rho^{k+1}_{AB}$
$\displaystyle=\frac{1}{8}\sum_{x_{0},x_{1},y=0}^{1}\Bigg{[}\sum_{a^{k}_{x_{0}x_{1}},b^{k}_{y}=0}^{1}\Bigg{(}\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}}}_{x_{0}x_{1}}}\otimes\sqrt{E^{k,\,b^{k}_{y}}_{y}}\Bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\rho_{AB}^{k}\Bigg{(}\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}}}_{x_{0}x_{1}}}\otimes\sqrt{E^{k,\,b^{k}_{y}}_{y}}\Bigg{)}^{\dagger}\Bigg{]}$
$\displaystyle=\frac{1}{4}\left(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k+1}_{ii}\,\sigma_{i}\otimes\sigma_{i}\right).$
(11)

The average is taken since we have assumed that multiple Alices or multiple
Bobs act independently of each other. Here, we have also used the assumption
that Alicek and Bobk perform measurements with unbiased inputs. Similarly,
when Alicek-Bobk performs the $3\rightarrow 1$ RAC, it is observed that the
average post-measurement state $\rho^{k+1}_{AB}$ received by Alicek+1-Bobk+1
has the Bell-diagonal form (11) (see Appendix C for details).

Moreover, when Alicek-Bobk performs the $n\rightarrow 1$ RAC task (where $n=2$
or $n=3$) with the state (2), it can be shown that $\min\limits_{i\leq
n}\left[(t^{k+1}_{ii})^{2}\right]\neq 0$ if $\min\limits_{i\leq
n}\left[(t^{k}_{ii})^{2}\right]\neq 0$ (For details, see Appendix B and
Appendix C).

Now, consider that the same $n\rightarrow 1$ RAC (i.e., either the
$2\rightarrow 1$ or the $3\rightarrow 1$ RAC) is performed by each of the
pairs. In such scenario, combining the above results, we can present the
following: if Alice1-Bob1 initially shares the singlet state, then this pair
achieves $P^{n\rightarrow 1}_{\texttt{Min}}>\frac{1}{2}$ (with $n=2$ or $n=3$)
using the aforementioned unsharp measurements. Moreover, the average post-
measurement state $\rho^{2}_{AB}$ received by Alice2-Bob2 is the Bell-diagonal
state (2) with $k=2$ and $\min\limits_{i\leq
n}\left[(t^{2}_{ii})^{2}\right]\neq 0$. Hence, Alice2-Bob2 also achieves
$P^{n\rightarrow 1}_{\texttt{Min}}>\frac{1}{2}$. Subsequently, Alice3-Bob3
receives the Bell-diagonal state (2) with $k=3$ and $\min\limits_{i\leq
n}\left[(t^{3}_{ii})^{2}\right]\neq 0$ and exhibits $P^{n\rightarrow
1}_{\texttt{Min}}>\frac{1}{2}$ as well. This process continues for arbitrarily
many pairs. Therefore, we can present the following theorem,

######  Theorem 1.

Unbounded pairs of Alice and Bob can demonstrate quantum advantage either in
$2\rightarrow 1$ RAC task assisted with two bits shared from a common source
and having maximally mixed marginal at the receiver’s end, or in $3\rightarrow
1$ RAC task assisted with two correlated bits.

Importantly, the statements of Theorem 1 hold for all values of $\lambda^{k}$
$\in$ $(0,1]$ and $\eta^{k}$ $\in$ $(0,1]$ for all possible $k$ $\in$
$\\{1,2,\cdots\\}$. Moreover, for the aforementioned $n\rightarrow 1$ RAC with
$n=2$ or $n=3$, starting with any Bell-diagonal two-qubit (entangled or
separable) state given by Eq.(2) with $k=1$ and $\min\limits_{i\leq
n}\left[(t^{1}_{ii})^{2}\right]\neq 0$, one gets the same result as stated in
Theorem 1. Hence, the following corollary can be stated,

######  Corollary 1.

Unbounded pairs of Alice and Bob can exhibit quantum advantage in some
particular $n\rightarrow 1$ RAC task (with $n=2$ or $n=3$) even when each of
the observers performs suitable projective measurements and the initially
shared two-qubit state belongs to a particular subset of separable states.

When Alice1-Bob1 initially shares the singlet state and performs the
aforementioned $n\rightarrow 1$ RAC (where $n=2$ or $n=3$) using the
measurements described earlier with $\lambda^{1}=\eta^{1}=1$ (i.e., projective
measurements), then this pair achieves $P^{n\rightarrow
1}_{\texttt{Min}}=\frac{1}{2}\left(1+\frac{1}{\sqrt{n}}\right)$. This is the
maximum permissible value of $P^{n\rightarrow 1}_{\texttt{Min}}$ with quantum
resources [4]. In this case also, the residual quantum correlation in the
average post-measurement state is sufficient for demonstrating quantum
advantage in the $n\rightarrow 1$ RAC by unbounded pairs of Alice and Bob.
Hence, a single pair of qubits can be utilized indefinitely to gain quantum
advantage in some particular RAC even when the optimal quantum advantage is
exhibited in the first round.

Remark: We observe that when an arbitrary pair gains a large amount of quantum
advantage, then only few numbers of subsequent pairs will get ‘significant’
quantum advantage. On the other hand, when a pair gets a small amount of
quantum advantage, a larger number of subsequent pairs can achieve
‘significant’ quantum advantage. Here, ‘significant’ quantum advantage implies
that $\left(P_{\texttt{Min}}^{n\rightarrow 1}-\frac{1}{2}\right)$ is positive
and large enough to be detected in a real experiment. Hence, there may exist a
trade-off relation between the amount of quantum advantage gained by an
arbitrary pair and the number of subsequent pairs exhibiting considerable
amount of quantum advantage. Moreover, either of these two quantities can be
increased at the expense of the other by suitably choosing the sharpness
parameters of the measurements (See Appendix D). In practical scenario, a
large but finite number of sequential pairs of observers may be required to
perform some communication tasks with only one pair of qubits. The number of
sequential pairs required to exhibit quantum advantage depends on the
particular context under consideration and that can be realized by fine-tuning
the unsharpness of the measurements.

Next, we consider a more general scenario where an arbitrary pair Alicek-Bobk
performs the aforementioned $2\rightarrow 1$ RAC task with probability $p_{k}$
and the aforementioned $3\rightarrow 1$ RAC with probability $(1-p_{k})$,
where $0\leq p_{k}\leq 1$. For example, Alicek and Bobk can fix the task to be
performed in each experimental run prior to the initiation of sequential RAC
and, during the execution of sequential RAC, they perform the two different
tasks accordingly. This type of scenario is particularly relevant when a
sequence of RAC tasks is implemented as an intermediate step in commercial
quantum computation. In such cases, different tasks may be required to be
performed by the same pair of particles in different steps depending on the
choices of users. In this scenario, if a singlet state or any Bell-diagonal
two-qubit (entangled or separable) state given by Eq.(2) with $k=1$ and
$\min\left[(t^{1}_{11})^{2},(t^{1}_{22})^{2},(t^{1}_{33})^{2}\right]\neq 0$ is
initially shared, then the following result is attained (see Appendix E for
details),

######  Corollary 2.

Unbounded pairs of Alice and Bob can demonstrate quantum advantage when an
arbitrary pair Alicek-Bobk performs a $2\rightarrow 1$ RAC (assisted with two
correlated bits with maximally mixed marginal at the receiver’s end) with
probability $p_{k}$ and a $3\rightarrow 1$ RAC (assisted with two bits shared
from a common source) with probability $1-p_{k}$ independent of other pairs.

When Alicek-Bobk performs projective measurements and $p_{k}=1$ (i.e.,
performs $2\rightarrow 1$ RAC with certainty), then the condition:
$\min\left[(t^{x}_{11})^{2},(t^{x}_{22})^{2},(t^{x}_{33})^{2}\right]\neq 0$
will not be satisfied for the average post-measurement state received by all
subsequent pairs (i.e., for all $x\in\\{k+1,k+2,\cdots\\}$). Hence, all these
pairs will not achieve quantum advantage in $3\rightarrow 1$ RAC. Hence, only
under unsharp measurements (with the sharpness parameters being strictly less
than $1$), we can state the following corollary (see Appendix E for details),

######  Corollary 3.

Unbounded pairs of Alice and Bob can demonstrate quantum advantage when an
arbitrary pair Alicek-Bobk performs a $2\rightarrow 1$ RAC with certainty and
another arbitrary pair Alice${}^{\tilde{k}}$-Bob${}^{\tilde{k}}$ performs a
$3\rightarrow 1$ RAC with certainty for all choices of
$k\neq\tilde{k}\in\\{1,2,\cdots\\}$.

##  IV Conclusions

Here we have considered a scenario involving multiple independent pairs of
Alice and Bob sharing a single pair of qubits and performing some particular
$2\rightarrow 1$ and $3\rightarrow 1$ RAC tasks with unbiased inputs
sequentially. In this scenario, we have shown that unbounded pairs can gain
quantum advantage even when all observers perform projective measurements.
These results address the issue of recycling a single copy of a quantum
resource in performing information processing tasks multiple times
sequentially. This is of utmost importance since, in reality, preparing
quantum correlations and preserving them against inevitable environmental
interactions are difficult.

Our results point out that quantum correlations present in separable states
[68] can be preserved indefinitely in spite of utilizing it in each step.
Furthermore, weak measurements are not necessary for this purpose; suitable
projective measurements can serve for this. Note that this is not the case for
entanglement or Bell-nonlocality. Hence, these results signify one fundamental
difference between the quantum correlations present in entanglement and that
present in separable states: the first one is destroyed only after one cycle
of projective measurements while the second one is retained even after
infinite cycles. The advantage of quantum information processing tasks
assisted with separable states [5, 66] is thus pointed out by our present
study. In fact, our results open up the possibility of implementing unbounded
sequence of any task, for which quantum advantage can be demonstrated even
using separable states (say, for example, remote state preparation [69]), with
only one pair of qubits.

There exists a complementarity between the question addressed here and the
one-way communication complexity problem [70, 71]. In one-way communication
complexity problem, Alice and Bob are given inputs
$x\in\left\\{0,1\right\\}^{n}$ and $y\in\left\\{0,1\right\\}^{m}$
respectively. The goal for Bob is to calculate a binary function $f(x,y)$.
Alice is allowed to send limited classical communications to Bob. This game
can be thought as a number of parallel RACs taking place simultaneously. The
main goal of any communication complexity problem is to minimize the amount of
classical communication. However, there is no restriction on the shared
entanglement. On the contrary, the present study is aimed to reduce the amount
of shared correlation, but does not focus on reducing the number of
communicating bits.

Recently, measurement protocols have been proposed to demonstrate arbitrary
many Bell-CHSH inequality [40] violations with various independent Bobs and a
single Alice using unbiased inputs when a pure entangled two-qubit state is
initially shared [39]. The result, however, requires arbitrarily high
precision engineering for the measurement apparatus and, hence, is too
strenuous to implement in a reality. On the other hand, the unsharp
measurements chosen in the present study can be realized in photonic systems
based on the techniques adopted in [62, 63]. Moreover, our results are valid
for any range of sharpness parameters and do not require any entanglement.
Hence, for experimental implementation of large sequence of detecting quantum
correlation with a single two-qubit state, our results are less demanding.

To the best of our knowledge, this study points out for the first time that
there exist some communication tasks in which unbounded pairs of observers can
exhibit quantum supremacy even if a single quantum resource is used. Finding
out different communication tasks with the above feature merits further
investigation. Next, it is worth to fully characterize the set of two-qubit
states for which Theorem 1 holds. It is also interesting to find out whether
there exists any two-qubit state for which weak measurements are necessary for
satisfying Theorem 1.

## Acknowledgements

DD acknowledges fruitful discussions with Somshubhro Bandyopadhyay, Manik
Banik and Debashis Saha. DD acknowledges Science and Engineering Research
Board (SERB), Government of India for financial support through National Post
Doctoral Fellowship (File No.: PDF/2020/001358). AG acknowledges Bose
Institute, Kolkata for financial support. SK thanks the Department of Science
and Technology (DST), Government of India for the financial assistance through
the QuEST project

## References

  * [1] S. Wiesner, ACM SIGACT News 15, 78 (1983).
  * [2] A. Ambainis, A. Nayak, A. Ta-Shma, and U. Vazirani, in Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC’99) (ACM, New York, 1999), pp. 376–383.
  * [3] A. Ambainis, A. Nayak, A. Ta-Shma, and U. Vazirani, J. ACM 49, 496 (2002).
  * [4] M. Pawlowski, and M. Zukowski, Phys. Rev. A 81, 042326 (2010).
  * [5] T. K. C. Bobby, and T. Paterek, New J. Phys. 16, 093063 (2014).
  * [6] A. Holevo, Problems Inform. Transmission 9, 177 (1973).
  * [7] A. Nayak, in 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039) (IEEE, New York, 1999), pp. 369–376.
  * [8] H. Klauck, in Proceedings of the IEEE Symposium on Foundations of Computer Science (IEEE Computer Society, NW Washington, DC, 2001), p. 288; in Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 2000), pp. 644–651.
  * [9] S. Aaronson, Theory of Computing 1, 1 (2005).
  * [10] D. Gavinsky, J. Kempe, O. Regev, and R. de Wolf, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC’06) (Association for Computing Machinery, New York, 2006), pp. 594–603.
  * [11] H. Buhrman, and R. de Wolf, in Proceedings 16th Annual IEEE Conference on Computational Complexity (IEEE Computer Society, Los Alamitos, California, 2001), pp. 120–130.
  * [12] D. Martínez, A. Tavakoli, M. Casanova, G. Cañas, B. Marques, and G. Lima, Phys. Rev. Lett. 121, 150504 (2018).
  * [13] S. Muhammad, A. Tavakoli, M. Kurant, M. Pawłowski, M. Żukowski, and M. Bourennane, Phys. Rev. X 4, 021047 (2014).
  * [14] M. Hayashi, K. Iwama, H. Nishimura, R. Raymond, and S. Yamashita, New J. Phys. 8, 129 (2006).
  * [15] M. Hayashi, K. Iwama, H. Nishimura, R. Raymond, and S. Yamashita, Proc. 24th Int. Symp. on Theoretical Aspects of Computer Science (STACS 2007), Lecture Notes in Computer Science 4393 pp 610-21 (2007).
  * [16] I. Kerenidis, and R. de Wolf, J. Comput. Syst. Sci., 69, 395–420, (2004).
  * [17] S. Wehner, and R. de Wolf, in Automata, Languages and Programming. ICALP 2005, Lecture Notes in Computer Science, Vol. 3580 (Springer, Berlin, Heidelberg, 2005), pp. 1424–1436.
  * [18] A. Ben-Aroya, O. Regev, and R. de Wolf, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS’08) (IEEE Computer Society, NWWashington, DC, United States, 2008), pp. 477–486.
  * [19] S. Wehner, M. Christandl, and A. C. Doherty, Phys. Rev. A 78, 062112 (2008).
  * [20] J. Ahrens, P.Badziąg, M. Pawłowski, M. Żukowski, and M. Bourennane Phys. Rev. Lett. 112, 140401 (2014).
  * [21] A. Tavakoli, A. Hameedi, B. Marques, and M. Bourennane, Phys. Rev. Lett. 114, 170502 (2015).
  * [22] M. Czechlewski, D. Saha, A. Tavakoli, and M. Pawłowski Phys. Rev. A 98, 062305 (2018).
  * [23] S. Aaronson, Proc. R. Soc. London Ser. A 463, 3089 (2007).
  * [24] A. Tavakoli, J. Kaniewski, T. Vertesi, D. Rosset, and N. Brunner, Phys. Rev. A 98, 062307 (2018).
  * [25] M. Farkas, and J. Kaniewski, Phys. Rev. A 99, 032316 (2019).
  * [26] A. Tavakoli, M. Smania, T. Vertesi, N. Brunner, and M. Bourennane, Science Advances 6, 16 (2020).
  * [27] K. Mohan, A. Tavakoli, and N. Brunner, New J. Phys. 21 083034 (2019).
  * [28] H.-W. Li, Z.-Q. Yin, Y.-C. Wu, X.-B. Zou, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, Phys. Rev. A 84, 034301 (2011).
  * [29] M. Pawlowski, and N. Brunner, Phys. Rev. A 84, 010302(R) (2011).
  * [30] A. Grudka, K. Horodecki, M. Horodecki, W. Kłobus, and M. Pawłowski, Phys. Rev. Lett. 113, 100401 (2014); A. Grudka, M. Horodecki, R. Horodecki, and A. Wójcik, Phys. Rev. A 92, 052312 (2015).
  * [31] M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Zukowski, Nature 461, 1101 (2009).
  * [32] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009).
  * [33] X.-R. Wang, L.-Y. Wu, C.-X. Liu, T.-J. Liu, J. Li, and Q. Wang, Phys. Rev. A 99, 052313 (2019).
  * [34] R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, Phys. Rev. Lett. 114, 250401 (2015).
  * [35] S. Mal, A. S. Majumdar, and D. Home, Mathematics 4 48 (2016).
  * [36] M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, npj Quantum Information 4, 63 (2018).
  * [37] M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, Quantum Sci. Technol. 2 015010 (2017).
  * [38] D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, Phys. Rev. A 99, 022305 (2019).
  * [39] P. J. Brown, and R. Colbeck, Phys. Rev. Lett. 125, 090401 (2020).
  * [40] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
  * [41] S. Roy, A. Kumari, S. Mal, and A. Sen De, arXiv:2012.12200 [quant-ph].
  * [42] S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, Phys. Rev. A 98, 012305 (2018).
  * [43] A. Shenoy H., S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, Phys. Rev. A 99, 022317 (2019).
  * [44] Y.-H. Choi, S. Hong, T. Pramanik, H.-T. Lim, Y.-S. Kim, H. Jung, S.-W. Han, S. Moon, and Y.-W. Cho, Optica 7, 675 (2020).
  * [45] A. Bera, S. Mal, A. Sen De, and U. Sen, Phys. Rev. A 98, 062304 (2018).
  * [46] G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, Phys. Rev. Applied 13, 044008 (2020).
  * [47] C. Srivastava, S. Mal, A. Sen De, and U. Sen, Phys. Rev. A 103, 032408 (2021).
  * [48] S. Datta, and A. S. Majumdar, Phys. Rev. A 98, 042311 (2018).
  * [49] C. Ren, T. Feng, D. Yao, H. Shi, J. Chen, and X. Zhou, Phys. Rev. A 100, 052121 (2019).
  * [50] T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, Phys. Rev. A 102, 032220 (2020).
  * [51] H. Anwer, N.Wilson, R. Silva, S. Muhammad, A. Tavakoli, and M. Bourennane, Quantum 5, 551 (2021).
  * [52] A. Kumari, and A. K. Pan, Phys. Rev. A 100, 062130 (2019).
  * [53] R. D. Baldijao, and M. Terra Cunha, Phys. Rev. A 102, 052226 (2020).
  * [54] S. Saha, D. Das, S. Sasmal, D. Sarkar, K. Mukherjee, A. Roy, and S. S. Bhattacharya, Quantum Inf Process 18, 42 (2019).
  * [55] A. G. Maity, D. Das, A. Ghosal, A. Roy, and A. S. Majumdar, Phys. Rev. A 101, 042340 (2020).
  * [56] S. Gupta, A. G. Maity, D. Das, A. Roy, and A. S. Majumdar, Phys. Rev. A 103, 022421 (2021)..
  * [57] F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acin, Phys. Rev. A 95, 020102(R) (2017).
  * [58] A. Tavakoli, and A. Cabello, Phys. Rev. A 97, 032131 (2018).
  * [59] H.-W. Li, Y.-S. Zhang, X.-B. An, Z.-F. Han, and G.-C. Guo, Commun. Phys. 1, 10 (2018).
  * [60] X.-B. An, H.-W. Li, Z.-Q. Yin, M.-J. Hu, W. Huang, B.-J. Xu, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, Opt. Lett. 43, 3437 (2018).
  * [61] S. Roy, A. Bera, S. Mal, A. Sen De, and U. Sen, Phys. Lett. A 392, 127143 (2021).
  * [62] H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Phys. Rev. Lett. 125, 080403 (2020).
  * [63] G. Foletto, L. Calderaro, G. Vallone, and P. Villoresi, Phys. Rev. Research 2, 033205 (2020).
  * [64] S. Lloyd, Phys. Rev. Lett. 88, 237901 (2002).
  * [65] D. Das, B. Bhattacharya, C. Datta, A. Roy, C. Jebaratnam, A. S. Majumdar, and R. Srikanth, Phys. Rev. A 97, 062335 (2018).
  * [66] C. Jebarathinam, D. Das, S. Kanjilal, R. Srikanth, D. Sarkar, I. Chattopadhyay, and A. S. Majumdar, Phys. Rev. A 100, 012344 (2019).
  * [67] P. Busch, M. Grabowski, and P. J. Lathi, _Operational Quantum Physics_ (Springer-Verlag, Berlin, 1997).
  * [68] H. Ollivier, and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001).
  * [69] B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, Nat. Phys. 8, 666 (2012).
  * [70] G. Brassard, Foundations of Physics 33, 1593-1616 (2003).
  * [71] R. Cleve, and H. Buhrman, Phys. Rev. A 56, 1201 (1997).

##  Appendix A Proof of Lemma 1

For the $2\rightarrow 1$ RAC: Let Alicek-Bobk shares the following Bell-
diagonal two-qubit state,

$\rho^{k}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(12)

with $(t^{k}_{uu})^{2}\geq(t^{k}_{vv})^{2}\geq(t^{k}_{ww})^{2}$ for an
arbitrary choice of $u\neq v\neq w\in\\{1,2,3\\}$.

Alicek and Bobk perform the $2\rightarrow 1$ RAC task contingent upon using
the unsharp measurements mentioned in Eqs.(3,4,5,6). Next, let us compute the
expression for a typical guessing probability $P(a^{k}_{x_{0}x_{1}}\oplus
b^{k}_{y}=x_{y})$. Using Born’s rule, one can write that

$\displaystyle P(a^{k}_{x_{0}x_{1}}\oplus b^{k}_{y}=x_{y})$
$\displaystyle=\sum_{z=0}^{1}P\left(a^{k}_{x_{0}x_{1}}=z,b^{k}_{y}=|x_{y}-z|\,\,\Big{|}\,\,A^{k}_{x_{0}x_{1}},B^{k}_{y}\right),$
(13)

where
$P\left(a^{k}_{x_{0}x_{1}}=z,b^{k}_{y}=|x_{y}-z|\,\,\Big{|}\,\,A^{k}_{x_{0}x_{1}},B^{k}_{y}\right)$
denotes the joint probability with which Alicek and Bobk get the outcomes $z$
and $|x_{y}-z|$ contingent upon performing the measurements
$A^{k}_{x_{0}x_{1}}$ and $B^{k}_{y}$ respectively. From Eq.(13), we have the
following,

$\displaystyle P(a^{k}_{x_{0}x_{1}}\oplus b^{k}_{y}=x_{y})$
$\displaystyle=\sum_{z=0}^{1}\text{Tr}\left[\rho^{k}_{AB}\left(E^{k,\,z}_{x_{0}x_{1}}\otimes
E^{k,\,|x_{y}-z|}_{y}\right)\right]$
$\displaystyle=\frac{1}{2}\left[1+(-1)^{x_{y}}\,\lambda^{k}\eta^{k}\,\left(t_{(y+1)\,(y+1)}\right)\,\left(\hat{u}_{x_{0}x_{1}}^{k}\cdot\hat{v}_{y}^{k}\right)\right],$
(14)

where $\hat{u}^{k}_{x_{0}x_{1}}$ and $\hat{v}^{k}_{y}$ are mentioned in
Eqs.(5,6). Now, a straightforward calculation leads to the following,

$P(a^{k}_{00}\oplus b^{k}_{0}=0)=P(a^{k}_{01}\oplus
b^{k}_{0}=0)=P(a^{k}_{10}\oplus b^{k}_{0}=1)=P(a^{k}_{11}\oplus
b^{k}_{0}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{11})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}}}\right],$
(15a) $P(a^{k}_{00}\oplus b^{k}_{1}=0)=P(a^{k}_{01}\oplus
b^{k}_{1}=1)=P(a^{k}_{10}\oplus b^{k}_{1}=0)=P(a^{k}_{11}\oplus
b^{k}_{1}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{22})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}}}\right].$
(15b)

Hence, the minimum success probability is given by,

$P_{\texttt{Min}}^{2\rightarrow
1}=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2}\right]}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}}}\right].$
(16)

Thus, for all values of $\lambda^{k}$ $\in$ $(0,1]$ and $\eta^{k}$ $\in$
$(0,1]$, $P_{\texttt{Min}}^{2\rightarrow 1}>\frac{1}{2}$ if
$\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2}\right]\neq 0$.

For the $3\rightarrow 1$ RAC: Let Alicek-Bobk performs the $3\rightarrow 1$
RAC task using the shared Bell-diagonal state given by Eq. (12). Alicek and
Bobk perform the unsharp measurements given by Eqs.(7,8,9,10) of the main
paper. The expression for a typical guessing probability
$P(a^{k}_{x_{0}x_{1}x_{2}}\oplus b^{k}_{y}=x_{y})$ can be calculated using
Born’s rule as follows

$\displaystyle P(a^{k}_{x_{0}x_{1}x_{2}}\oplus b^{k}_{y}=x_{y})$
$\displaystyle=\sum_{z=0}^{1}\text{Tr}\left[\rho^{k}_{AB}\left(E^{k,\,z}_{x_{0}x_{1}x_{2}}\otimes
E^{k,\,|x_{y}-z|}_{y}\right)\right]$
$\displaystyle=\frac{1}{2}\left[1+(-1)^{x_{y}}\,\lambda^{k}\eta^{k}\,\left(t_{(y+1)\,(y+1)}\right)\,\left(\hat{u}_{x_{0}x_{1}x_{2}}^{k}\cdot\hat{v}_{y}^{k}\right)\right],$
(17)

where $\hat{u}^{k}_{x_{0}x_{1}x_{2}}$ and $\hat{v}^{k}_{y}$ are mentioned
earlier in Eqs.(9,10) of the main paper. Hence, we have the following,

$\displaystyle P(a^{k}_{000}\oplus b^{k}_{0}=0)=P(a^{k}_{001}\oplus
b^{k}_{0}=0)=P(a^{k}_{010}\oplus b^{k}_{0}=0)=P(a^{k}_{011}\oplus
b^{k}_{0}=0)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{11})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right],$
$\displaystyle P(a^{k}_{100}\oplus b^{k}_{0}=1)=P(a^{k}_{101}\oplus
b^{k}_{0}=1)=P(a^{k}_{110}\oplus b^{k}_{0}=1)=P(a^{k}_{111}\oplus
b^{k}_{0}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{11})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right],$
$\displaystyle P(a^{k}_{000}\oplus b^{k}_{1}=0)=P(a^{k}_{001}\oplus
b^{k}_{1}=0)=P(a^{k}_{010}\oplus b^{k}_{1}=1)=P(a^{k}_{011}\oplus
b^{k}_{1}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{22})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right],$
$\displaystyle P(a^{k}_{100}\oplus b^{k}_{1}=0)=P(a^{k}_{101}\oplus
b^{k}_{1}=0)=P(a^{k}_{110}\oplus b^{k}_{1}=1)=P(a^{k}_{111}\oplus
b^{k}_{1}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{22})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right],$
$\displaystyle P(a^{k}_{000}\oplus b^{k}_{2}=0)=P(a^{k}_{001}\oplus
b^{k}_{2}=1)=P(a^{k}_{010}\oplus b^{k}_{2}=0)=P(a^{k}_{011}\oplus
b^{k}_{2}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{33})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right],$
$\displaystyle P(a^{k}_{100}\oplus b^{k}_{2}=0)=P(a^{k}_{101}\oplus
b^{k}_{2}=1)=P(a^{k}_{110}\oplus b^{k}_{2}=0)=P(a^{k}_{111}\oplus
b^{k}_{2}=1)=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{(t^{k}_{33})^{2}}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right].$
(18)

From the above equations, the minimum success probability is given by,

$P_{\texttt{Min}}^{3\rightarrow
1}=\frac{1}{2}\left[1+\lambda^{k}\eta^{k}\frac{\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2},(t^{k}_{33})^{2}\right]}{\sqrt{(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}}}\right].$
(19)

Therefore, for all values of $\lambda^{k}$ $\in$ $(0,1]$ and $\eta^{k}$ $\in$
$(0,1]$, $P_{\texttt{Min}}^{3\rightarrow 1}>\frac{1}{2}$ if
$\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2},(t^{k}_{33})^{2}\right]\neq 0$.

##  Appendix B Calculating $\rho^{k+1}_{AB}$ in case of $2\rightarrow 1$ RAC

Let Alicek-Bobk performs the $2\rightarrow 1$ RAC task with the following
Bell-diagonal two-qubit state,

$\rho^{k}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(20)

where $(t^{k}_{uu})^{2}\geq(t^{k}_{vv})^{2}\geq(t^{k}_{ww})^{2}$ for an
arbitrary choice of $u\neq v\neq w\in\\{1,2,3\\}$.

Alicek and Bobk perform the aforementioned unsharp measurements. The average
post-measurement state $\rho_{AB}^{k+1}$ received by Alicek+1-Bobk+1 from
Alicek-Bobk can be obtained using the generalized von Neumann-Lüder’s
transformation rule as follows,

$\displaystyle\rho^{k+1}_{AB}=\frac{1}{8}\sum_{x_{0},x_{1},y=0}^{1}$
$\displaystyle\Bigg{[}\sum_{a^{k}_{x_{0}x_{1}},b^{k}_{y}=0}^{1}\Bigg{(}\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}}}_{x_{0}x_{1}}}\otimes\sqrt{E^{k,\,b^{k}_{y}}_{y}}\Bigg{)}\rho^{k}_{AB}\Bigg{(}\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}}}_{x_{0}x_{1}}}\otimes\sqrt{E^{k,\,b^{k}_{y}}_{y}}\Bigg{)}^{\dagger}\Bigg{]},$
(21)

where

$\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}}}_{x_{0}x_{1}}}=\frac{\sqrt{1+\lambda^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}+(-1)^{a^{k}_{x_{0}x_{1}}}\,\left(\hat{u}^{k}_{x_{0}x_{1}}\cdot\vec{\sigma}\right)\right]+\frac{\sqrt{1-\lambda^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}-(-1)^{a^{k}_{x_{0}x_{1}}}\,\left(\hat{u}^{k}_{x_{0}x_{1}}\cdot\vec{\sigma}\right)\right],$
(22)

and

$\sqrt{E^{k,\,b^{k}_{y}}_{y}}=\frac{\sqrt{1+\eta^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}+(-1)^{b^{k}_{y}}\,\left(\hat{v}^{k}_{y}\cdot\vec{\sigma}\right)\right]+\frac{\sqrt{1-\eta^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}-(-1)^{b^{k}_{y}}\,\left(\hat{v}^{k}_{y}\cdot\vec{\sigma}\right)\right],$
(23)

with $\hat{u}^{k}_{x_{0}x_{1}}$ and $\hat{v}^{k}_{y}$ being mentioned in
Eqs.(5,6). Using Eqs.(21-23), one has the following,

$\rho^{k+1}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k+1}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(24)

with

$\displaystyle t^{k+1}_{11}$
$\displaystyle=\frac{t^{k}_{11}\,\left[1+\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\sqrt{1-(\lambda^{k})^{2}}\right]}{2\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]},$
$\displaystyle t^{k+1}_{22}$
$\displaystyle=\frac{t^{k}_{22}\,\left[1+\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{22})^{2}+(t^{k}_{11})^{2}\sqrt{1-(\lambda^{k})^{2}}\right]}{2\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]},$
$\displaystyle t^{k+1}_{33}$
$\displaystyle=t^{k}_{33}\,\sqrt{1-(\eta^{k})^{2}}\sqrt{1-(\lambda^{k})^{2}}.$
(25)

Hence, the average post-measurement state $\rho^{k+1}_{AB}$ is Bell-diagonal.
Further, we have $\min\left[(t^{k+1}_{11})^{2},(t^{k+1}_{22})^{2}\right]\neq
0$ if $\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2}\right]\neq 0$ for all
possible values of $\lambda^{k}$ $\in$ $(0,1]$ and $\eta^{k}$ $\in$ $(0,1]$.

In particular, when $\lambda^{k}=\eta^{k}=1$, we have

$\displaystyle t^{k+1}_{11}$
$\displaystyle=\frac{(t^{k}_{11})^{3}}{2\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]},$
$\displaystyle t^{k+1}_{22}$
$\displaystyle=\frac{(t^{k}_{22})^{3}}{2\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]},$
$\displaystyle t^{k+1}_{33}$ $\displaystyle=0.$ (26)

Therefore, the state $\rho^{k+1}_{AB}$ remains to be Bell-diagonal with
$\min\left[(t^{k+1}_{11})^{2},(t^{k+1}_{22})^{2}\right]\neq 0$ if
$\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2}\right]\neq 0$ even when Alicek-
Bobk performs projective measurements.

##  Appendix C Calculating $\rho^{k+1}_{AB}$ in case of $3\rightarrow 1$ RAC

Let Alicek-Bobk performs the $3\rightarrow 1$ RAC task with the following
Bell-diagonal two-qubit state,

$\rho^{k}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(27)

where $(t^{k}_{uu})^{2}\geq(t^{k}_{vv})^{2}\geq(t^{k}_{ww})^{2}$ for an
arbitrary choice of $u\neq v\neq w\in\\{1,2,3\\}$.

Alicek and Bobk perform the unsharp measurements mentioned in Eqs.(7,8,9,10).
The average post-measurement state $\rho_{AB}^{k+1}$ received by
Alicek+1-Bobk+1 from Alicek-Bobk can be obtained using the generalized von
Neumann-Lüder’s transformation rule and it is given by,

$\displaystyle\rho^{k+1}_{AB}=\frac{1}{24}\sum_{x_{0},x_{1},x_{2}=0}^{1}\,\,\,\,\,\sum_{y=0}^{1}$
$\displaystyle\Bigg{[}\sum_{a^{k}_{x_{0}x_{1}x_{2}}=0}^{1}\,\,\,\,\,\,\sum_{b^{k}_{y}=0}^{1}\Bigg{(}\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}x_{2}}}_{x_{0}x_{1}x_{2}}}\otimes\sqrt{E^{k,\,b^{k}_{y}}_{y}}\Bigg{)}\rho^{k}_{AB}\Bigg{(}\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}x_{2}}}_{x_{0}x_{1}x_{2}}}\otimes\sqrt{E^{k,\,b^{k}_{y}}_{y}}\Bigg{)}^{\dagger}\Bigg{]},$
(28)

where

$\sqrt{E^{k,\,a^{k}_{x_{0}x_{1}x_{2}}}_{x_{0}x_{1}x_{2}}}=\frac{\sqrt{1+\lambda^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}+(-1)^{a^{k}_{x_{0}x_{1}x_{2}}}\,\left(\hat{u}^{k}_{x_{0}x_{1}x_{2}}\cdot\vec{\sigma}\right)\right]+\frac{\sqrt{1-\lambda^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}-(-1)^{a^{k}_{x_{0}x_{1}x_{2}}}\,\left(\hat{u}^{k}_{x_{0}x_{1}x_{2}}\cdot\vec{\sigma}\right)\right],$
(29)

and

$\sqrt{E^{k,\,b^{k}_{y}}_{y}}=\frac{\sqrt{1+\eta^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}+(-1)^{b^{k}_{y}}\,\left(\hat{v}^{k}_{y}\cdot\vec{\sigma}\right)\right]+\frac{\sqrt{1-\eta^{k}}}{2\sqrt{2}}\left[\mathbf{I}_{2}-(-1)^{b^{k}_{y}}\,\left(\hat{v}^{k}_{y}\cdot\vec{\sigma}\right)\right]$
(30)

with $\hat{u}^{k}_{x_{0}x_{1}x_{2}}$ and $\hat{v}^{k}_{y}$ are mentioned in
Eq.(9) and Eq.(10) respectively. Using Eqs.(28-30), we have the following,

$\rho^{k+1}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k+1}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(31)

with

$\displaystyle t^{k+1}_{11}$
$\displaystyle=\frac{t^{k}_{11}\,\left[1+2\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{11})^{2}+\left[(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]\sqrt{1-(\lambda^{k})^{2}}\right]}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]},$
$\displaystyle t^{k+1}_{22}$
$\displaystyle=\frac{t^{k}_{22}\,\left[1+2\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{22})^{2}+\left[(t^{k}_{33})^{2}+(t^{k}_{11})^{2}\right]\sqrt{1-(\lambda^{k})^{2}}\right]}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]},$
$\displaystyle t^{k+1}_{33}$
$\displaystyle=\frac{t^{k}_{33}\,\left[1+2\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{33})^{2}+\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]\sqrt{1-(\lambda^{k})^{2}}\right]}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]}.$
(32)

Hence, the average post-measurement state $\rho^{k+1}_{AB}$ is Bell-diagonal.
Furthermore, we notice that
$\min\left[(t^{k+1}_{11})^{2},(t^{k+1}_{22})^{2},(t^{k+1}_{33})^{2}\right]\neq
0$ if $\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2},(t^{k}_{33})^{2}\right]\neq
0$ for all possible values of $\lambda^{k}$ $\in$ $(0,1]$ and $\eta^{k}$ $\in$
$(0,1]$.

When $\lambda^{k}=\eta^{k}=1$, we have

$\displaystyle t^{k+1}_{11}$
$\displaystyle=\frac{(t^{k}_{11})^{3}}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]},$
$\displaystyle t^{k+1}_{22}$
$\displaystyle=\frac{(t^{k}_{22})^{3}}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]},$
$\displaystyle t^{k+1}_{33}$
$\displaystyle=\frac{(t^{k}_{33})^{3}}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]}.$
(33)

Therefore, the state $\rho^{k+1}_{AB}$ remains to be Bell-diagonal with
$\min\left[(t^{k+1}_{11})^{2},(t^{k+1}_{22})^{2},(t^{k+1}_{33})^{2}\right]\neq
0$ if $\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2},(t^{k}_{33})^{2}\right]\neq
0$ even when Alicek-Bobk performs projective measurements.

For $\rho_{AB}^{1}=|\psi^{-}\rangle\langle\psi^{-}|$
---
Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{2\rightarrow 1}$ | Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{2\rightarrow 1}$
with $k=$ |  |  |  | with $k=$ |  |  |
$1$ | $1$ | $1$ | $0.854$ | $1$ | $0.340$ | $0.340$ | $0.541$
$2$ | $1$ | $1$ | $0.588$ | $2$ | $1$ | $1$ | $0.833$
$3$ | $1$ | $1$ | $0.522$ | $3$ | $1$ | $1$ | $0.583$
$4,5,6,\cdots$ | $1$ | $1$ | $\frac{1}{2}<P_{\texttt{Min}}^{2\rightarrow 1}<0.520$ | $4$ | $1$ | $1$ | $0.521$
| $5,6,7,\cdots$ | $1$ | $1$ | $\frac{1}{2}<P_{\texttt{Min}}^{2\rightarrow 1}<0.520$
For $\rho_{AB}^{1}=|\psi^{-}\rangle\langle\psi^{-}|$
Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{2\rightarrow 1}$ | Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{2\rightarrow 1}$
with $k=$ |  |  |  | with $k=$ |  |  |
$1$ | $0.340$ | $0.340$ | $0.541$ | $1$ | $0.340$ | $0.340$ | $0.541$
$2$ | $0.340$ | $0.340$ | $0.538$ | $2$ | $0.340$ | $0.340$ | $0.538$
$3$ | $1$ | $1$ | $0.813$ | $3$ | $0.340$ | $0.340$ | $0.536$
$4$ | $1$ | $1$ | $0.578$ | $4$ | $0.340$ | $0.340$ | $0.534$
$5$ | $1$ | $1$ | $0.520$ | $5$ | $0.340$ | $0.340$ | $0.532$
$6,7,8,\cdots$ | $1$ | $1$ | $\frac{1}{2}<P_{\texttt{Min}}^{2\rightarrow 1}<0.520$ | $6$ | $0.340$ | $0.340$ | $0.530$
| $7$ | $0.340$ | $0.340$ | $0.528$
| $8$ | $0.340$ | $0.340$ | $0.527$
| $9$ | $0.340$ | $0.340$ | $0.525$
| $10$ | $0.340$ | $0.340$ | $0.524$
| $11$ | $0.340$ | $0.340$ | $0.522$
| $12$ | $0.340$ | $0.340$ | $0.521$
| $13$ | $0.340$ | $0.340$ | $0.520$
| $14,15,16,\cdots$ | $0.340$ | $0.340$ | $\frac{1}{2}<P_{\texttt{Min}}^{2\rightarrow 1}<0.520$
Table 1: Minimum success probabilities in $2\rightarrow 1$ RAC task (assisted
with two bits, shared from a common source and having maximally mixed marginal
at the receiver’s end) by different consecutive pairs for different choices of
$\lambda^{k}$, $\eta^{k}$ ($k\in\\{1,2,\cdots\\}$). In each of the above
tables, the initially shared two-qubit state is assumed to be singlet state,
$\rho_{AB}^{1}=|\psi^{-}\rangle\langle\psi^{-}|$. Here, all the numerical
values are rounded to the third decimal places.

##  Appendix D Trade-off between the amount of quantum advantage gained by an
arbitrary pair and the number of subsequent pairs exhibiting considerable
amount of quantum advantage

We have shown that an unbounded pairs of Alice and Bob can, in principle,
demonstrate quantum advantage in some particular $n\rightarrow 1$ RAC task
(with $n=2$ or $n=3$). Here, quantum advantage implies that the magnitude of
$(P_{\texttt{Min}}^{n\rightarrow 1}-\frac{1}{2})$ is positive. However, for
experimental implementation, the magnitude of $(P_{\texttt{Min}}^{n\rightarrow
1}-\frac{1}{2})$ should not only be positive, but also be large enough to be
detected in real experiment. In order to explore this issue relevant for
practical realization, we consider that the quantum advantage is ‘significant’
if $P_{\texttt{Min}}^{n\rightarrow 1}\geq 0.520$. Note, here the bound $0.520$
is chosen as an example. One can choose any other bound depending on the
precision of the experimental apparatus and the characteristics of the results
described below will not change.

We observe that when an arbitrary pair gains a large amount of quantum
advantage by performing projective measurements or unsharp measurements with
the sharpness parameters being close to unity, then only few number of
subsequent pairs will get ‘significant’ quantum advantage. On the other hand,
when a pair gets a small amount of quantum advantage by performing unsharp
measurements with the sharpness parameters being much less than unity, a
larger number of subsequent pairs can achieve ‘significant’ quantum advantage.
We will describe this aspect through a number of examples.

Let us consider that the singlet state is initially shared by the pair
Alice1-Bob1 and each of the multiple pairs performs $2\rightarrow 1$ RAC task
assisted with two bits, shared from a common source and having maximally mixed
marginal at the receiver’s end. The results in this case are presented in
Table 1. From this table, we observe that at most three consecutive pairs can
show $P_{\texttt{Min}}^{2\rightarrow 1}\geq 0.520$ when all observers perform
projective measurements. Next, consider the case when the first pair performs
measurements with $\lambda^{1}=0.340$ and $\eta^{1}=0.340$ and all other pairs
perform projective measurements. In this case, the minimum success probability
for the first pair is reduced compared to the previous case (with
$\lambda^{1}=1$ and $\eta^{1}=1$), but the maximum number of pairs exhibiting
$P_{\texttt{Min}}^{2\rightarrow 1}\geq 0.520$ is increased to four. Now,
consider that the first and second pairs perform unsharp measurements with
$\lambda^{1}=0.340$, $\eta^{1}=0.340$, $\lambda^{2}=0.340$, $\eta^{2}=0.340$
and all other pairs perform projective measurements. In this case, the minimum
success probability for the second pair is reduced compared to the case with
$\lambda^{2}=1$ and $\eta^{2}=1$, and the maximum number of pairs exhibiting
$P_{\texttt{Min}}^{2\rightarrow 1}\geq 0.520$ is further increased to five.
Proceeding in this way, we find out that when all the pairs perform
measurements with $\lambda^{k}=0.340$ and $\eta^{k}=0.340$
($k\in\\{1,2,\cdots\\}$), then the maximum number of pairs exhibiting
$P_{\texttt{Min}}^{2\rightarrow 1}\geq 0.520$ is increased to thirteen.

Next, let us focus on the $3\rightarrow 1$ RAC task assisted with two bits
shared from a common source. Here also, consider that the singlet state is
initially shared by the pair Alice1-Bob1. The results in this case are
presented in Table 2. From this table, it can be noticed that at most two
consecutive pairs can achieve $P_{\texttt{Min}}^{3\rightarrow 1}\geq 0.520$
contingent upon performing projective measurements by all pairs. Next,
consider that the first pair performs unsharp measurements associated with
$\lambda^{1}=0.370$ and $\eta^{1}=0.370$ and each of the other pairs performs
projective measurements. In this case, we observe that the minimum success
probability for the first pair is reduced compared to the previous case with
$\lambda^{1}=1$ and $\eta^{1}=1$, but the maximum number of pairs exhibiting
$P_{\texttt{Min}}^{3\rightarrow 1}\geq 0.520$ is increased to three. Next, we
consider that the first pair and second pair perform unsharp measurements with
$\lambda^{1}=0.370$, $\eta^{1}=0.370$ and $\lambda^{2}=0.370$,
$\eta^{2}=0.370$ respectively. All other pairs perform projective
measurements. In this case, the minimum success probability for the second
pair is decreased compared to the case with $\lambda^{2}=1$ and $\eta^{2}=1$.
On the other hand, the maximum number of pairs exhibiting
$P_{\texttt{Min}}^{3\rightarrow 1}\geq 0.520$ is further increased to four.
Following this approach, we find out that when all the pairs perform unsharp
measurements with $\lambda^{k}=0.370$ and $\eta^{k}=0.370$ for all
$k\in\\{1,2,\cdots\\}$, the maximum number of pairs exhibiting
$P_{\texttt{Min}}^{3\rightarrow 1}\geq 0.520$ becomes eight.

It can be easily checked that the above aspects remain unaltered when the
sequential execution of the aforementioned $n\rightarrow 1$ RAC task (with
$n=2$ or $n=3$) is initiated with an appropriate Bell-diagonal two-qubit
(entangled or separable) state given by Eq.(2) with $k=1$ and
$\min\limits_{i\leq n}\left[(t^{1}_{ii})^{2}\right]\neq 0$ and for different
choices of the sharpness parameters. Therefore, we can conclude that there may
exist a trade-off relation between the amount of quantum advantage gained by
an arbitrary pair and the number of subsequent pairs exhibiting
$P_{\texttt{Min}}^{n\rightarrow 1}\geq 0.520$. Importantly, any of these two
quantities can be increased at the expense of the other by suitably choosing
the sharpness parameters associated with the measurements.

For $\rho_{AB}^{1}=|\psi^{-}\rangle\langle\psi^{-}|$
---
Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{3\rightarrow 1}$ | Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{3\rightarrow 1}$
with $k=$ |  |  |  | with $k=$ |  |  |
$1$ | $1$ | $1$ | $0.789$ | $1$ | $0.370$ | $0.370$ | $0.540$
$2$ | $1$ | $1$ | $0.532$ | $2$ | $1$ | $1$ | $0.762$
$3,4,5,\cdots$ | $1$ | $1$ | $\frac{1}{2}<P_{\texttt{Min}}^{3\rightarrow 1}<0.520$ | $3$ | $1$ | $1$ | $0.529$
| $4,5,6,\cdots$ | $1$ | $1$ | $\frac{1}{2}<P_{\texttt{Min}}^{3\rightarrow 1}<0.520$
For $\rho_{AB}^{1}=|\psi^{-}\rangle\langle\psi^{-}|$
Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{3\rightarrow 1}$ | Alicek-Bobk | $\lambda^{k}$ | $\eta^{k}$ | $P_{\texttt{Min}}^{3\rightarrow 1}$
with $k=$ |  |  |  | with $k=$ |  |  |
$1$ | $0.370$ | $0.370$ | $0.540$ | $1$ | $0.370$ | $0.370$ | $0.540$
$2$ | $0.370$ | $0.370$ | $0.536$ | $2$ | $0.370$ | $0.370$ | $0.536$
$3$ | $1$ | $1$ | $0.738$ | $3$ | $0.370$ | $0.370$ | $0.532$
$4$ | $1$ | $1$ | $0.526$ | $4$ | $0.370$ | $0.370$ | $0.530$
$5,6,7,\cdots$ | $1$ | $1$ | $\frac{1}{2}<P_{\texttt{Min}}^{3\rightarrow 1}<0.520$ | $5$ | $0.370$ | $0.370$ | $0.527$
| $6$ | $0.370$ | $0.370$ | $0.524$
| $7$ | $0.370$ | $0.370$ | $0.522$
| $8$ | $0.370$ | $0.370$ | $0.520$
| $9,10,11,\cdots$ | $0.370$ | $0.370$ | $\frac{1}{2}<P_{\texttt{Min}}^{3\rightarrow 1}<0.520$
Table 2: Minimum success probabilities in $3\rightarrow 1$ RAC task (assisted
with two bits shared from a common source) by different consecutive pairs for
different choices of $\lambda^{k}$, $\eta^{k}$ ($k\in\\{1,2,\cdots\\}$). In
each of the above tables, the initially shared two-qubit state is assumed to
be singlet state, $\rho_{AB}^{1}=|\psi^{-}\rangle\langle\psi^{-}|$. Here, all
the numerical values are rounded to the third decimal places.

##  Appendix E Proof of Corollary 2 and Corollary 3

Suppose Alicek-Bobk performs the aforementioned $2\rightarrow 1$ RAC task with
probability $p_{k}$ and the aforementioned $3\rightarrow 1$ RAC with
probability $1-p_{k}$ (with $0\leq p_{k}\leq 1$) using the following Bell-
diagonal two-qubit state,

$\rho^{k}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(34)

where $(t^{k}_{uu})^{2}\geq(t^{k}_{vv})^{2}\geq(t^{k}_{ww})^{2}>0$ for an
arbitrary choice of $u\neq v\neq w\in\\{1,2,3\\}$. This implies that
$\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2}\right]\neq 0$ and
$\min\left[(t^{k}_{11})^{2},(t^{k}_{22})^{2},(t^{k}_{33})^{2}\right]\neq 0$.
Hence, while performing the $2\rightarrow 1$ RAC, this state will give
$P_{\texttt{Min}}^{2\rightarrow 1}>\frac{1}{2}$. On the other hand, while
performing the $3\rightarrow 1$ RAC, this state will give
$P_{\texttt{Min}}^{3\rightarrow 1}>\frac{1}{2}$. Hence, overall, Alicek-Bobk
will achieve quantum advantage with this state.

Next, the average post-measurement state $\rho_{AB}^{k+1}$ received by
Alicek+1-Bobk+1 from Alicek-Bobk is given by,

$\rho^{k+1}_{AB}=p_{k}\,\rho^{k+1}_{AB_{2\rightarrow
1}}+(1-p_{k})\,\rho^{k+1}_{AB_{3\rightarrow 1}},$ (35)

where the average is taken since each pair act independently of other pairs.
In the above equation, $\rho^{k+1}_{AB_{2\rightarrow 1}}$ is the average post-
measurement state received by Alicek+1-Bobk+1 when Alicek-Bobk performs the
aforementioned $2\rightarrow 1$ RAC with certainty. One can evaluate
$\rho^{k+1}_{AB_{2\rightarrow 1}}$ using Eq.(21). On the other hand,
$\rho^{k+1}_{AB_{3\rightarrow 1}}$ is the average post-measurement state
received by Alicek+1-Bobk+1 when Alicek-Bobk performs the aforementioned
$3\rightarrow 1$ RAC with certainty. It can be evaluated using Eq.(28). Hence,
we can infer that the state $\rho_{AB}^{k+1}$ has the following Bell-diagonal
form,

$\rho^{k+1}_{AB}=\frac{1}{4}(\mathbf{I}_{4}+\sum_{i=1}^{3}t^{k+1}_{ii}\,\sigma_{i}\otimes\sigma_{i}),$
(36)

where $t^{k+1}_{ii}$ for $i=1,2,3$ can be calculated using Eqs.(25), (32),
(35). Hence, we have

$\displaystyle t^{k+1}_{11}=$ $\displaystyle
p_{k}\frac{t^{k}_{11}\,\left[1+\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\sqrt{1-(\lambda^{k})^{2}}\right]}{2\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]}$
$\displaystyle\quad+(1-p_{k})\frac{t^{k}_{11}\,\left[1+2\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{11})^{2}+\left[(t^{k}_{22})^{2}+(t^{k}_{33})^{2}\right]\sqrt{1-(\lambda^{k})^{2}}\right]}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+t^{k}_{33})^{2}\right]},$
(37) $\displaystyle t^{k+1}_{22}=$ $\displaystyle
p_{k}\frac{t^{k}_{22}\,\left[1+\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{22})^{2}+(t^{k}_{11})^{2}\sqrt{1-(\lambda^{k})^{2}}\right]}{2\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]}$
$\displaystyle\quad+(1-p_{k})\frac{t^{k}_{22}\,\left[1+2\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{22})^{2}+\left[(t^{k}_{33})^{2}+(t^{k}_{11})^{2}\right]\sqrt{1-(\lambda^{k})^{2}}\right]}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+t^{k}_{33})^{2}\right]},$
(38) $\displaystyle t^{k+1}_{33}=$ $\displaystyle
p_{k}t^{k}_{33}\,\sqrt{1-(\eta^{k})^{2}}\sqrt{1-(\lambda^{k})^{2}}+(1-p_{k})\frac{t^{k}_{33}\,\left[1+2\sqrt{1-(\eta^{k})^{2}}\right]\left[(t^{k}_{33})^{2}+\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}\right]\sqrt{1-(\lambda^{k})^{2}}\right]}{3\left[(t^{k}_{11})^{2}+(t^{k}_{22})^{2}+t^{k}_{33})^{2}\right]}.$
(39)

From the above equations, it is observed that when $p_{k}\in[0,1)$, then
$t_{11}^{k+1}\neq 0$, $t_{22}^{k+1}\neq 0$, $t_{33}^{k+1}\neq 0$ for all
possible values of $\lambda^{k}$ $\in$ $(0,1]$ and $\eta^{k}$ $\in$ $(0,1]$.
Hence, in this case, Alicek+1-Bobk+1 will get quantum advantage in both
$2\rightarrow 1$ RAC and $3\rightarrow 1$ RAC. Now, suppose that
Alicek+1-Bobk+1 performs $2\rightarrow 1$ RAC task with probability $p_{k+1}$
and $3\rightarrow 1$ RAC with probability $1-p_{k+1}$ (with $0\leq p_{k+1}\leq
1$). Following the aforementioned argument, we can, therefore, conclude that
Alicek+1-Bobk+1 will also get quantum advantage overall.

Next, consider that $p_{k}=1$. In this case, $t_{11}^{k+1}\neq 0$,
$t_{22}^{k+1}\neq 0$, for all possible values of $\lambda^{k}$ $\in$ $(0,1]$
and $\eta^{k}$ $\in$ $(0,1]$. But $t_{33}^{k+1}\neq 0$ only for $\lambda^{k}$
$\in$ $(0,1)$ and $\eta^{k}$ $\in$ $(0,1)$; $t_{33}^{k+1}=0$ if
$\lambda^{k}=1$ and $\eta^{k}=1$. Subsequently, Alicek+1-Bobk+1 will not get
quantum advantage while performing $3\rightarrow 1$ RAC if Alicek-Bobk
performs the $2\rightarrow 1$ RAC with certainty (i.e., $p_{k}=1$) under
projective measurements. Hence, in order to ensure overall quantum advantage
by Alicek+1-Bobk+1 for all values of $p_{k+1}\in[0,1]$, we need the following:
$\lambda^{k}\neq 1$ and $\eta^{k}\neq 1$.

To summarize, if both the pairs- Alicek-Bobk and Alicek+1-Bobk+1 perform
$2\rightarrow 1$ RAC task with some non-zero probability and $3\rightarrow 1$
RAC with some non-zero probability, then both the pairs will get overall
quantum advantage for all values of $\lambda^{k}$ $\in$ $(0,1]$ and $\eta^{k}$
$\in$ $(0,1]$. Applying this argument to arbitrary number of pairs with
$p_{k}$ being equal to or not equal to $p_{\tilde{k}}$ for all choices of
$k\neq\tilde{k}\in\\{1,2,\cdots\\}$, we get Corollary 2.

On the other hand, if Alicek-Bobk performs $2\rightarrow 1$ RAC task with
certainty, then we have $p_{k}=1$. In this case, if Alicek+1-Bobk+1 performs
$3\rightarrow 1$ RAC with certainty, then this pair will not get quantum
advantage if $\lambda^{k}=1$ and $\eta^{k}=1$. However, for all values of
$\lambda^{k}$ $\in$ $(0,1)$ and $\eta^{k}$ $\in$ $(0,1)$, Alicek+1-Bobk+1 will
get quantum advantage. Applying this argument to arbitrary number of pairs, we
get Corollary 3.