# Local State Discrimination and Ordering of Multipartite Entangled States

Sumit Rout1, Ananda G. Maity2, Amit Mukherjee2,
Saronath Halder3 and Manik Banik4
1Integrated Science Education and Research Centre, Visva Bharati University,
Santiniketan 731235, India.
2S. N. Bose National Center for Basic Sciences,
Block JD, Sector III, Salt Lake, Kolkata 700098, India.
3Harish-Chandra Research Institute, HBNI,
Chhatnag Road, Jhunsi, Allahabad 211019, India.
4School of Physics, IISER Thiruvananthapuram,
Vithura, Kerala 695551, India

###### Abstract

Local operations assisted with classical communication (LOCC) and/or its
stochastic generalization induce an ordering relation among bipartite
entangled states based on their inter-convertibility. This, in multipartite
scenario, leads to different classes of entanglement where states within a
class can be inter-converted but conversion among different classes is not
possible, and thus results in incomparable multipartite entangled resources.
Here we show that a bona-fide ordering relation, under the operational
paradigm of LOCC, can be established between two such incomparable states
based on their performance in some operational task. Entanglement assisted
local state discrimination provides one such platform for resource comparison
and the notion of genuinely nonlocal product states becomes quite pertinent in
this regard. We provide the detailed framework along with explicit examples
exhibiting how incomparable resources can be put into well defined ordered
relation.

##  1 Introduction

Entanglement is the key resource for a number of quantum protocols [1, 2, 3,
4, 5, 6]. Resource comparison of different such states is thus quite desirable
from practical perspective. Inter-convertibility among different entangled
states under local operations assisted with classical communication (LOCC)
leads to a natural framework to define equivalence relations among the set of
multipartite entangled states. In bipartite scenario, the majorization theory
[7] plays crucial role in characterizing the possibility of LOCC conversions
among pure states [8, 9, 10]. A more generic framework of state inter-
convertibility considers stochastic local operations and classical
communication (SLOCC) that deals with conversion of the states under LOCC but
without imposing that it has to be achieved with certainty [11]. However, in
multipartite scenario, still there exist different classes of states that can
be inter-converted within a class but not outside the class [12, 13, 14].
Later works indeed established that LOCC induces a trivial ordering as not all
pure entangled multipartite states are comparable, i.e., LOCC transformations
among them are almost never possible [15, 16, 17]. Recently, the authors in
Ref. [18] have considered a generalized class of operations, namely,
biseparability-preserving (BSP) operations and have shown that the LOCC
inequivalent classification of multipartite states collapses under this
general framework leading to a resource theory of entanglement with a unique
multipartite maximally entangled state . However, the lack of operational
motivation is the main shortcoming of this framework. Not all BSP operations
are locally implementable – the spatially separated parties need to come
together to implement such a generic BSP operation, whereas in all practical
scenarios, entanglement is considered as resource with the parties being in
different locations. Comparison of inequivalent multipartite entangled states
is therefore still demanding from the operational perspective.

The aim in this present work is to establish bona-fide ordering relation(s)
among LOCC/SLOCC incomparable entangled states based on some operational
task(s). The intended task should be nonlocal in nature involving spatially
separated players who are empowered with LOCC to accomplish the task. An
inherent prerequisite of this task is the suboptimal performance by the
players when assisted with LOCC only; which opens up the possibility to
achieve a better performance with additional entangled resource(s). Such a
task can introduce an ordering relation among the resource content of
different multipartite states. A state $\rho$ is said to be as good as or
better than the state $\sigma$ for performing the task $\tau$, i.e.,
$\rho\succcurlyeq_{\tau}\sigma$, if and only if
$\mathcal{P}_{\tau}(\rho)\geq\mathcal{P}_{\tau}(\sigma)$; where
$\mathcal{P}_{\tau}(x)\in\mathbb{R}$ denotes the payoff obtained in the task
$\tau$ performed with the resource state $x$ shared among the players. Strict
inequality ‘$\succ_{\tau}$’ indicates superiority of $\rho$ over $\sigma$ in
the concerned task $\tau$. The problem of local state discrimination (LSD) is
a good candidate for the aforesaid purpose which we consider in this work. In
LSD, an intriguing scenario arises when one considers discrimination of
multipartite product states. In a pioneering work, Bennett et al. constructed
orthogonal product bases for multipartite Hilbert spaces that can not be
perfectly discriminated by LOCC where perfect global discrimination is assured
through mutual orthogonality [19]. Such a product basis is called nonlocal
product basis (NPB). The work by Bennett et al. motivates a huge research
interest in the problem of LSD which also has engrossing connections with the
theory of quantum entanglement (see [20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38] and references therein). However, it is
only recently, a nontrivial generalization, namely genuinely nonlocal product
basis (GNPB), is proposed for multipartite Hilbert spaces where perfect
discrimination of the product basis requires all the parties to come together
(or share sufficient entanglement) to perform global
measurement.111Henceforth, this requirement will be briefly referred to as
necessity of the parties to come together. [39, 40]. Interestingly, here, we
show that the concept of GNPB as well as a related concept called genuinely
nonlocal set of product states (GNPS) turn out to be quite functional for the
purpose of the present work. We construct several sets of GNPS and use them
for resource comparison of different multipartite entangled states that
otherwise are LOCC/SLOCC incomparable. In the following, we first present a
few relevant observations regarding LSD.

##  2 Local state discrimination

Although the history of quantum state discrimination dates back to early
1970’s with an initial attempt to formulate information protocols using
quantum optical devices [41, 42, 43], LSD problem gained research interest
much later [44, 45, 19]. It asks to identify the correct state chosen randomly
from an apriori known ensemble of states
$\\{p_{i},\ket{\psi_{i}}\\}_{i=1}^{m}$ under the restriction that the
spatially separated parties can perform only LOCC;
$\forall~{}i~{},\ket{\psi_{i}}\in\otimes_{j=1}^{n}\mathcal{H}_{j}$ with
$\mathcal{H}_{j}$ being the Hilbert space of the $j^{th}$ subsystem (note that
only one copy of the system is provided). Wherever possible we will avoid the
party index for brevity. All the systems considered in this work are finite
dimensional and hence $\mathcal{H}_{j}$ will be isomorphic to some complex
euclidean space $\mathbb{C}^{d_{j}}$. In a product LSD problem, all
$\ket{\psi_{i}}$’s are considered to be fully product state, i.e.,
$\forall~{}i,~{}\ket{\psi_{i}}=\otimes_{j=1}^{n}\ket{\phi^{j}_{i}}$ with
$\ket{\phi^{j}_{i}}\in\mathbb{C}^{d_{j}}$. Here we recall an example of a set
of nonlocal product states (NPS), i.e., set of pairwise orthogonal product
states that can not be perfectly distinguished under LOCC [19]:

$\mathbb{S}_{Ben}\equiv\left\\{\ket{0}\ket{\eta_{\pm}},\ket{\eta_{\pm}}\ket{2},\ket{2}\ket{\xi_{\pm}},\ket{\xi_{\pm}}\ket{0}\right\\},$
(1)

where $\ket{\eta_{\pm}}:=\frac{1}{\sqrt{2}}\left(\ket{0}\pm\ket{1}\right)$ and
$\ket{\xi_{\pm}}:=\frac{1}{\sqrt{2}}\left(\ket{1}\pm\ket{2}\right)$. As noted
in [19], deletion of any element from this set makes it local, i.e., if the
prior probability of any of the states in the ensemble is zero then the state
given from the ensemble can be perfectly distinguished under LOCC. However,
inclusion of the orthogonal state $\ket{1}\ket{1}$ along with the ensemble
$\mathbb{S}_{Ben}$ retains the nonlocal feature of the resulting ensemble.
This observation leads us to the following generalization.

######  Observation 1.

Let $\mathbb{S}\subset\otimes_{j=1}^{n}\mathbb{C}^{d_{j}}$ be a multipartite
NPS. Then the set of states $\mathbb{A}:=\mathbb{S}\cup\mathbb{S}^{\prime}$ is
again an NPS for any set of mutually orthogonal states $\mathbb{S}^{\prime}$
such that $\mathbb{S}\perp\mathbb{S}^{\prime}$.

Two sets of states $\mathbb{S}$ and $\mathbb{S}^{\prime}$ are called
orthogonal iff
$\braket{\phi}{\phi^{\prime}}=0,~{}\forall~{}\ket{\phi}\in\mathbb{S},~{}{and}~{}\ket{\phi^{\prime}}\in\mathbb{S}^{\prime}$.
The proof simply follows an argument of reductio ad absurdum. If $\mathbb{A}$
were a locally distinguishable set then for every $\ket{\psi}\in\mathbb{A}$
chosen at random, it is possible to perfectly identify it under LOCC. This
should hold even when the state lies in the nonlocal set $\mathbb{S}$ which
leads to a contradiction.

Let us now consider a new set of states
$\mathbb{S}^{\prime}_{Ben}:=\mathbb{S}_{Ben}\otimes\ket{\phi_{0}}_{C}$ which
means,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathbb{S}^{\prime}_{Ben}\equiv$}\left\\{\\!\begin{aligned}
\ket{0}\ket{\eta_{\pm}}\ket{\phi_{0}}_{C},\ket{\eta_{\pm}}\ket{2}\ket{\phi_{0}}_{C},\\\
\ket{2}\ket{\xi_{\pm}}\ket{\phi_{0}}_{C},\ket{\xi_{\pm}}\ket{0}\ket{\phi_{0}}_{C}\end{aligned}\right\\}.$

Here, the fixed state $\ket{\phi_{0}}_{C}$ is possessed either by Alice or by
Bob. This new set $\mathbb{S}^{\prime}_{Ben}$ is also an NPS between Alice and
Bob as such a fixed state of another system they can always be created locally
[46]. Following observation states this in a more generic way.

######  Observation 2.

Let $\mathbb{S}=\\{\ket{\psi_{i}}_{A_{1}\cdots A_{n}}\\}_{i=1}^{k}$ be an NPS
shared among $n$ parties. Consider the set
$\mathbb{S^{\prime}}:=\\{\ket{\psi_{i}}_{A_{1}\cdots
A_{n}}\\}_{i=1}^{k}\otimes\ket{\phi_{0}}_{a_{1}\cdots a_{m}}$, where
$\ket{\phi_{0}}_{a_{1}\cdots a_{m}}$ is some fully separable state with some
of the subsystems $\\{a_{i}\\}$ being in possession of some parties. The
resulting set $\mathbb{S^{\prime}}$ is an NPS among those $n$ parties.

If we consider the two-party task $\tau_{2}$ of local discrimination of the
set $\mathbb{S}_{Ben}$ then a protocol proposed by Cohen [47] showed that two-
qubit maximally entangled state $\ket{\phi^{+}}:=(\ket{00}+\ket{11})/\sqrt{2}$
suffices for perfect discrimination even though
$\mathbb{S}_{Ben}\subset\mathbb{C}^{3}\otimes\mathbb{C}^{3}$. Note that
Cohen’s protocol is nontrivial as it consumes less amount of entanglement
($1$-ebit) than the naive teleportation protocol which in this case requires
$\log_{2}3$-ebit of entanglement. Cohen also provided an argument that
following his protocol the task $\tau_{2}$ can not be perfectly achieved if
any two-qubit nonmaximally entangled state
$\ket{\chi}:=\alpha\ket{00}+\beta\ket{11}$ (with $\alpha\neq\beta$) is
provided as resource. He further conjectured that $\ket{\phi^{+}}$ may be the
necessary resource for accomplishment of the task $\tau_{2}$ independent of
the fact whichever local protocol is followed. In the language of present work
we can thus say the following.

######  Lemma 1.

(Cohen [47]) The task $\tau_{2}$ induces the ordering relation
$\ket{\phi^{+}}\succ_{\tau_{2}}\ket{\chi}$. In fact,
$\ket{\phi^{+}}\succ_{\tau_{2}}\rho$ for any
$\rho\in\mathcal{D}\left(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\right)$;
$\mathcal{D}(\mathcal{H})$ denotes the collection of all density operator
acting on $\mathcal{H}$.

Of course the aforesaid result demands an affirmative proof of Cohen’s
conjecture which remains unresolved till date. At this point it is noteworthy
that a protocol independent ordering relation between $\ket{\phi^{+}}$ and
$\ket{\chi}$ can be obtained if one considers generic LSD problem that
involves discrimination of entangled states. For instance, local
discrimination of two-qubit Bell states constitutes such a task [22]. For
optimal resource requirement of some LSD problems involving multipartite
entangled states see the Refs. [48, 49].

Although the ordering relation in Lemma 1 is protocol dependent till date (as
Cohen’s conjecture is not yet resolved), in the following we will show that
consideration of product LSD problem can induce protocol independent ordering
relation among multipartite entanglement resources that otherwise are LOCC
incomparable/inequivalent. For that the notion of GNPB/GNPS plays the crucial
role which has been recently introduced in [39]. A GNPB of a multipartite
Hilbert space (involving more that two subsystems) requires entangled
resources across every cut for perfect discrimination of the states. Such
product bases also implies existence of separable measurement whose
implementation requires entangled resources across all bipartitions. Later,
classification of GNPSs as well as their entanglement assisted discrimination
protocol have also been studied [40]. Next we provide construction of such a
nontrivial GNPS.

######  Proposition 1.

The set of states $\mathbb{G}_{3}$ defined below is a GNPS in
$\mathbb{C}^{4}_{A}\otimes\mathbb{C}^{3}_{B}\otimes\mathbb{C}^{3}_{C}$.

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathbb{G}_{3}\equiv$}\left\\{\\!\begin{aligned}
\ket{\zeta_{\pm}^{0}}~{}&:=\ket{\epsilon_{\pm}}\ket{1}\ket{2},~{}~{}\ket{\zeta^{1,1}_{\pm}}:=\ket{1}\ket{\gamma_{\pm}^{1}}\ket{2},\\\
\ket{\zeta^{1,2}_{\pm}}&:=\ket{\gamma_{\pm}^{1}}\ket{p}\ket{2},~{}\ket{\zeta^{1,3}_{\pm}}:=\ket{p}\ket{\epsilon_{\pm}}\ket{2},\\\
\ket{\zeta^{2,1}_{\pm}}&:=\ket{2}\ket{1}\ket{\gamma_{\pm}^{2}},~{}\ket{\zeta^{2,2}_{\pm}}:=\ket{\gamma_{\pm}^{2}}\ket{1}\ket{p},\\\
&~{}~{}~{}~{}~{}~{}~{}\ket{\zeta^{2,3}_{\pm}}:=\ket{p}\ket{1}\ket{\epsilon_{\pm}}~{}\end{aligned}\right\\},$

where $\\{\ket{p},\ket{q},\ket{1},\ket{2}\\}$ are mutually orthonormal,
$\ket{\epsilon_{\pm}}:=\frac{1}{\sqrt{2}}(\ket{p}\pm\ket{q})$ and
$\ket{\gamma_{\pm}^{i}}:=\frac{1}{\sqrt{2}}(\ket{q}\pm\ket{i})$, with
$i\in\\{1,2\\}$.

###### Proof.

Consider the subset of states
$\$^{i}\equiv\left\\{\ket{\zeta_{\pm}^{0}},\ket{\zeta^{i,1}_{\pm}},\ket{\zeta^{i,2}_{\pm}},\ket{\zeta^{i,3}_{\pm}}\right\\}\subset\mathbb{G}_{3}$,
for $i\in\\{1,2\\}$. The set $\$^{1}$ has the structure as of the set
$\mathbb{S}_{Ben}$ (with a notation change, $0\rightarrow p,1\rightarrow
q,2\rightarrow 1$) between Alice and Bob while Charlie has the fixed state
$\ket{2}$. This along with Observation 1 and Observation 2 assures that the
set $\mathbb{G}_{3}$ cannot be locally discriminated even when Charlie groups
with either Alice or Bob. Similarly the set $\$^{2}$ prohibits perfect local
discrimination of $\mathbb{G}_{3}$ even when Alice and Bob are grouped
together. This completes the proof. ∎

##  3 Comparing inequivalent entangled resources

In this section, we first provide a sufficient resource for local
discrimination of the set $\mathbb{G}_{3}$.

######  Theorem 1.

The set of states $\mathbb{G}_{3}$ can be perfectly discriminated locally when
the state $\ket{g_{3}}:=(\ket{000}+\ket{111})/\sqrt{2}$ is shared as resource.

The protocol constituting proof of the above Theorem is provided in the
Appendix. Theorem 1 immediately leads us to the following ordering relation.

######  Corollary 1.

The task $\tau_{3}$ of local discrimination of $\mathbb{G}_{3}$ induces the
order $\ket{g_{3}}\succ_{\tau_{3}}\rho$ for any
$\rho\in\mathcal{D}(\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}})$ with
arbitrary local dimension $d_{1}$ and $d_{2}$.

Note that, whenever Schmidt number of $\rho$ is strictly greater than $2$, the
two entangled resources in Corollary 1 are incomparable as Schmidt number can
not be increased neither under SLOCC operation [9, 50] even nor under
separable operation [51]. Here it should be mentioned that, in this work, we
are concerned with the entanglement properties of single copy of the state,
and thus asymptotic results do not apply. The resource state used in Theorem 1
achieves the minimal dimension. The question arises whether GHZ state is the
only state in $\left(\mathbb{C}^{2}\right)^{\otimes 3}$ to accomplish the task
$\tau_{3}$. Following a similar reasoning as in [47] it can be argued that the
state $\alpha\ket{000}+\beta\ket{111}$, with $\alpha\neq\beta$ will not
suffice the purpose. However the question remains open whether the $3$-qubit
W-state $\ket{w_{3}}:=(\ket{001}+\ket{010}+\ket{100})/\sqrt{3}$ can perform
the task $\tau_{3}$. At this point we conjecture that
$\ket{g_{3}}\succ_{\tau_{3}}\ket{w_{3}}$. An affirmative answer to our
conjecture will induce a bona-fide ordering relation between the SLOCC
incomparable resources $\ket{g_{3}}$ and $\ket{w_{3}}$. Here it should be kept
in mind that the entanglement properties of $\ket{g_{3}}$ and $\ket{w_{3}}$
are incomparable at single copy level [12]. Such resources may be comparable
under SLOCC when asymptotically many copies are considered (see [52] and
reference therein).

A generalization of the construction in Proposition 1 is provided in the
Appendix for arbitrary number of parties. For $n$-partite case the
construction $\mathbb{G}_{n}$ resides in
$\mathbb{C}^{n+1}\otimes\left(\mathbb{C}^{3}\right)^{\otimes n-1}$ and the
$n$-partite GHZ state $\ket{g_{n}}:=(\ket{0^{\otimes n}}+\ket{1^{\otimes
n}})/\sqrt{2}$ turns out to be a minimal dimensional resource for prefect
discrimination of the set. This construction further generalizes Corollary 1.

######  Corollary 2.

The task $\tau_{n}$ of local discrimination of $\mathbb{G}_{n}$ induces the
order $\ket{g_{n}}\succ_{\tau_{n}}\rho$, for any
$\rho\in\mathcal{D}\left(\otimes_{j=1}^{n-1}\mathbb{C}^{d_{j}}\right)$ with
arbitrary local dimensions.

We now consider two tri-partite resources, namely, two copies of three-qubit
GHZ state, i.e.,
$\ket{\psi_{3}}=\ket{g_{3}}_{ABC}\otimes\ket{g_{3}}_{A^{\prime}B^{\prime}C^{\prime}}$
and three copies of (symmetrically distributed) two-qubit maximally entangled
states (EPR state), i.e.,
$\ket{\phi_{3}}=\ket{\phi^{+}}_{AB}\otimes\ket{\phi^{+}}_{B^{\prime}C}\otimes\ket{\phi^{+}}_{A^{\prime}C^{\prime}}$.
Both these states live in $\left(\mathbb{C}^{4}\right)^{\otimes 3}$ with Alice
holding the subsystem $A$ and $A^{\prime}$, Bob holding $B$ and $B^{\prime}$,
and Charlie holding $C$ and $C^{\prime}$. It is well known that two bipartite
pure entangled states have same amount of entanglement if their single party
marginals have identical spectral, and in such case the two states are local
unitary (LU) equivalent. However, the tripartite states $\ket{\psi_{3}}$ and
$\ket{\phi_{3}}$ are not LU equivalent even though they have identical single
party marginals. In fact these two resources are not LOCC comparable, i.e.,
under LOCC $\ket{\psi_{3}}\not\leftrightarrows\ket{\phi_{3}}$ [11]. Later it
has been shown that this incomparability also holds in the asymptotic level as
it is not possible to convert $2N$ three-party GHZ states into $3N$ EPR states
even in the limit $N\rightarrow\infty$ [53]. Our next result attributes a task
based ordering relation between these two resources.

######  Theorem 2.

There exists a tri-partite product LSD task $\tau_{3}^{\prime}$ such that
$\ket{\psi_{3}}\succcurlyeq_{\tau_{3}^{\prime}}\ket{\phi_{3}}$.

See the appendix for explicit description of the task $\tau_{3}^{\prime}$ and
for the proof. While proving this theorem we show that the defined task can be
perfectly done with the shared resource state $\ket{\psi_{3}}$. Furthermore,
under the known state discrimination protocols we show that $\ket{\phi_{3}}$
fails. However, a strict ordering relation222The protocol dependent nature of
argument for the inferior peformance of $\ket{\phi_{3}}$ as compared to
$\ket{\psi_{3}}$ in order to accomplish the task $\tau_{3}^{\prime}$ implies
that the former cannot be superior to the latter in the context of this task.
in the above theorem demands a protocol independent proof that
$\ket{\phi_{3}}$ is no good to accomplish $\tau_{3}^{\prime}$ perfectly, which
we leave here as an open question. We believe that a protocol independent
proof of Lemma 1 will provide useful insight to address the aforesaid open
problem. Interestingly, we come up with a different product LSD problem to
obtain a strict ordering relation between these two resources.

######  Theorem 3.

There exists a tripartite product LSD task $\tau_{3}^{\prime\prime}$ such that
$\ket{\psi_{3}}\succ_{\tau_{3}^{\prime\prime}}\ket{\phi_{3}}$.

###### Proof.

The task $\tau_{3}^{\prime\prime}$ considers the following LSD problem. A
state randomly chosen from the set $\mathbb{G}_{3}$ is given to Alice, Bob,
and Charlie for discrimination. Post successful discrimination another
randomly chosen state from $\mathbb{G}_{3}$ is given again for discrimination.
The second copy is chosen independent of the first one and hence knowledge of
the first copy does not provide any information about the second one. More
formally, the task $\tau_{3}^{\prime\prime}$ considers state discrimination of
the ordered tuple $\left(\ket{\zeta_{i}},\ket{\zeta_{j}}\right)$ chosen
randomly from $\mathbb{G}_{3}\times\mathbb{G}_{3}$ with uniform probability.
Here, the state $\ket{\zeta_{j}}$ is given after discrimination of the state
$\ket{\zeta_{i}}$. The argument in Proposition 1 clearly obliterates the
possibility of perfect success of $\tau_{3}^{\prime\prime}$ under LOCC. An
additional resource $\ket{\phi_{3}}$ also fails to achieve the desired
objective perfectly. The set $\mathbb{G}_{3}$ being a GNPS necessitates
consumption of at least two of the three symmetrically distributed EPR states
for perfect discrimination of the first element of the ordered pair
$\left(\ket{\zeta_{i}},\ket{\zeta_{j}}\right)$. Since identification of the
first element does not provide any information regarding the second, thus the
it can not be perfectly discriminated using the remaining one EPR state.
However, given the resource $\ket{\psi_{3}}$, i.e., symmetrically distributed
two copies of three qubit GHZ, the players can use the first and second copy
sequentially to perfectly identify $\ket{\zeta_{i}}$ and $\ket{\zeta_{j}}$
respectively. This is achieved by following the protocol supplied in Theorem
1. This completes the proof. ∎

At this point we want to emphasis that, unlike the ordering relation in Lemma
1, those in Corollary 1, Corollary 2, and Theorem 3 are protocol independent.
This demonstrates enormous useful implication of the notion GNPB/GNPS in
comparison of LOCC/SLOCC inequivalent multipartite entanglement.

##  4 Discussions

Comparison of the nonlocal strength of different entangled states through
their performance in some operational task is not new in literature. In fact,
the seminal no-go theorem of John S Bell is one such well studied programme
[54]. Bell scenario can be thought as a game involving spatially separated
multiple players. Success probability in these games with entangled resources
introduces a hierarchy of nonlocal content of different states [54, 55, 56,
57] as well as classifies the type of nonlocality in multipartite states [58,
59]. However, the crucial difference between Bell framework to the present one
lies in the operational paradigm. Bell nonlocality is a resource under
‘wirings and classical communication prior to the inputs’ [60]. Post-input
communication can enhance as well as change the type of nonlocality. Although
Bell scenario captures some nonlocal aspects of multipartite quantum states
(which finds useful applications in device-independent protocols [61, 62, 63,
64, 65]), its entanglement characteristics are more naturally captured in the
operational framework of LOCC, which has been adopted in the present work.

It is noteworthy that the ordering relation studied here is not absolute,
rather it is a relative ordering. If $\rho\succ_{\tau}\sigma$ with respect to
some task $\tau$, it is quite possible that there exists another task
$\tau^{\prime}$ for which the ordering gets reversed, i.e.,
$\sigma\succ_{\tau^{\prime}}\rho$. Similar situation occurs in Bell scenario
too. While the two-qubit maximally entangled state is the optimal resource for
a particular Bell game [56], in a different game some non maximally entangled
state becomes optimal [66, 67]. In fact, every two-qubit non-maximally
entangled state can be shown to be the optimal resource in some game [68, 69].
In regard to the present work, we also obtain a task
$\tau_{3}^{\prime\prime\prime}$ which imposes an opposite ordering relation
that of Theorem 2 between the states $\ket{\psi_{3}}$ and $\ket{\phi_{3}}$
(see the Appendix).

##  5 Conclusions and future perspectives

Our work provides a bona-fide method to compare the entanglement content in
different quantum states that otherwise are incomparable under the paradigm of
LOCC inter-convertibility. In particular, we have shown that the local state
discrimination problem turns out to be utilitarian in this regard. Unlike the
recent work of Ref. [18] our approach is operational as it is based on some
task. To this aim we have constructed several sets of multipartite GNPSs and
subsequently shown that entanglement assisted local discrimination protocols
of those sets induce legitimate ordering relations among LOCC incomparable
entangled states.

Our work welcomes further research regarding resource comparison of
multipartite entangled states through their performance in quantum state
discrimination problem(s). First of all, proving the conjecture
$\ket{g_{3}}\succ_{\tau_{3}}\ket{w_{3}}$ will certainly provide nontrivial
information about genuine resource content of GHZ and W class of states.
Obtaining some task that will reverse the ordering relation in Theorem 3 is
also very much welcome. We have considered only product LSD problems. However,
for the purpose of resource comparison this is not a necessary requirement.
One may consider general LSD problem that involves discrimination of entangled
states too. In fact, we believe that this general consideration will reveal a
much richer structure of quantum entanglement. For instance, the intriguing
entanglement catalyst discrimination phenomena as observed in [32] requires
this generic consideration. However, product LSD problem has some interesting
features of its own. For instance, appending suitably chosen product states
one can construct a GNPB from a GNPS. Such a GNPB results in multiparty
separable measurement that requires entangled resources across all
bipartitions. The sufficient entanglement required for discriminating the
GNPS/GNPB provides only a lower bound on the amount of entanglement required
for implementing the corresponding separable measurement. Finding necessary
and sufficient amount of entanglement for implementing them is itself an
interesting research question.

## Acknowledgments

We thankfully acknowledge the discussions with Guruprasad Kar and Arup Roy. We
would also like to thank Bihalan Bhattacharya for detailed presentation of the
work [18] at ‘Bi-monthly discussion series on Quantum Information and Quantum
Foundations’ at S. N. Bose National Center for Basic Sciences. Part of the
work has been done when MB was a INSPIRE Faculty at S. N. Bose National Center
for Basic Sciences. MB acknowledges support through INSPIRE Faculty fellowship
by the Department of Science and Technology, Government of India.

## Appendix

##  Appendix A Proof of Theorem 1

###### Proof.

We associate block letter party index with the states that need to be
distinguished and denote the resource state as
$\ket{g_{3}}_{abc}=(\ket{000}_{abc}+\ket{111}_{abc})/\sqrt{2}$. Local
distinguishability of the set $\mathbb{G}_{3}$ boils down to identify the
pairs $\left\\{\ket{\zeta_{\pm}}\right\\}$ preserving the post-measurement
orthogonality between $\ket{\zeta_{+}}$ and $\ket{\zeta_{-}}$, as the result
of [21] assures local distinguishability between any two orthogonal states.
The protocol proceeds as follows.

Step-1: Alice performs the measurement
$\mathcal{M}\equiv\\{M,\mathbb{I}-M\\}$, where

$M:=\mathbb{P}\left[\ket{p}_{A};\ket{0}_{a}\right]+\mathbb{P}\left[\left(\ket{q},\ket{1},\ket{2}\right)_{A};\ket{1}_{a}\right].$

Here we use the notation
$\mathbb{P}\left[\left(\ket{e},\ket{f},\cdots\right)_{K};\left(\ket{x},\ket{y},\cdots\right)_{k}\right]:=\left(\ket{e}\bra{e}+\ket{f}\bra{f}+\cdots\right)_{K}\otimes\left(\ket{x}\bra{x}+\ket{y}\bra{y}+\cdots\right)_{k}$.
Suppose the projector $M$ clicks. The state
$\ket{\zeta}_{ABC}\otimes\ket{g}_{abc}$ evolves to either
$\ket{\zeta}_{ABC}\otimes\ket{000}_{abc}$, or
$\ket{\zeta}_{ABC}\otimes\ket{111}_{abc}$, or it becomes entangled, where
$\ket{\zeta}_{ABC}\in\mathbb{G}_{3}$. Complete list of the evolved states are
given below,

$\displaystyle\rotatebox[origin={c}]{0.0}{}\left\\{\\!\begin{aligned}
\left\\{\ket{\zeta^{1,3}_{\pm}},\ket{\zeta^{2,3}_{\pm}}\right\\}_{ABC}\otimes\ket{000}_{abc},\\\
\left\\{\ket{\zeta^{1,1}_{\pm}},\ket{\zeta^{1,2}_{\pm}},\ket{\zeta^{2,1}_{\pm}},\ket{\zeta^{2,2}_{\pm}}\right\\}_{ABC}\otimes\ket{111}_{abc},\\\
\ket{\zeta_{\pm}^{0}}_{ABC}\Rightarrow\left(\ket{p}_{A}\ket{000}_{abc}\pm\ket{q}_{A}\ket{111}_{abc}\right)\ket{1}_{B}\ket{2}_{C}\end{aligned}\right\\},$

Step-2: Bob and Charlie respectively perform the measurement,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{K}\equiv$}\left\\{\\!\begin{aligned}
K_{1}:=\mathbb{I}-K_{2}-K_{3},~{}K_{2}:=\mathbb{P}\left[\ket{p}_{B};\ket{1}_{b}\right],~{}\\\
K_{3}:=\mathbb{P}\left[\left(\ket{p},\ket{q}\right)_{B};\ket{0}_{b}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\},$
$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{N}\equiv$}\left\\{\\!\begin{aligned}
N_{1}:=\mathbb{I}-N_{2}-N_{3},~{}N_{2}:=\mathbb{P}\left[\ket{p}_{C};\ket{1}_{c}\right],~{}\\\
N_{3}:=\mathbb{P}\left[\left(\ket{p},\ket{q}\right)_{C};\ket{0}_{c}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

If $K_{3}$ clicks the state is one of
$\left\\{\ket{\zeta^{1,3}_{\pm}}\right\\}$, if $K_{2}$ clicks the state is one
of $\left\\{\ket{\zeta^{1,2}_{\pm}}\right\\}$, if $N_{3}$ clicks the state is
one of $\left\\{\ket{\zeta^{2,3}_{\pm}}\right\\}$, if $N_{2}$ clicks the state
is one of $\left\\{\ket{\zeta^{2,2}_{\pm}}\right\\}$. When both $K_{1}$ and
$N_{1}$ click the state is one of
$\left\\{\ket{\zeta_{\pm}^{0}},\ket{\zeta^{1,1}_{\pm}},\ket{\zeta^{2,1}_{\pm}}\right\\}$.
Obtaining the outcome results from Bob and Charlie Alice performs the
measurement,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{M}^{\prime}\equiv$}\left\\{\\!\begin{aligned}
M^{\prime}_{1}:=\mathbb{P}\left[\ket{1}_{A};\mathbb{I}_{a}\right],~{}M^{\prime}_{2}:=\mathbb{P}\left[\ket{2}_{A};\mathbb{I}_{a}\right],\\\
M^{\prime}_{0}:=\mathbb{I}-M^{\prime}_{1}-M^{\prime}_{2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

If $M^{\prime}_{1}$ clicks the state is one of
$\left\\{\ket{\zeta^{1,1}_{\pm}}\right\\}$, if $M^{\prime}_{2}$ clicks the
state is one of $\left\\{\ket{\zeta^{2,1}_{\pm}}\right\\}$, else it is one of
$\left\\{\ket{\zeta_{\pm}^{0}}\right\\}$. If $\mathbb{I}-M$ clicks in Step-1 a
similar protocol will follow. ∎

##  Appendix B Construction of GNPS for arbitrary party

######  Proposition 2.

Consider the set of states
$\mathbb{G}_{m+1}\equiv\left\\{\ket{\zeta_{\pm}^{0}},\ket{\zeta_{\pm}^{i,1}},\ket{\zeta_{\pm}^{i,2}},\ket{\zeta_{\pm}^{i,3}}\right\\}_{i=1}^{m}$
given in the Table 1. This set is a GNPS in
$\mathbb{C}^{m+2}\otimes\left(\mathbb{C}^{3}\right)^{\otimes m}$ with Alice
holding the subsystem $\mathbb{C}^{m+2}$ and each Bob holding a qutrit
subsystem.

Alice | Bob-1 | Bob-2 | $\cdots$ | Bob-m |
---|---|---|---|---|---
$\ket{\epsilon_{\pm}}$ | $\ket{1}$ | $\ket{2}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{0}}$
$\ket{1}$ | $\ket{\gamma_{\pm}^{1}}$ | $\ket{2}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{1,1}}$
$\ket{\gamma_{\pm}^{1}}$ | $\ket{p}$ | $\ket{2}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{1,2}}$
$\ket{p}$ | $\ket{\epsilon_{\pm}}$ | $\ket{2}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{1,3}}$
$\ket{2}$ | $\ket{1}$ | $\ket{\gamma_{\pm}^{2}}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{2,1}}$
$\ket{\gamma_{\pm}^{2}}$ | $\ket{1}$ | $\ket{p}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{2,2}}$
$\ket{p}$ | $\ket{1}$ | $\ket{\epsilon_{\pm}}$ | $\cdots$ | $\ket{m}$ | $\Rightarrow\ket{\zeta_{\pm}^{2,3}}$
|  |  | $\vdots$ |  |
$\ket{m}$ | $\ket{1}$ | $\ket{2}$ | $\cdots$ | $\ket{\gamma_{\pm}^{m}}$ | $\Rightarrow\ket{\zeta_{\pm}^{m,1}}$
$\ket{\gamma_{\pm}^{m}}$ | $\ket{1}$ | $\ket{2}$ | $\cdots$ | $\ket{p}$ | $\Rightarrow\ket{\zeta_{\pm}^{m,2}}$
$\ket{p}$ | $\ket{1}$ | $\ket{2}$ | $\cdots$ | $\ket{\epsilon_{\pm}}$ | $\Rightarrow\ket{\zeta_{\pm}^{m,3}}$
Table 1: Set of states $\mathbb{G}_{m+1}$. Here
$\ket{\epsilon_{\pm}}:=\frac{1}{\sqrt{2}}(\ket{p}\pm\ket{q})$,
$\ket{\gamma_{\pm}^{i}}:=\frac{1}{\sqrt{2}}(\ket{q}\pm\ket{i})$ and
$\braket{i}{j}=\delta_{ij}$ with $i,j\in\\{p,q,1,\cdots,m\\}$

###### Proof.

Consider the subset of states
$\$^{i}\equiv\left\\{\ket{\zeta_{\pm}^{0}},\ket{\zeta^{i,1}_{\pm}},\ket{\zeta^{i,2}_{\pm}},\ket{\zeta^{i,3}_{\pm}}\right\\}\subset\mathbb{G}_{m+1}$,
for $i\in\\{1,\cdots,m\\}$. The set $\$^{i}$ has similar structure as of the
set $\mathbb{S}_{Ben}$ between Alice and $i^{th}$ Bob while other Bobs have
fixed states tagged with this set. This fact along with Observation 1 and
Observation 2 assure that the set $\mathbb{G}_{m+1}$ cannot be locally
discriminated even if any $m$ parties come together. ∎

######  Proposition 3.

The set of states $\mathbb{G}_{m+1}$ can be perfectly discriminated locally
with the resource state $\ket{g_{m+1}}_{ab_{1}\cdots
b_{m}}=\frac{1}{\sqrt{2}}\left(\ket{0^{\otimes m+1}}+\ket{1^{\otimes
m+1}}\right)_{ab_{1}\cdots b_{m}}$.

###### Proof.

In Step-1 Alice performs the measurement
$\mathcal{M}\equiv\\{M,\mathbb{I}-M\\}$, where

$\displaystyle
M:=\mathbb{P}\left[\ket{p}_{A};\ket{0}_{a}\right]+\mathbb{P}\left[\left(\ket{q},\ket{1},\cdots,\ket{m}\right)_{A};\ket{1}_{a}\right].$

The evolved states are given by,

$\displaystyle\rotatebox[origin={c}]{0.0}{}\left\\{\\!\begin{aligned}
\left\\{\ket{\zeta^{1,3}_{\pm}},\cdots,\ket{\zeta^{m,3}_{\pm}}\right\\}\otimes\ket{0^{\otimes
m+1}}_{ab_{1}\cdots b_{m}},\\\
\left\\{\ket{\zeta^{1,1}_{\pm}},\ket{\zeta^{1,2}_{\pm}},\cdots,\ket{\zeta^{m,1}_{\pm}},\ket{\zeta^{m,2}_{\pm}}\right\\}\otimes\ket{1^{\otimes
m+1}}_{ab_{1}\cdots b_{m}},\\\
\ket{\zeta_{\pm}^{0}}\Rightarrow\ket{\tilde{\zeta}^{0}_{\pm}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\},$

where,

$\displaystyle\ket{\tilde{\zeta}^{0}_{\pm}}$ $\displaystyle=$
$\displaystyle\left(\ket{p}_{A}\ket{0^{\otimes m+1}}_{ab_{1}\cdots
b_{m}}\pm\ket{q}_{A}\ket{1^{\otimes m+1}}_{ab_{1}\cdots
b_{m}}\right)\otimes\ket{1}_{B_{1}}\cdots\ket{m}_{B_{m}}.$ (2)

Step-2: $i^{th}$ Bob performs the similar measurement as in Theorem 1. If
$K^{i}_{3}$ clicks the state is one of
$\left\\{\ket{\zeta^{i,3}_{\pm}}\right\\}$, if $K^{i}_{2}$ clicks the state is
one of $\left\\{\ket{\zeta^{i,2}_{\pm}}\right\\}$, if all $K^{i}_{1}$’s click
the state is one of
$\left\\{\ket{\zeta_{\pm}^{0}},\ket{\zeta^{i,1}_{\pm}}\right\\}_{i=1}^{m}$.
Alice then performs the measurement,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{M}^{\prime}\equiv$}\left\\{\\!\begin{aligned}
M^{\prime}_{1}:=\mathbb{P}\left[\ket{1}_{A};\mathbb{I}_{a}\right],\cdots
M^{\prime}_{m}:=\mathbb{P}\left[\ket{m}_{A};\mathbb{I}_{a}\right],\\\
M^{\prime}_{0}:=\mathbb{I}-\left(M^{\prime}_{1}+\cdots+M^{\prime}_{2}\right).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

If $M^{\prime}_{i}$ clicks the state is one of
$\left\\{\ket{\zeta^{i,1}_{\pm}}\right\\}$, else it is one of
$\left\\{\ket{\zeta_{\pm}^{0}}\right\\}$. ∎

Proposition 2 implies that the task $\tau_{m+1}$ of distinguishing the set
$\mathbb{G}_{m+1}$ can not be locally accomplished even when arbitrary $m$
parties come at one location. This fact together with Proposition 3 imply the
ordering relation $\ket{g_{m+1}}\succ_{\tau_{m+1}}\rho$, for any
$\rho\in\mathcal{D}\left(\otimes_{j=1}^{m}\mathbb{C}^{d_{j}}\right)$ with
arbitrary local dimensions (Corollary 2).

##  Appendix C Proof of Theorem 2

The NPS $\mathbb{S}_{Ben}\subset\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ can be
expressed in the following generic form,

$\mathbb{S}_{g}^{3}\equiv\left\\{\ket{a}\ket{s_{\pm}},\ket{s_{\pm}}\ket{c},\ket{c}\ket{t_{\pm}},\ket{t_{\pm}}\ket{a}\right\\},$
(3)

where $\\{\ket{a},\ket{b},\ket{c}\\}$ are pairwise orthonormal states and
$\ket{s_{\pm}}:=\frac{1}{\sqrt{2}}\left(\ket{a}\pm\ket{b}\right)$ and
$\ket{t_{\pm}}:=\frac{1}{\sqrt{2}}\left(\ket{b}\pm\ket{c}\right)$. A
generalization of $\mathbb{S}_{g}^{3}$ in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ is given by,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathbb{S}_{g}^{5}\equiv$}\left\\{\\!\begin{aligned}
\ket{\Gamma^{1}_{\pm}}:=\ket{a}\ket{s_{\pm}},~{}~{}\ket{\Gamma^{2}_{\pm}}:=\ket{s_{\pm}}\ket{c},~{}~{}\\\
\ket{\Gamma^{3}_{\pm}}:=\ket{c}\ket{t_{\pm}},~{}~{}\ket{\Gamma^{4}_{\pm}}:=\ket{t_{\pm}}\ket{a},~{}~{}\\\
\ket{\Gamma^{5}_{ijk}}:=\ket{d}\ket{u_{ijk}},\ket{\Gamma^{6}_{ijk}}:=\ket{u_{ijk}}\ket{e},\\\
\ket{\Gamma^{7}_{ijk}}:=\ket{e}\ket{v_{ijk}},\ket{\Gamma^{8}_{ijk}}:=\ket{v_{ijk}}\ket{d}~{}~{}\end{aligned}\right\\},$
(4)

where $\\{\ket{a},\ket{b},\ket{c},\ket{d},\ket{e}\\}$ is an orthonormal basis
of $\mathbb{C}^{5}$ and $\ket{u_{ijk}}\in\mathcal{S}_{abcd}$ and
$\ket{v_{ijk}}\in\mathcal{S}_{abce}$, with

$\displaystyle\rotatebox[origin={c}]{0.0}{$\bar{\mathcal{S}}_{\alpha\beta\delta\gamma}\equiv$}\left\\{\\!\begin{aligned}
\ket{\alpha}+(-1)^{i}\ket{\beta}+(-1)^{j}\ket{\delta}+(-1)^{k}\ket{\gamma},\\\
\mbox{with}~{}i,j,k\in\\{0,1\\}~{}{and}~{}i\oplus_{2}j\oplus_{2}k=0\end{aligned}\right\\}.$

$\bar{\mathcal{S}}_{\alpha\beta\delta\gamma}$ contains the unnormalized states
of $\mathcal{S}_{\alpha\beta\delta\gamma}$. The NPS $\mathbb{S}_{g}^{5}$ has a
layered tile structure (see Fig. 1). This has been recently studied to
understand the intricate geometrical structure of the set of bipartite states
having positive partial transpose, i.e., the Peres set [37]. Furthermore, from
Ref.[70] it is evident that the set (4) can be locally distinguished if a
$2$-qutrit maximally entangled state is shared as resource. Note that the
protocol in [70] is resource efficient compared to the teleportation based
protocol as the later requires a maximally entangled state of
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$

Consider now the following set of states in
$\mathbb{C}^{6}_{A}\otimes\mathbb{C}^{5}_{B_{1}}\otimes\mathbb{C}^{5}_{B_{2}}$,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\Sigma\equiv$}\left\\{\\!\begin{aligned}
\ket{\Omega^{1}_{\pm}}&:=\ket{1}\ket{\alpha_{\pm}}\ket{4},~{}\ket{\Omega^{2}_{\pm}}:=\ket{\alpha_{\pm}}\ket{3}\ket{4},\\\
\ket{\Omega^{3}_{\pm}}&:=\ket{3}\ket{\beta_{\pm}}\ket{4},~{}\ket{\Omega^{4}_{\pm}}:=\ket{\beta_{\pm}}\ket{1}\ket{4},\\\
\ket{\Omega^{5}_{\pm}}&:=\ket{4}\ket{3}\ket{\gamma_{\pm}},~{}\ket{\Omega^{6}_{\pm}}:=\ket{\gamma_{\pm}}\ket{3}\ket{1},\\\
\ket{\Omega^{7}_{\pm}}&:=\ket{1}\ket{3}\ket{\alpha_{\pm}},~{}\ket{\Omega^{8}_{ijk}}:=\ket{0}\ket{\Psi_{ijk}}\ket{3},~{}\\\
\ket{\Omega^{9}_{ijk}}&:=\ket{\Psi_{ijk}}\ket{4}\ket{3},\ket{\Omega^{10}_{ijk}}:=\ket{4}\ket{\Phi_{ijk}}\ket{3},\\\
\ket{\Omega^{11}_{ijk}}&:=\ket{\Phi_{ijk}}\ket{0}\ket{3},\ket{\Omega^{12}_{ijk}}:=\ket{5}\ket{0}\ket{\Phi_{ijk}},\\\
\ket{\Omega^{13}_{ijk}}&:=\ket{\varUpsilon_{ijk}}\ket{0}\ket{5},\ket{\Omega^{14}_{ijk}}:=\ket{3}\ket{0}\ket{\varUpsilon_{ijk}}\end{aligned}\right\\},$
(5)

where $\ket{\alpha_{\pm}}:=\frac{1}{\sqrt{2}}\ket{1\pm
2},\ket{\beta_{\pm}}:=\frac{1}{\sqrt{2}}\ket{2\pm
3},\ket{\gamma_{\pm}}:=\frac{1}{\sqrt{2}}\ket{2\pm 4}$ and
$\ket{\Psi_{ijk}}\in\mathcal{S}_{0123},~{}\ket{\Phi_{ijk}}\in\mathcal{S}_{1234}~{}{and}~{}\ket{\varUpsilon_{ijk}}\in\mathcal{S}_{1245}$.

Figure 1: Tile structure of the set $\mathbb{S}_{g}^{5}$. Cardinality of the
set is $24$. Each inner layered tile contains $2$ mutually orthonormal states,
while each outer layered tile contains $4$ mutually orthonormal states.
Orthogonality among the states from different tiles be evident from the
structure.

Figure 2: [Color on-line] Tile structure of the set $\$_{AB_{1}}$ (left) and
$\$_{AB_{2}}$ (right). With all the states in outer layered (grey) tiles of
$\$_{AB_{1}}$ Bob-2’s state is $\ket{4}_{B_{2}}$, while for inner layer (blue)
his state is $\ket{3}_{B_{2}}$. In $\$_{AB_{2}}$, Bob-1’s state tagged with
outer layer is $\ket{0}_{B_{1}}$ and for inner layer it is $\ket{3}_{B_{1}}$.

Before proceeding further, let us first analyze the structure of the set
$\Sigma$. The subset
$\$_{AB_{1}}\equiv\left\\{\ket{\Omega^{1}_{\pm}},\ket{\Omega^{2}_{\pm}},\ket{\Omega^{3}_{\pm}},\ket{\Omega^{4}_{\pm}},\ket{\Omega^{8}_{ijk}},\ket{\Omega^{9}_{ijk}},\ket{\Omega^{10}_{ijk}},\ket{\Omega^{11}_{ijk}}\right\\}$
has a kind of analogous structure as of (4) between Alice and Bob-1 (see
Fig.2). Please note here an important point: Bob-2 has the state
$\ket{4}_{B_{2}}$ tagged with
$\\{\ket{\Omega^{1}_{\pm}},\ket{\Omega^{2}_{\pm}},\ket{\Omega^{3}_{\pm}},\ket{\Omega^{4}_{\pm}}\\}$,
while with
$\\{\ket{\Omega^{8}_{ijk}},\ket{\Omega^{9}_{ijk}},\ket{\Omega^{10}_{ijk}},\ket{\Omega^{11}_{ijk}}\\}$
Bob-2’s state $\ket{3}_{B_{2}}$ is tagged.
Similarly,
$\$_{AB_{2}}\equiv\left\\{\ket{\Omega^{2}_{\pm}},\ket{\Omega^{5}_{\pm}},\ket{\Omega^{6}_{\pm}},\ket{\Omega^{7}_{\pm}},\ket{\Omega^{11}_{ijk}},\ket{\Omega^{12}_{ijk}},\ket{\Omega^{13}_{ijk}},\ket{\Omega^{14}_{ijk}}\right\\}$
has a kind of similar structure as of (4) between Alice and Bob-2 with Bob-1
having the tagged state $\ket{3}_{B_{1}}$ with
$\\{\ket{\Omega^{2}_{\pm}},\ket{\Omega^{5}_{\pm}},\ket{\Omega^{6}_{\pm}},\ket{\Omega^{7}_{\pm}}\\}$
and having the tagged state $\ket{0}_{B_{1}}$ with
$\\{\ket{\Omega^{11}_{ijk}},\ket{\Omega^{12}_{ijk}},\ket{\Omega^{13}_{ijk}},\ket{\Omega^{14}_{ijk}}\\}$.
This structure, along with the Observations (1) and (2), leads us to the
following proposition.

######  Proposition 4.

The set of states $\Sigma$ is a GNPS in
$\mathbb{C}^{6}\otimes\mathbb{C}^{5}\otimes\mathbb{C}^{5}$.

This Proposition assures that for perfect local discrimination of the set
$\Sigma$ requires multipartite entanglement resource, and the task
$\tau_{3}^{\prime}$ in Theorem 2 is the local discrimination of the set
$\Sigma$. To prove Theorem 2 we first prove the following proposition.

######  Proposition 5.

Two copies of $3$-qubit GHZ states are sufficient for perfect local
discrimination of the set $\Sigma$.

###### Proof.

Two copies of $3$-qubit GHZ state is local unitary equivalent to one copy of
$3$-ququad GHZ state
$\ket{g(4)}_{ab_{1}b_{2}}:=\frac{1}{2}(\ket{000}+\ket{111}+\ket{222}+\ket{333})_{ab_{1}b_{2}}\in\mathbb{C}^{4}_{a}\otimes\mathbb{C}^{4}_{b_{1}}\otimes\mathbb{C}^{4}_{b_{2}}$.
Thus the problem reduced to distinguish the state
$\ket{\Omega}_{AB_{1}B_{2}}\in\Sigma$ under LOCC given the resource state
$\ket{g(4)}_{ab_{1}b_{2}}$. The protocol goes as follows:

Step-1: Alice performs the measurement,
$\mathcal{M}\equiv\left\\{M_{1},M_{2},M_{3},M_{4}\right\\}$, where

$\displaystyle M_{1}$
$\displaystyle:=\mathbb{P}\left[\left(\ket{0},\ket{5}\right)_{A};\ket{0}_{a}\right]+\mathbb{P}\left[\left(\ket{1},\ket{2}\right)_{A};\ket{1}_{a}\right]+\mathbb{P}\left[\left(\ket{3}\right)_{A};\ket{2}_{a}\right]+\mathbb{P}\left[\left(\ket{4}\right)_{A};\ket{3}_{a}\right],$
$\displaystyle M_{2}$
$\displaystyle:=\mathbb{P}\left[\left(\ket{0},\ket{5}\right)_{A};\ket{1}_{a}\right]+\mathbb{P}\left[\left(\ket{1},\ket{2}\right)_{A};\ket{2}_{a}\right]+\mathbb{P}\left[\left(\ket{3}\right)_{A};\ket{3}_{a}\right]+\mathbb{P}\left[\left(\ket{4}\right)_{A};\ket{0}_{a}\right],$
$\displaystyle M_{3}$
$\displaystyle:=\mathbb{P}\left[\left(\ket{0},\ket{5}\right)_{A};\ket{2}_{a}\right]+\mathbb{P}\left[\left(\ket{1},\ket{2}\right)_{A};\ket{3}_{a}\right]+\mathbb{P}\left[\left(\ket{3}\right)_{A};\ket{0}_{a}\right]+\mathbb{P}\left[\left(\ket{4}\right)_{A};\ket{1}_{a}\right],$
$\displaystyle M_{4}$
$\displaystyle:=\mathbb{P}\left[\left(\ket{0},\ket{5}\right)_{A};\ket{3}_{a}\right]+\mathbb{P}\left[\left(\ket{1},\ket{2}\right)_{A};\ket{0}_{a}\right]+\mathbb{P}\left[\left(\ket{3}\right)_{A};\ket{1}_{a}\right]+\mathbb{P}\left[\left(\ket{4}\right)_{A};\ket{2}_{a}\right].$

Suppose that outcome corresponding to the projector $M_{1}$ clicks. The
composite state evolves as,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\Sigma^{\prime}\equiv$}\left\\{\\!\begin{aligned}
&\left\\{\ket{\Omega^{8}_{ijk}},\ket{\Omega^{12}_{ijk}}\right\\}_{AB_{1}B_{2}}\otimes\ket{000}_{ab_{1}b_{2}},~{}~{}~{}~{}\left\\{\ket{\Omega^{1}_{\pm}},\ket{\Omega^{2}_{\pm}},\ket{\Omega^{7}_{\pm}}\right\\}_{AB_{1}B_{2}}\otimes\ket{111}_{ab_{1}b_{2}},\\\
&\left\\{\ket{\Omega^{3}_{\pm}},\ket{\Omega^{14}_{ijk}}\right\\}_{AB_{1}B_{2}}\otimes\ket{222}_{ab_{1}b_{2}},~{}~{}~{}\left\\{\ket{\Omega^{5}_{\pm}},\ket{\Omega^{10}_{ijk}}\right\\}_{AB_{1}B_{2}}\otimes\ket{333}_{ab_{1}b_{2}},\\\
&\ket{\Omega^{4}_{\pm}}_{AB_{1}B_{2}}\otimes\ket{g(4)}_{ab_{1}b_{2}}\Rightarrow\left(\ket{2}_{A}\ket{111}_{ab_{1}b_{2}}\pm\ket{3}_{A}\ket{222}_{ab_{1}b_{2}}\right)\ket{1}_{B_{1}}\ket{4}_{B_{2}},\\\
&\ket{\Omega^{6}_{\pm}}_{AB_{1}B_{2}}\otimes\ket{g(4)}_{ab_{1}b_{2}}\Rightarrow\left(\ket{2}_{A}\ket{111}_{ab_{1}b_{2}}\pm\ket{4}_{A}\ket{333}_{ab_{1}b_{2}}\right)\ket{3}_{B_{1}}\ket{1}_{B_{2}},\\\
&\ket{\Omega^{9}_{ijk}}\otimes\ket{g(4)}\Rightarrow\ket{\mu},\\\
&\ket{\Omega^{11}_{ijk}}\otimes\ket{g(4)}\Rightarrow\ket{\nu},\\\
&\ket{\Omega^{13}_{ijk}}\otimes\ket{g(4)}\Rightarrow\ket{\lambda}\end{aligned}\right\\}.$
(6)

where,

$\ket{\mu}:=\left(\ket{0}_{A}\ket{000}_{ab_{1}b_{2}}+\ket{(-1)^{i}1+(-1)^{j}2}_{A}\ket{111}_{ab_{1}b_{2}}+(-1)^{k}\ket{3}_{A}\ket{222}_{ab_{1}b_{2}}\right)\ket{4}_{B_{1}}\ket{3}_{B_{2}},$
$\ket{\nu}:=\left(\ket{1+(-1)^{i}2}_{A}\ket{111}_{ab_{1}b_{2}}+(-1)^{j}\ket{3}_{A}\ket{222}_{ab_{1}b_{2}}+(-1)^{k}\ket{4}_{A}\ket{333}_{ab_{1}b_{2}}\right)\ket{0}_{B_{1}}\ket{3}_{B_{2}},$
$\ket{\lambda}:=\left(\ket{1+(-1)^{i}2}_{A}\ket{111}_{ab_{1}b_{2}}+(-1)^{j}\ket{4}_{A}\ket{333}_{ab_{1}b_{2}}+(-1)^{k}\ket{5}_{A}\ket{000}_{ab_{1}b_{2}}\right)\ket{0}_{B_{1}}\ket{5}_{B_{2}}.$

Note that, the post-measurement evolved states in (6) are mutually orthogonal
and this is a necessary requirement for further perusing the discriminating
protocol. In the subsequent steps also either the given states is identified
or some states are eliminated keeping mutual orthogonality among the remaining
states. In fact, the form of the states remain identical as of (6).

Step-2: Bob-1 first performs the measurement

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{K}^{(1)}\equiv$}\left\\{\\!\begin{aligned}
K_{1}^{(1)}&:=\mathbb{P}\left[(\ket{2},\ket{3})_{B_{1}};\ket{2}_{b_{1}}\right],\\\
K_{2}^{(1)}&:=\mathbb{I}-K_{1}^{(1)}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

If $K_{1}^{(1)}$ he concludes the state as one of
$\left\\{\ket{\Omega^{3}_{\pm}}\right\\}$ and they can be further
distinguished by LOCC [21], otherwise the state is one of the remaining states
and Bob-1 performs the measurement

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{K}^{(2)}\equiv$}\left\\{\\!\begin{aligned}
K^{(2)}_{1}&:=\mathbb{P}\left[(\ket{1},\ket{2})_{B_{1}};(\ket{1},\ket{2})_{b_{1}}\right],\\\
K^{(2)}_{2}&:=\mathbb{I}-K^{(2)}_{1}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

Note that the measurements $\mathcal{K}^{(2)}$ does not commute with
$\mathcal{K}^{(1)}$ and thus cannot be performed simultaneously in a single
step with $\mathcal{K}^{(1)}$. If $K^{(2)}_{1}$ clicks the state is one of
$\left\\{\ket{\Omega^{1}_{\pm}},\ket{\Omega^{4}_{\pm}}\right\\}$. To
distinguish them, Alice performs the measurement
$\\{\mathbb{P}\left[\ket{1}_{A};\mathbb{I}_{a}\right],~{}\mathbb{I}-\mathbb{P}\left[\ket{1}_{A};\mathbb{I}_{a}\right]\\}$.
If the first projector clicks the state is one of
$\left\\{\ket{\Omega^{1}_{\pm}}\right\\}$, else it is one of
$\left\\{\ket{\Omega^{4}_{\pm}}\right\\}$. If $K^{(2)}_{2}$ clicks the state
is one of the remaining states on which Bob performs another nontrivial
measurement

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{K}^{(3)}\equiv$}\left\\{\\!\begin{aligned}
K^{(3)}_{1}&:=\mathbb{P}\left[\ket{4}_{B_{1}};(\ket{0},\ket{1},\ket{2})_{b_{1}}\right],\\\
K^{(3)}_{2}&:=\mathbb{I}-K^{(3)}_{1}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

If $K^{(3)}_{1}$ clicks the state is one of
$\left\\{\ket{\Omega^{9}_{ijk}}\right\\}$ and to distinguish them Alice
performs a measurement in $\\{\ket{\Psi_{ijk}}\\}$ basis on her ‘A’ subsystem;
otherwise the state is one of remaining states, and Bob performs another
(nontrivial) measurement

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{K}^{(4)}\equiv$}\left\\{\\!\begin{aligned}
K^{(4)}_{1}&:=\mathbb{P}\left[(\ket{1},\ket{2},\ket{3},\ket{4})_{B_{1}};(\ket{1},\ket{2},\ket{3})_{b_{1}}\right],\\\
K^{(4)}_{2}&:=\mathbb{I}-K^{(4)}_{1}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

If $K^{(4)}_{1}$ clicks the state is one from the set
$\left\\{\ket{\Omega^{2}_{\pm}},\ket{\Omega^{5}_{\pm}},\ket{\Omega^{6}_{\pm}},\ket{\Omega^{7}_{\pm}},\ket{\Omega^{10}_{ijk}}\right\\}$,
otherwise it belongs to the set
$\left\\{\ket{\Omega^{8}_{ijk}},\ket{\Omega^{12}_{ijk}},\ket{\Omega^{11}_{ijk}},\ket{\Omega^{13}_{ijk}},\ket{\Omega^{14}_{ijk}}\right\\}$.

If $K^{(4)}_{1}$ clicks, then Bob-2 performs the measurement,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{N}^{(1)}\equiv$}\left\\{\\!\begin{aligned}
N^{(1)}_{1}&:=\mathbb{P}\left[(\ket{2},\ket{4})_{B_{2}};\ket{3}_{b_{2}}\right],\\\
N^{(1)}_{2}&:=\mathbb{P}\left[\ket{3}_{B_{2}};\ket{3}_{b_{2}}\right],\\\
N^{(1)}_{3}&:=\mathbb{P}\left[\ket{4}_{B_{2}};\ket{1}_{b_{2}}\right],\\\
N^{(1)}_{4}&:=\mathbb{I}-N^{(1)}_{1}-N^{(1)}_{2}-N^{(1)}_{3}\end{aligned}\right\\}.$

If $N^{(1)}_{1}$ the state is one of
$\left\\{\ket{\Omega^{5}_{\pm}}\right\\}$, if $N^{(1)}_{2}$ the state is one
of $\left\\{\ket{\Omega^{10}_{ijk}}\right\\}$ (locally distinguishable by
Bob-1 by performing $\left\\{\ket{\Phi_{ijk}}\right\\}$ basis measurement), if
$N^{(1)}_{3}$ the state is one of $\left\\{\ket{\Omega^{2}_{\pm}}\right\\}$,
else it is one of
$\left\\{\ket{\Omega^{6}_{\pm}},\ket{\Omega^{7}_{\pm}}\right\\}$, which can be
further grouped into $\left\\{\ket{\Omega^{6}_{\pm}}\right\\}$ and
$\left\\{\ket{\Omega^{7}_{\pm}}\right\\}$ locally by Alice.

If $K^{(4)}_{2}$ clicks, then Bob-2 performs the measurement,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{N}^{(2)}\equiv$}\left\\{\\!\begin{aligned}
N^{(2)}_{1}&:=\mathbb{P}\left[(\ket{1},\ket{2},\ket{4},\ket{5})_{B_{2}};\ket{2}_{b_{2}}\right],\\\
N^{(2)}_{2}&:=\mathbb{P}\left[(\ket{1},\ket{2},\ket{3},\ket{4})_{B_{2}};\ket{0}_{b_{2}}\right],\\\
N^{(2)}_{3}&:=\mathbb{I}-N^{(2)}_{1}-N^{(2)}_{2}\end{aligned}\right\\}.$

If $N^{(2)}_{1}$ the state is one of
$\left\\{\ket{\Omega^{14}_{ijk}}\right\\}$, if $N^{(2)}_{2}$ the state is one
of $\left\\{\ket{\Omega^{8}_{ijk}},\ket{\Omega^{12}_{ijk}}\right\\}$ (can be
further grouped into $\left\\{\ket{\Omega^{8}_{ijk}}\right\\}$ and
$\left\\{\ket{\Omega^{12}_{ijk}}\right\\}$ locally by Alice), else it is one
of $\left\\{\ket{\Omega^{11}_{ijk}},\ket{\Omega^{13}_{ijk}}\right\\}$ which
can further be grouped into $\left\\{\ket{\Omega^{11}_{ijk}}\right\\}$ and
$\left\\{\ket{\Omega^{13}_{ijk}}\right\\}$ locally by Bob-2.

In Step-1, instead of $M_{1}$ if some other $M_{i}$ is clicked a similar
procedure follows in Step-2 which completes the proof. ∎

One can raise the question whether the resource state in Theorem-2 be optimal
for local discrimination of the corresponding GNPS. Although this remains an
open question, however, we are able to come up with the following observation.

######  Observation 3.

Three copies of $2$-qubit EPR state
$\left(\frac{\ket{00}+\ket{11}}{\sqrt{2}}\right)^{\otimes
3}\in\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ symmetrically
distributed among Alice, Bob-1, and Bob-2 fail to perfectly discriminate the
set $\Sigma$ under local protocols.

Figure 3: Red lines indicate the twist breaking steps by Bob-1. For instance,
measurement for $L_{1}$ line is given by
$\mathcal{M}^{L_{1}}\equiv\left\\{M^{L_{1}},\mathbb{I}-M^{L_{1}}\right\\}$
where,
$M^{L_{1}}:=\mathbb{P}\left[\left(\ket{1},\ket{2},\ket{3},\ket{4}\right)_{B_{1}};\ket{1}_{b_{1}}\right]+\mathbb{P}\left[\left(\ket{0}\right)_{B_{1}};\ket{0}_{b_{1}}\right]$.

Note that in the entanglement assisted discrimination protocol within the
framework of Cohen [47] the first step is the twist-breaking (TB) step (i.e.
Step-1 in all the aforesaid protocols) that creates multiple images of the set
of states distinctly observed by all the parties sharing entanglement.

As already discussed $\$_{AB_{1}}\subset\Sigma$ has a layered tile structure
with outer and inner layers states having two different fixed state of Bob-2
as tag. Suppose Bob-1 performs a TB measurement along line $L_{1}$ as shown in
Fig. 3, i.e., he performs the measurement
$\mathcal{M}^{L_{1}}\equiv\left\\{M^{L_{1}},\mathbb{I}-M^{L_{1}}\right\\}$
where,
$M^{L_{1}}:=\mathbb{P}\left[\left(\ket{1},\ket{2},\ket{3},\ket{4}\right)_{B_{1}};\ket{1}_{b_{1}}\right]+\mathbb{P}\left[\left(\ket{0}\right)_{B_{1}};\ket{0}_{b_{1}}\right]$.
Here the shared EPR state between Alice and Bob-1 is
$\ket{\phi^{+}}_{a_{1}b_{1}}$. Irrespective of the outcome the post-
measurement states corresponding to the inner layer have fixed ancillary tag
of system $b_{1}$. Thus nonlocality remains between Alice and Bob-1 (due to
inner layer). It thus requires additional entanglement between Alice and Bob-1
for perfect discrimination. This nonlocality can be removed if Bob-2 creates
another EPR between Alice and Bob-1 using the remaining two EPR pairs and
following the entanglement swapping protocol. However this will leave
nonlocality of $\$_{AB_{2}}$ intact and thus the set $\Sigma$ remains locally
indistinguishable. Using similar kind of arguments it can be shown that for
each of the TB measurement (i.e. along line $L_{2},~{}L_{3}$ or $L_{4}$) some
states remain indistinguishable. This argument can be extended even if other
parties starts the TB measurement. At this point the recent work of Ref.[70]
is worth mentioning. From this result it follows that two-qutrit maximally
entangled states suffices for perfect discrimination of the NPS of Fig. 1.

Thus Proposition 5 together with Observation 3 implies Theorem 2.

##  Appendix D Reverse ordering of Theorem 2

Here we aim to show the reverse phenomena of Example-1, i.e., $3$ copies of
symmetrically distributed $2$-qubit EPR state perfectly discriminate a GNPS
while $2$ copies of $3$-qubit GHZ state fails to do so. For that we recall the
example of GNPB from Ref.[37].

######  Proposition 6.

(Halder et al. [37]) The set of states,

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathbb{H}\equiv$}\left\\{\\!\begin{aligned}
\ket{0}\ket{1}\ket{\eta_{\pm}},~{}\ket{1}\ket{\eta_{\pm}}\ket{0},~{}\ket{\eta_{\pm}}\ket{0}\ket{1},\\\
\ket{0}\ket{2}\ket{\kappa_{\pm}},~{}\ket{2}\ket{\kappa_{\pm}}\ket{0},~{}\ket{\kappa_{\pm}}\ket{0}\ket{2},\\\
\ket{1}\ket{2}\ket{\eta_{\pm}},~{}\ket{2}\ket{\eta_{\pm}}\ket{1},~{}\ket{\eta_{\pm}}\ket{1}\ket{2},\\\
\ket{2}\ket{1}\ket{\kappa_{\pm}},~{}\ket{1}\ket{\kappa_{\pm}}\ket{2},~{}\ket{\kappa_{\pm}}\ket{2}\ket{1},\\\
\ket{k}\ket{k}\ket{k}~{}~{}|~{}~{}k\in\\{0,1,2\\}~{}~{}~{}~{}\end{aligned}\right\\},$

is a GNPB in $(\mathbb{C}^{3})^{\otimes 3}$;
$\ket{\eta_{\pm}}:=(\ket{0}\pm\ket{1})/\sqrt{2}$,
$\ket{\kappa_{\pm}}:=(\ket{0}\pm\ket{2})/\sqrt{2}$.

In fact, the authors in [37] have proven a stronger result. Not only the above
basis can not be perfectly distinguished when any two of the parties come
together, even a single state can not be eliminated from this set by
orthogonality preserving measurement without all the parties coming together.
This particular phenomena is termed as strong quantum nonlocality without
entanglement. A more recent result [40] discuss about the nonlocal resource
for perfect local discrimination of this set.

######  Proposition 7.

(Rout et al. [40]) Three copies of symmetrical distributed $2$-qubit maximally
entangled state are sufficient for perfect local discrimination of the set
$\mathbb{H}$.

Now we come up with the following observation about local distinguishability
of the above set under the assistance of two copies of $3$-qubit GHZ state.

######  Observation 4.

Two copies of $3$-qubit GHZ shared among Alice, Bob-1, and Bob-2 fail to
perfectly discriminate the GNPB $\mathbb{H}$.

While proving the Proposition 7 in Ref.[40] we observe that if the
discrimination protocol is started with two copies of EPR pair shared between
Alice and Bob-1 and Alice and Bob-2 (among which cuts these EPR pairs are
shared that is irrelevant as the construction is party symmetric) then in the
best case scenario the given state can be perfectly discriminated locally if
it does not belong to the subset $\mathbb{H}^{\prime}\subset\mathbb{H}$, where

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathbb{H}^{\prime}\equiv$}\left\\{\\!\begin{aligned}
\ket{0}\ket{1}\ket{\eta_{\pm}},~{}\ket{1}\ket{\eta_{\pm}}\ket{0},~{}\ket{\eta_{\pm}}\ket{0}\ket{1},\\\
\ket{0}\ket{0}\ket{0}~{},\ket{1}\ket{1}\ket{1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{aligned}\right\\}.$

Note that the set $\mathbb{H}^{\prime}$ is the NPB corresponding to the SHIFTS
unextendible product basis in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ [20]. One further
EPR pair shared between any two parties will suffice for perfect local
discrimination of the given state. Instead of two initial pairs of EPR state
if the protocol is started with two copies of three-qubit GHZ state shared
among the three parties a similar situation arises.

If we now consider the tripartite task $\tau_{3}^{\prime\prime\prime}$ of
local distinguishability the of set $\mathbb{H}$, then Proposition 7 along
with Observation 4 imply the ordering relation
$\ket{\phi_{3}}\succcurlyeq_{\tau_{3}^{\prime\prime\prime}}\ket{\psi_{3}}$.

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