# Activating Strongest Possible Nonlocality from Local Sets: An Elimination
Paradigm

Subhendu B. Ghosh  Physics and Applied Mathematics Unit, Indian Statistical
Institute, Kolkata, 203 B. T. Road, Kolkata 700108, India  Tathagata Gupta
Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata,
203 B. T. Road, Kolkata 700108, India  Ardra A. V  School of Physics, IISER
Thiruvananthapuram, Vithura, Kerala 695551, India  Anandamay Das Bhowmik
Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata,
203 B. T. Road, Kolkata 700108, India  Sutapa Saha  Physics and Applied
Mathematics Unit, Indian Statistical Institute, Kolkata, 203 B. T. Road,
Kolkata 700108, India  Tamal Guha  Department of Computer Science, The
University of Hong Kong, Pokfulam Road, Hong Kong  Amit Mukherjee  S. N. Bose
National Center for Basic Sciences, Block JD, Sector III, Saltlake, Kolkata
700098, India

###### Abstract

Apart from the Bell nonlocality, which deals with the correlations generated
from the local input-output statistics, quantum theory exhibits another kind
of nonlocality that involves the indistiguishability of the locally preparable
set of multipartite states. While Bell-type nonlocality cannot be distilled
from a given local correlation, it is already reported that the latter kind of
nonlocality can be activated from a "local", i.e., locally distinguishable set
of states. Although, recently it is shown that a stronger notion of such a
nonlocality, which deals with elimination instead of discrimination, can be
activated from locally preparable bipartite states of dimension $7\times 8$,
the present work observes that the same notion can be demonstrated even in
lower dimensional multipartite systems. Importantly, the strongest possible
version of such an activation is further depicted here, where none of the
transformed product states can be eliminated, even if all but one of the
parties come together.

##  I Introduction

Nonlocality is the most curious feature of the quantum world. A celebrated
nonlocal phenomenon - Bell nonlocality [1] arises in quantum physics due to
entanglement. This phenomenon actually does not allow any local realistic
explanation. Apart from its fundamental implications, Bell nonlocality is also
immensely useful in some intriguing applications [2, 3, 4, 5, 6, 7, 8].
However, besides Bell nonlocality, quantum theory admits a different kind of
multipartite nonclassicality, which identifies a set of orthogonal states
which are impossible to distinguish under local operations and classical
communications (LOCC) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46]. Consequently, a stronger version of state
distinguishability has recently been introduced in [47], where the task is to
eliminate some elements from a set of orthogonal states (preserving their
orthogonality), shared between the distant parties, under LOCC and such an
impossibility is termed as local irreducibility. For example, four two-qubit
Bell states or, complete three-qubit GHZ basis are locally irreducible [47].
Note that, any locally irreducible set of quantum states are surely locally
indistinguishable, while the converse is not true.

Hitherto, we have discussed nonlocal features that are predominantly
manifested only by entangled states. Surprisingly, entanglement is not always
necessary to exhibit those nonlocal characteristics. Bennett et al., in their
pioneering work [48], showed that there exist sets of orthogonal product
states which are not perfectly locally distinguishable. This intriguing non-
classical feature is termed as ‘quantum nonlocality without entanglement’.
More interestingly, it has also been shown that the set of locally
indistinguishable product basis in $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ [48]
is also locally irreducible [47].

However, for multipartite scenarios, manifestation of nonlocality gets more
complex. For example, in case of locally irreducible three qubit GHZ basis, it
can be shown that whenever any two of the players come together (that is,
allowed to perform joint or global measurement) they are able to eliminate
some states from the set [47]. The same question of collaboration between all
but one party can also be considered in case of multipartite quantum
nonlocality without entanglement phenomenon. It can be shown that there is a
set of orthogonal tripartite product states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ [48] which are
locally indistinguishable when all the players are spatially separated. But if
any two of the players are allowed to come together to perform some joint
measurement on their composite subsystems, the set will turn out to be a
locally distinguishable one. However, [47], Halder et al. have demonstrated
strongest form of quantum nonlocality without entanglement in multipartite
scenario by introducing a set of tripartite product states that are locally
irreducible even when all but one players assemble together. Naturally, this
set of states is locally indistinguishable in bipartition. This phenomena is
termed as ‘strong quantum nonlocality without entanglement’. Moreover, a
characterization of sets of multipartite nonlocal product states has also been
introduced recently [49]. These recent works of quantum nonlocality without
entanglement in multipartite systems have been fueling a plethora of
interesting studies [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60].

Apart from revealing the elegant intricacies of state space structure, local
indistinguishability and irreducibility of quantum states also indicate the
prospect of locking of information such that unlocking requires entangled
resources. This characteristic certainly has a crucial significance in various
quantum cryptographic schemes [61, 62, 63, 64, 65]. It is, therefore, an
intriguing question to transform a locally distinguishable set of multipartite
states to a set with such nonlocal features via local operations and classical
communications.

In a recent work, Bandyopadhyay and Halder [66] showed that there exist some
locally distinguishable sets of multipartite orthogonal states which can be
deterministically transformed to locally indistinguishable sets via
orthogonality preserving measurements (OPM) on the local sites of the players.
They called this phenomenon genuine activation of nonlocality. The term
‘genuine’ denotes that the set must be free from local redundancy.111Presence
of local redundancy in a set of orthogonal states actually signifies that
discarding of one or more subsystems from initial system keeps the
orthogonality of the set of reduced states intact. We are, thus, interested in
such sets which will be nonorthogonal after discarding one or more subsystems
[66]. Apparently, it may seem that the phenomenon of genuine activation is
somewhat parallel to the distillation of Bell nonlocality[67, 68]. However,
there is a fundamental difference. In case of genuine activation of
nonlocality, the initial set does not need to show any nonlocal feature
whereas for distillation of Bell-type nonlocality the initial correlation must
not be local-realistic. Indeed, in a recent result, Li and Zheng have further
extended the search by exploring genuine activation of local irreducibility
for bipartite product states [69] and come up with a locally distinguishable
set of bipartite product states which via local OPM can be deterministically
converted to a locally irreducible set of bipartite product states. However,
the question of genuine activation of nonlocality without entanglement in case
of multipartite scenario has yet not been explored in its entirety. In this
article, we show that a stronger form of nonlocality of a locally
distinguishable set of tripartite orthogonal product states can be genuinely
activated. More precisely, the set can be deterministically transformed via
local OPMs to a set of orthogonal product states which is locally irreducible
even when any two of the three parties come together. Further, we come up with
a stronger example using a set of locally distinguishable states in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{6}$, where the
measurement performed on the third party’s possession can activate the
strongest possible genuine nonlocality without entanglement.

In the following sections, we discuss genuine activation of multipartite
quantum nonlocality without entanglement case by case.

##  II Genuine Activation of Multipartite Quantum Nonlocality Without
Entanglement

We will start our exploration of genuinely activable sets by showing that a
locally distinguishable set of bipartite product states can be transformed to
a set of locally indistinguishable set of orthogonal states via local
orthogonality preserving measurements.

Consider the set
$\mathcal{G}_{1}\equiv\\{\ket{\psi_{i}}_{AB}\\}_{i=1}^{5}(\subset\mathbb{C}^{3}\otimes\mathbb{C}^{6})$,
where,

$\displaystyle\ket{\psi_{1}}_{AB}$ $\displaystyle=$
$\displaystyle\ket{0}_{A}\ket{\mathbf{0}-\mathbf{1}+\mathbf{4}-\mathbf{5}}_{B}$
(1a) $\displaystyle\ket{\psi_{2}}_{AB}$ $\displaystyle=$
$\displaystyle\ket{2}_{A}\ket{\mathbf{1}-\mathbf{2}+\mathbf{5}-\mathbf{3}}_{B}$
(1b) $\displaystyle\ket{\psi_{3}}_{AB}$ $\displaystyle=$
$\displaystyle\ket{1-2}_{A}\ket{\mathbf{0}-\mathbf{4}}_{B}$ (1c)
$\displaystyle\ket{\psi_{4}}_{AB}$ $\displaystyle=$
$\displaystyle\ket{0-1}_{A}\ket{\mathbf{2}-\mathbf{3}}_{B}$ (1d)
$\displaystyle\ket{\psi_{5}}_{AB}$ $\displaystyle=$
$\displaystyle\ket{0+1+2}_{A}\ket{\mathbf{0}+\mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}}_{B}.$
(1e)

######  Proposition 1.

The set $\mathcal{G}_{1}$ is locally distinguishable and free from local
redundancy.

###### Proof.

It is quite straightforward to prove that the set $\mathcal{G}_{1}$ is without
local redundancy. Here, Bob’s system can be considered to be the composition
of qubit and qutrit subsystems. Precisely,
$\ket{\mathbf{0}}_{B}:=\ket{00}_{b_{1}b_{2}},\ket{\mathbf{1}}_{B}:=\ket{01}_{b_{1}b_{2}},\ket{\mathbf{2}}_{B}:=\ket{02}_{b_{1}b_{2}},\ket{\mathbf{3}}_{B}:=\ket{10}_{b_{1}b_{2}},\ket{\mathbf{4}_{B}}:=\ket{11}_{b_{1}b_{2}},\ket{\mathbf{5}}_{B}:=\ket{12}_{b_{1}b_{2}}$.
Take two states, $\ket{\psi_{3}}_{AB}$ and $\ket{\psi_{4}}_{AB}$. When any of
the sub-parts (qubit or qutrit) of Bob’s system for both states is discarded
the reduced states will be non-orthogonal.

Furthermore, $\mathcal{G}_{1}$ can also be shown to be locally
distinguishable. The players pursue the following distinguishability protocol.
First Bob performs a measurement
$N_{B}\equiv\\{N_{1}:=P[\ket{\mathbf{0}-\mathbf{4}}_{B}],N_{2}:=P[\ket{\mathbf{2}-\mathbf{3}}_{B}],N_{3}:=P[\ket{\mathbf{0}+\mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}}_{B}],N_{4}:=\mathbb{I}-(N_{1}+N_{2}+N_{3})\\}$.
Here, $P[(\ket{i},\ket{j})_{\\#}]:=(\ket{i}\bra{i}+\ket{j}\bra{j})_{\\#}$, and
$\\#$ denotes the party. When $N_{1}$ clicks, the given state must be
$\ket{\psi_{3}}$. Similarly, for the click $N_{2}$, the state is
$\ket{\psi_{4}}$, and for $N_{3}$ it is $\ket{\psi_{5}}$. Whenever $N_{4}$
clicks the given state can be either $\ket{\psi_{1}}$ or $\ket{\psi_{2}}$.
However, in that case, Alice can perform a measurement to distinguish between
these two [11, 70]. This concludes the local distinguishability protocol for
the set $\mathcal{G}_{1}$. ∎

In the following, we will demonstrate a protocol to activate the nonlocality
without entanglement from the set $\mathcal{G}_{1}$.

######  Theorem 1.

The locally distinguishable set $\mathcal{G}_{1}$ can be transformed
deterministically to a locally indistinguishable set via local OPM.

###### Proof.

Consider that Bob performs a local OPM on the subsystem $B$:
$K_{B}\equiv\\{K_{1}:=P[(\ket{\mathbf{0}},\ket{\mathbf{1}},\ket{\mathbf{2}})_{B}],K_{2}:=P[(\ket{\mathbf{3}},\ket{\mathbf{4}},\ket{\mathbf{5}})_{B}]\\}$.
If $K_{1}$ clicks they end up in one of

$\displaystyle\left\\{\\!\begin{aligned}
\ket{0}_{A}\ket{\mathbf{0}-\mathbf{1}}_{B},\leavevmode\nobreak\
\ket{2}_{A}\ket{\mathbf{1}-\mathbf{2}}_{B},\\\
\ket{1-2}_{A}\ket{\mathbf{0}}_{B},\leavevmode\nobreak\
\ket{0-1}_{A}\ket{\mathbf{2}}_{B},\\\ \leavevmode\nobreak\
\ket{0+1+2}_{A}\ket{\mathbf{0}+\mathbf{1}+\mathbf{2}}_{B}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \end{aligned}\right\\}$

On the other hand, if Bob gets $K_{2}$, they are then left with one of the
following five states:

$\displaystyle\left\\{\\!\begin{aligned}
\ket{0}_{A}\ket{\mathbf{4}-\mathbf{5}}_{B},\leavevmode\nobreak\
\ket{2}_{A}\ket{\mathbf{5}-\mathbf{3}}_{B},\\\
\ket{1-2}_{A}\ket{\mathbf{4}}_{B},\leavevmode\nobreak\
\ket{0-1}_{A}\ket{\mathbf{3}}_{B},\\\ \leavevmode\nobreak\
\ket{0+1+2}_{A}\ket{\mathbf{3}+\mathbf{4}+\mathbf{5}}_{B}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \end{aligned}\right\\}$

It is clear that the five updated states when $K_{1}$ clicks form the
celebrated unextendible product basis (UPB) [71, 72] in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}$. The states in case of $K_{2}$ outcome
also form the same UPB where
$\\{\ket{\mathbf{3}}_{B},\ket{\mathbf{4}}_{B},\ket{\mathbf{5}}_{B}\\}$ span
Bob’s three dimensional Hilbert space. It has been well established that this
orthogonal set of product states is locally indistinguishable [48, 72]. ∎

This is certainly an example of genuine activation of bipartite quantum
nonlocality without entanglement. However, orthogonal sets of bipartite
product states that show activable nonlocality have already been reported
[69]. But in some of the protocols, mentioned there, different outcome of a
single local OPM provides different dimensional sets of nonlocal product
states. Moreover, the dimension requirement to activate such nonlocality in
[69] is minimum $7\times 8$ for bipartite systems, while our elegant example
shows that such an activation is possible even with lower dimensional quantum
systems.

The question of genuine activation of multipartite quantum nonlocality from
orthogonal sets of product states also has not been explored yet. In the
following, we delve into this question and answer in affirmative with an
explicit example. Consider the orthogonal set of tripartite product states
$\mathcal{G}_{2}\equiv\\{\ket{\phi_{i}}_{ABC}\\}_{i=1}^{4}(\subset\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{4})$
where,

$\displaystyle\ket{\phi_{1}}_{ABC}$ $\displaystyle=$
$\displaystyle\ket{0}_{A}\ket{0-1}_{B}\ket{\mathbf{1}+\mathbf{2}}_{C}$ (4a)
$\displaystyle\ket{\phi_{2}}_{ABC}$ $\displaystyle=$
$\displaystyle\ket{0-1}_{A}\ket{1}_{B}\ket{\mathbf{0}+\mathbf{3}}_{C}$ (4b)
$\displaystyle\ket{\phi_{3}}_{ABC}$ $\displaystyle=$
$\displaystyle\ket{1}_{A}\ket{0}_{B}\ket{\mathbf{0}-\mathbf{1}+\mathbf{2}-\mathbf{3}}_{C}$
(4c) $\displaystyle\ket{\phi_{4}}_{ABC}$ $\displaystyle=$
$\displaystyle\ket{0+1}_{A}\ket{0+1}_{B}\ket{\mathbf{0}+\mathbf{1}+\mathbf{2}+\mathbf{3}}_{C},$
(4d)

######  Proposition 2.

The set $\mathcal{G}_{2}$ is free from local redundancy and discriminable
under LOCC.

###### Proof.

It is straightforward to show that the set $\mathcal{G}_{2}$ is free form
local redundancy. Here, Charlie’s system can be considered as two composite
qubits. Let us denote those subsystems by $c_{1}$ and $c_{2}$:
$\ket{\mathbf{0}}_{C}:=\ket{00}_{c_{1}c_{2}},\ket{\mathbf{1}}_{C}:=\ket{01}_{c_{1}c_{2}},\ket{\mathbf{2}}_{C}:=\ket{10}_{c_{1}c_{2}},\ket{\mathbf{3}}:=\ket{11}_{c_{1}c_{2}}$.
Consider the states $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$. Note that
discarding the subsystem $c_{i}$ we will have
$\rho^{k}_{c_{j}}:=Tr_{c_{i}}\ket{\phi_{k}}\bra{\phi_{k}}$, for
$i,j,k\in\\{1,2\\}$. It is quite evident that $\rho^{1}_{c_{j}}$ and
$\rho^{2}_{c_{j}}$ are non-orthogonal for $j=1,2$.

We will now show that the set $\mathcal{G}_{2}$ is locally distinguishable.
The distinguishability protocol is as follows. First, Charlie performs a
measurement
$K_{C}\equiv\\{K_{1}:=P[\ket{\mathbf{0}+\mathbf{3}}_{C}],K_{2}:=P[\ket{\mathbf{0}-\mathbf{3}}_{C}],K_{3}:=P[\ket{\mathbf{1}+\mathbf{2}}_{C}],K_{4}:=P[\ket{\mathbf{1}-\mathbf{2}}_{C}]\\}$.
If $K_{1}$ clicks the given state must be one of
$\\{\ket{\phi_{2}}_{ABC},\ket{\phi_{4}}_{ABC}\\}$ which are perfectly locally
distinguishable [11, 70]. If $K_{2}$ clicks, the given state must be
$\ket{\phi_{2}}_{ABC}$. When $K_{3}$ clicks, the given state is one of
$\\{\ket{\phi_{1}}_{ABC},\ket{\phi_{4}}_{ABC}\\}$ which can always be
perfectly distinguished via LOCC. For the click $K_{4}$, the given state is
certainly $\ket{\phi_{3}}_{ABC}$. ∎

Now, we are in a position to show that the set $\mathcal{G}_{2}$ can be
transformed, with certainty, to a set of orthogonal states which are
impossible to distinguish locally.

######  Theorem 2.

The set $\mathcal{G}_{2}$ can be converted to a set of tripartite locally
indistinguishable product states, a.k.a, the Shifts UPB [71] using local OPM.

###### Proof.

Let us consider that Charlie performs a local OPM on the subsystem $C$:
$R_{B}\equiv\\{R_{1}:=P[(\ket{\mathbf{0}},\ket{\mathbf{1}})_{C}],R_{2}:=P[(\ket{\mathbf{2}},\ket{\mathbf{3}})_{C}]\\}$.
If $R_{1}$ clicks they end up in one of

$\displaystyle\\{\ket{0}_{A}\ket{0-1}_{B}\ket{\mathbf{1}}_{C},\ket{0-1}_{A}\ket{1}_{B}\ket{\mathbf{0}}_{C},$
$\displaystyle\ket{1}_{A}\ket{0}_{B}\ket{\mathbf{0-1}}_{C},\ket{0+1}_{A}\ket{0+1}_{B}\ket{\mathbf{0+1}}_{C}\\}.$

On the other hand, if $R_{2}$ clicks, they will be left with one of

$\displaystyle\\{\ket{0}_{A}\ket{0-1}_{B}\ket{\mathbf{2}}_{C},\ket{0-1}_{A}\ket{1}_{B}\ket{\mathbf{3}}_{C},$
$\displaystyle\ket{1}_{A}\ket{0}_{B}\ket{\mathbf{2-3}}_{C},\ket{0+1}_{A}\ket{0+1}_{B}\ket{\mathbf{2+3}}_{C}\\}.$

It is evident that both the above sets are basically equivalent to the Shifts
UPB in $\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ [71, 72,
48]. Furthermore, states belonging to Shifts UPB are known to be perfectly
indistinguishable via LOCC [72]. This completes our proof. ∎

Note that, whenever any two parties of the above Shifts UPB come together, the
set turns out to be locally distinguishable. Any orthogonal multipartite set
of product states that can be thought of as a bipartition of
$\mathbf{C}^{2}\otimes\mathbb{C}^{d}$ with $d\geq 2$, can always be shown as
locally distinguishable [72]. Therefore, it is quite clear that this set must
not be locally irreducible in any bipartition.

##  III Genuine Activation of Strongest Possible Quantum Nonlocality Without
Entanglement

After demonstrating the activation of genuine nonlocality without entanglement
in the multipartite scenario, the pertinent question is whether or not there
exists any orthogonal set which can show genuine activation of the strongest
possible form of quantum nonlocality without entanglement as demonstrated in
[47]. Note that, strong quantum nonlocality without entanglement [47] can not
be obtained in mere three-qubit systems. The minimum dimension required to
show such phenomena is at least three qutrit. In the following, we provide a
set which answers this question in affirmation. Consider the orthogonal set
$\mathcal{G}_{3}$ that contains the following of $27$ tripartite product
states222For the sake of better readability here we drop the party notation
($A,B,C$) in the subscripts $\ket{\xi_{i}^{\pm}},\leavevmode\nobreak\
i\in\\{1,\cdots,4,6,\cdots,9,11,\cdots,14\\}$ and
$\ket{\xi_{j}},\leavevmode\nobreak\ j\in\\{5,10,15\\}$ in
$\mathbb{C}^{6}\otimes\mathbb{C}^{6}\otimes\mathbb{C}^{6}$.

$\displaystyle\ket{\xi_{1}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}-\mathbf{4}}\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}$
(5a) $\displaystyle\ket{\xi_{2}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}-\mathbf{4}}\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}$
(5b) $\displaystyle\ket{\xi_{3}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}$
(5c) $\displaystyle\ket{\xi_{4}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}$
(5d) $\displaystyle\ket{\xi_{5}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}-\mathbf{4}}\ket{\mathbf{0}-\mathbf{4}}\ket{\mathbf{0}-\mathbf{4}}$
(5e) $\displaystyle\ket{\xi_{6}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}\ket{\mathbf{0}-\mathbf{4}}$
(5f) $\displaystyle\ket{\xi_{7}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}\ket{\mathbf{0}-\mathbf{4}}$
(5g) $\displaystyle\ket{\xi_{8}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}\ket{\mathbf{1}-\mathbf{5}}$
(5h) $\displaystyle\ket{\xi_{9}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}\ket{\mathbf{2}-\mathbf{3}}$
(5i) $\displaystyle\ket{\xi_{10}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{1}-\mathbf{5}}$
(5j) $\displaystyle\ket{\xi_{11}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}\ket{\mathbf{0}-\mathbf{4}}\ket{\mathbf{1}-\mathbf{5}}$
(5k) $\displaystyle\ket{\xi_{12}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}\ket{\mathbf{0}-\mathbf{4}}\ket{\mathbf{2}-\mathbf{3}}$
(5l) $\displaystyle\ket{\xi_{13}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}\ket{\mathbf{1}-\mathbf{5}}\ket{\mathbf{2}-\mathbf{3}}$
(5m) $\displaystyle\ket{\xi_{14}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{1}-\mathbf{5}}$
(5n) $\displaystyle\ket{\xi_{15}}$ $\displaystyle=$
$\displaystyle\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{2}-\mathbf{3}}\ket{\mathbf{2}-\mathbf{3}},$
(5o)

######  Proposition 3.

The set $\mathcal{G}_{3}$ is not locally redundant and is distinguishable
under LOCC, even when all the parties are separated.

###### Proof.

We first provide a brief outline of the proof that the above set of states are
free from local redundancy. The detailed proof is given in the appendix. Note
that, the quantum system possessed by each individuals can only be composed of
a qubit and qutrit subsystem. Therefore, for each player we can write:
$\ket{\mathbf{0}}:=\ket{00},\ket{\mathbf{1}}:=\ket{01},\ket{\mathbf{2}}:=\ket{02},\ket{\mathbf{3}}:=\ket{10},\ket{\mathbf{4}}:=\ket{11},\ket{\mathbf{5}}:=\ket{12}$.
First consider the players discard their subsystems in such a way that they
ultimately get the dimension of the whole tripartite system below $27$. This
is possible when more than one player discard their qutrits (that is,
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, or,
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{6}$, or,
$\mathbb{C}^{2}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ etc.). In this
case, it is clear that all the states in (5) will not retain their
orthogonality. Other possible cases of discarding the sub-parts can be
branched as follows: any one player discards their qutrit, any one player
discards their qubit and more than one player discard their qubits. Though
cumbersome but it is quite straightforward to show that in all these cases the
reduced states’ set will not be orthogonal anymore. Therefore, we can conclude
that the set $\mathcal{G}_{3}$ does not have local redundancy.

Now, we move to the proof that the set $\mathcal{G}_{3}$ can be distinguished
with the help of LOCC alone. Due to the symmetries present in the set
$\mathcal{G}_{3}$, each player may need to perform any of the following three
measurements at different steps of the protocol.

$\displaystyle\mathcal{M}_{1}$
$\displaystyle\equiv\\{P_{1}:=P[\ket{\mathbf{0-4}}],P_{2}:={P}[\ket{\mathbf{1-5}}],P_{3}:={P}[\ket{\mathbf{2-3}}],$
$\displaystyle
P_{4}:=\mathbb{I}-({P}[\ket{\mathbf{0-4}}]+{P}[\ket{\mathbf{1-5}}]+{P}[\ket{\mathbf{2-3}}])\\}$
$\displaystyle\mathcal{M}_{2}$
$\displaystyle\equiv\\{Q_{1}:=P[\ket{\mathbf{0+1+4+5}}],Q_{2}:=P[\ket{\mathbf{0-1+4-5}}],$
$\displaystyle
Q_{3}:=\mathbb{I}-(P[\ket{\mathbf{0+1+4+5}}]+[\ket{\mathbf{0-1+4-5}}])\\}$
$\displaystyle\mathcal{M}_{3}$
$\displaystyle\equiv\\{R_{1}:=P[\ket{\mathbf{0+2+4+3}}],R_{2}:=P[\ket{\mathbf{0-2+4-3}}],$
$\displaystyle
R_{3}:=\mathbb{I}-(P[\ket{\mathbf{0+2+4+3}}]+{P}[\ket{\mathbf{0-2+4-3}}])\\}.$

The detailed protocol is pictorially described in the Appendix. ∎

######  Theorem 3.

The set $\mathcal{G}_{3}$ can be deterministically transformed via local OPMs
to a orthogonal set of tripartite product states which are locally irreducible
even if all but one player come together.

###### Proof.

Suppose, each player performs a specific orthogonality preserving local
measurement:
$\mathcal{K}\equiv\\{K_{1}:=P[\ket{\mathbf{0}},\ket{\mathbf{1}},\ket{\mathbf{2}}],K_{2}:=P[\ket{\mathbf{3}},\ket{\mathbf{4}},\ket{\mathbf{5}}]\\}$.
Here, the notation we follow is as follows: $K_{j}^{i}$ is the $j$th projector
($K_{j}$) that clicks when $i$th player performs the measurement
$\mathcal{K}$. Therefore, after the measurement a total of eight possibilities
can occur as each player can get any one of two possible outcomes $K_{1}^{i}$
or $K_{2}^{i}$. In each of these eight cases it is straightforward to see that
the updated set of $27$ states will be of the following generic form.

$\displaystyle\left\\{\\!\begin{aligned}
\ket{p}\ket{q}\ket{\eta_{\pm}},\leavevmode\nobreak\
\ket{q}\ket{\eta_{\pm}}\ket{p},\leavevmode\nobreak\
\ket{\eta_{\pm}}\ket{p}\ket{q},\\\
\ket{p}\ket{r}\ket{\kappa_{\pm}},\leavevmode\nobreak\
\ket{r}\ket{\kappa_{\pm}}\ket{p},\leavevmode\nobreak\
\ket{\kappa_{\pm}}\ket{p}\ket{r},\\\
\ket{q}\ket{r}\ket{\eta_{\pm}},\leavevmode\nobreak\
\ket{r}\ket{\eta_{\pm}}\ket{q},\leavevmode\nobreak\
\ket{\eta_{\pm}}\ket{q}\ket{r},\\\
\ket{r}\ket{q}\ket{\kappa_{\pm}},\leavevmode\nobreak\
\ket{q}\ket{\kappa_{\pm}}\ket{r},\leavevmode\nobreak\
\ket{\kappa_{\pm}}\ket{r}\ket{q},\\\
\ket{p}\ket{p}\ket{p},\leavevmode\nobreak\
\ket{q}\ket{q}\ket{q},\leavevmode\nobreak\
\ket{r}\ket{r}\ket{r}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \end{aligned}\right\\}$ (6)

Here, $\ket{\eta_{\pm}}:=(\ket{p}\pm\ket{q})/\sqrt{2}$ and
$\ket{\kappa_{\pm}}:=(\ket{p}\pm\ket{r})/\sqrt{2}$. In each of the eight
outcomes, for all $27$ states $p,q$ and $r$ will have some specific values
from $p\in\\{\mathbf{0,4}\\},q\in\\{\mathbf{1,5}\\}$ and
$r\in\\{\mathbf{2,3}\\}$. For example, if for all the players, the outcomes
are $K_{1}$ throughout then the reduced set of states will be of the above
form with $p=\\{\mathbf{0}\\},q=\\{\mathbf{1}\\}$ and $r=\\{\mathbf{2}\\}$.

Note that the above set of states is basically the orthogonal set that
manifests strong quantum nonlocality without entanglement [47]. ∎

At this end, one may be further curious to activate such a strongest possible
genuine quantum nonlocality without entanglement by performing a measurement
on the possession of a single party. This has vivid importance in the
framework of data hiding and secret sharing between all but one untrusted
parties. Precisely speaking, in such a scenario the particular trusted agent
(personified as Charlie) has full authority to judge how trustworthy are the
other parties and depending upon that he may compel others to meet him in
person to decode a hidden secret. As an example consider the following set
$\mathcal{G}_{4}$ of $27$ orthogonal product states
$\ket{\zeta_{i}^{\pm}},\leavevmode\nobreak\
i\in\\{1,\cdots,4,6,\cdots,9,11,\cdots,14\\}$ and
$\ket{\zeta_{j}},\leavevmode\nobreak\ j\in\\{5,10,15\\}$ in
$\mathbb{C}^{3^{\otimes 2}}\otimes\mathbb{C}^{6}$,

$\displaystyle\ket{\zeta_{1}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{0}\ket{1}\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}$
(7a) $\displaystyle\ket{\zeta_{2}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{0}\ket{2}\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}$
(7b) $\displaystyle\ket{\zeta_{3}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{1}}\ket{{2}}\ket{\mathbf{0}\pm\mathbf{1}+\mathbf{4}\pm\mathbf{5}}$
(7c) $\displaystyle\ket{\zeta_{4}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{2}}\ket{{1}}\ket{\mathbf{0}\pm\mathbf{2}+\mathbf{4}\pm\mathbf{3}}$
(7d) $\displaystyle\ket{\zeta_{5}}$ $\displaystyle=$
$\displaystyle\ket{{0}}\ket{{0}}\ket{\mathbf{0}-\mathbf{4}}$ (7e)
$\displaystyle\ket{\zeta_{6}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{1}}\ket{{0}\pm{1}}\ket{\mathbf{0}-\mathbf{4}}$ (7f)
$\displaystyle\ket{\zeta_{7}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{2}}\ket{{0}\pm{2}}\ket{\mathbf{0}-\mathbf{4}}$ (7g)
$\displaystyle\ket{\zeta_{8}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{2}}\ket{{0}\pm{1}}\ket{\mathbf{1}-\mathbf{5}}$ (7h)
$\displaystyle\ket{\zeta_{9}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{1}}\ket{{0}\pm{2}}\ket{\mathbf{2}-\mathbf{3}}$ (7i)
$\displaystyle\ket{\zeta_{10}}$ $\displaystyle=$
$\displaystyle\ket{{1}}\ket{{1}}\ket{\mathbf{1}-\mathbf{5}}$ (7j)
$\displaystyle\ket{\zeta_{11}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{0}\pm{1}}\ket{{0}}\ket{\mathbf{1}-\mathbf{5}}$ (7k)
$\displaystyle\ket{\zeta_{12}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{0}\pm{2}}\ket{{0}}\ket{\mathbf{2}-\mathbf{3}}$ (7l)
$\displaystyle\ket{\zeta_{13}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{0}\pm{1}}\ket{{1}}\ket{\mathbf{2}-\mathbf{3}}$ (7m)
$\displaystyle\ket{\zeta_{14}^{\pm}}$ $\displaystyle=$
$\displaystyle\ket{{0}\pm{2}}\ket{{2}}\ket{\mathbf{1}-\mathbf{5}}$ (7n)
$\displaystyle\ket{\zeta_{15}}$ $\displaystyle=$
$\displaystyle\ket{{2}}\ket{{2}}\ket{\mathbf{2}-\mathbf{3}},$ (7o)

######  Proposition 4.

The set $\mathcal{G}_{4}$ is distinguishable under LOCC and free from local
redundancy.

###### Proof.

The proof that the set $\mathcal{G}_{4}$ does not have redundancy is quite
straightforward. One may consider that the subsystem of Charlie
($\mathbb{C}^{6}$) consists of a qubit and qutrit. Now, if we discard any of
the qubit or qutrit parts, not all pairs that remain would be orthogonal. The
proof is quite evident from the proof of proposition 3.

We will now move to describe a local discrimination protocol of the set
$\mathcal{G}_{4}$. We will provide here a brief outline of the protocol.

Charlie first performs a measurement

$\displaystyle\mathcal{M}^{C}_{1}$
$\displaystyle\equiv\\{P_{1}:=P[\ket{\mathbf{0-4}}_{C}],P_{2}:={P}[\ket{\mathbf{1-5}}_{C}],P_{3}:={P}[\ket{\mathbf{2-3}}_{C}],$
$\displaystyle
P_{4}:=\mathbb{I}-({P}[\ket{\mathbf{0-4}}_{C}]+{P}[\ket{\mathbf{1-5}}_{C}]+{P}[\ket{\mathbf{2-3}}_{C}])\\}$

Now, depending upon different outcomes, Bob and Charlie will perform some
suitable measurements at their local sites to distinguish the set. A step by
step detailed analysis is provided in the Appendix. ∎

######  Theorem 4.

The set $\mathcal{G}_{4}$ can be deterministically transformed, via a single
local OPM at Charlie’s site, to an orthogonal set of tripartite product states
which is locally irreducible in every bipartition.

###### Proof.

Consider that Charlie performs a local OPM,
$\mathcal{K}^{C}\equiv\\{K_{1}^{C}:=P[\ket{\mathbf{0}},\ket{\mathbf{1}},\ket{\mathbf{2}}_{C}],K_{2}^{C}:=P[\ket{\mathbf{3}},\ket{\mathbf{4}},\ket{\mathbf{5}}_{C}]\\}$
For different outcomes of $\mathcal{K}^{C}$, the post measurement state will
turn out to be any of the following set:

$\displaystyle\left\\{\\!\begin{aligned}
\ket{0}\ket{1}\ket{{\tilde{\nu}_{\pm}}},\leavevmode\nobreak\
\ket{1}\ket{\nu_{\pm}}\ket{p},\leavevmode\nobreak\
\ket{\nu_{\pm}}\ket{o}\ket{q},\\\
\ket{0}\ket{2}\ket{\tilde{\tau}_{\pm}},\leavevmode\nobreak\
\ket{2}\ket{\tau_{\pm}}\ket{p},\leavevmode\nobreak\
\ket{\tau_{\pm}}\ket{o}\ket{r},\\\
\ket{1}\ket{2}\ket{\tilde{\nu}_{\pm}},\leavevmode\nobreak\
\ket{2}\ket{\nu_{\pm}}\ket{q},\leavevmode\nobreak\
\ket{\nu_{\pm}}\ket{1}\ket{r},\\\
\ket{2}\ket{1}\ket{\tilde{\tau}_{\pm}},\leavevmode\nobreak\
\ket{1}\ket{\tau_{\pm}}\ket{r},\leavevmode\nobreak\
\ket{\tau_{\pm}}\ket{2}\ket{q},\\\ \ket{0}\ket{0}\ket{p},\leavevmode\nobreak\
\ket{1}\ket{1}\ket{q},\leavevmode\nobreak\
\ket{2}\ket{2}\ket{r}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \end{aligned}\right\\}$ (8)

where, Here, $\ket{\nu_{\pm}}:=(\ket{0}\pm\ket{1})/\sqrt{2}$,
$\ket{\tau_{\pm}}:=(\ket{0}\pm\ket{2})/\sqrt{2}$,
$\ket{\tilde{\nu}_{\pm}}:=(\ket{p}\pm\ket{q})/\sqrt{2}$ and
$\ket{\tilde{\tau}_{\pm}}:=(\ket{p}\pm\ket{r})/\sqrt{2}$. $p,q$ and $r$ can
have any value $\\{\mathbf{0,4}\\}$, $\\{\mathbf{1,5}\\}$ and
$\\{\mathbf{2,3}\\}$ respectively. Now, when $K_{1}^{C}$ clicks, the post
measurement state can be any of the set (8) with $(p,q,r)=(\mathbf{0,1,2})$.
Otherwise, if $K_{2}^{C}$ clicks, they are left with any of set (8) where
$(p,q,r)=(\mathbf{4,5,3})$. It is evident that the set (8) shows strong
quantum nonlocality without entanglement [47, 49]. This completes our proof.

∎

##  IV Discussion

In summary, we have studied the genuine activation of nonlocality from several
sets of local states. However, the phrases "local" and "nonlocal" have been
used from the state discrimination perspective. More precisely, we have dealt
with two different sets of locally distinguishable multipartite product states
in $\mathbb{C}^{3}\otimes\mathbb{C}^{6}$ and $\mathbb{C}^{2^{\otimes
2}}\otimes\mathbb{C}^{4}$, which can be transformed to the set of locally
indistinguishable states in $\mathbb{C}^{3^{\otimes 2}}$ and
$\mathbb{C}^{2^{\otimes 3}}$ respectively, by choosing an appropriate
measurement in possession of one of the parties. Furthermore, we have
considered a stronger notion of state discrimination problem, namely the
orthogonality preserving reducibility and have shown to activate such a notion
from a set of multipartite locally distinguishable product states of dimension
$\mathbb{C}^{6^{\otimes 3}}$. It is observed that under LOCC, the set can be
transformed deterministically to a set of states in $\mathbb{C}^{3^{\otimes
3}}$, which is even irreducible in all possible bipartitions. Further, we have
moved to a stricter notion of such activation where the transformation is
possible to achieve by a single agent only. Such an example is demonstrated to
transform a locally distinguishable set of states of $\mathbb{C}^{3^{\otimes
2}}\otimes\mathbb{C}^{6}$ to a strongly irreducible one in
$\mathbb{C}^{3^{\otimes 3}}$. The elegance of state construction and the
transformation protocol makes it trivial to extend in any arbitrary higher
dimensional set of product states exhibiting nonlocality in terms of local
discrimination and orthogonality preserving elimination. Besides its
foundational interest to understand the topology of the state spaces of
composite quantum systems, our work deserves significant importance from the
practical perspective. It has mimicked an interesting framework of secured
data hiding between several parties, where the distributor is flexible to
update the distinguishibility of the secured data hidden in the correlation of
the given states.

##  V Acknowledgement

We would like to acknowledge stimulating discussions with Guruprasad Kar,
Manik Banik, Saronath Halder and Ramij Rahaman. Tamal Guha would like to
acknowledge his academic visit at Indian Statistical Institute, Kolkata,
during November-December of 2021.

## References

  * Bell [1966] J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966).
  * Brunner  _et al._ [2008] N. Brunner, S. Pironio, A. Acin, N. Gisin, A. A. Méthot, and V. Scarani, Testing the dimension of hilbert spaces, Phys. Rev. Lett. 100, 210503 (2008).
  * Pironio  _et al._ [2010] S. Pironio, A. Acín, S. Massar, and et al., Random numbers certified by bell’s theorem, Nature 464, 1021 (2010).
  * Acín  _et al._ [2006] A. Acín, N. Gisin, and L. Masanes, From bell’s theorem to secure quantum key distribution, Phys. Rev. Lett. 97, 120405 (2006).
  * Barrett  _et al._ [2005] J. Barrett, L. Hardy, and A. Kent, No signaling and quantum key distribution, Phys. Rev. Lett. 95, 010503 (2005).
  * Colbeck and Renner [2012] R. Colbeck and R. Renner, Free randomness can be amplified, Nature Physics 8, 450–453 (2012).
  * Brunner and Linden [2013] N. Brunner and N. Linden, Connection between bell nonlocality and bayesian game theory, Nat. Commun. 4, 2057 (2013).
  * Banik  _et al._ [2019] M. Banik, S. S. Bhattacharya, N. Ganguly, T. Guha, A. Mukherjee, A. Rai, and A. Roy, Two-qubit pure entanglement as optimal social welfare resource in bayesian game, Quantum 3, 185 (2019).
  * Virmani  _et al._ [2001] S. Virmani, M. F. Sacchi, M. B. Plenio, and D. Markham, Optimal local discrimination of two multipartite pure states, Phys. Lett. A 288, 62 (2001).
  * Groisman and Vaidman [2001] B. Groisman and L. Vaidman, Nonlocal variables with product-state eigenstates, J. Phys. A Math. Gen. 34, 6881 (2001).
  * Walgate  _et al._ [2000] J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local distinguishability of multipartite orthogonal quantum states, Phys. Rev. Lett. 85, 4972 (2000).
  * Horodecki  _et al._ [2003] M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Local indistinguishability: More nonlocality with less entanglement, Phys. Rev. Lett. 90, 047902 (2003).
  * Fan [2004] H. Fan, Distinguishability and indistinguishability by local operations and classical communication, Phys. Rev. Lett. 92, 177905 (2004).
  * Ghosh  _et al._ [2004] S. Ghosh, G. Kar, A. Roy, and D. Sarkar, Distinguishability of maximally entangled states, Phys. Rev. A 70, 022304 (2004).
  * De Rinaldis [2004] S. De Rinaldis, Distinguishability of complete and unextendible product bases, Phys. Rev. A 70, 022309 (2004).
  * Nathanson [2005] M. Nathanson, Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases, J. Math. Phys. 46, 062103 (2005).
  * Watrous [2005a] J. Watrous, Bipartite subspaces having no bases distinguishable by local operations and classical communication, Phys. Rev. Lett. 95, 080505 (2005a).
  * Watrous [2005b] J. Watrous, Bipartite subspaces having no bases distinguishable by local operations and classical communication, Phys. Rev. Lett. 95, 080505 (2005b).
  * Niset and Cerf [2006] J. Niset and N. J. Cerf, Multipartite nonlocality without entanglement in many dimensions, Phys. Rev. A 74, 052103 (2006).
  * Ye  _et al._ [2007] M.-Y. Ye, W. Jiang, P.-X. Chen, Y.-S. Zhang, Z.-W. Zhou, and G.-C. Guo, Local distinguishability of orthogonal quantum states and generators of su($\mathit{N}$), Phys. Rev. A 76, 032329 (2007).
  * Fan [2007] H. Fan, Distinguishing bipartite states by local operations and classical communication, Phys. Rev. A 75, 014305 (2007).
  * Duan  _et al._ [2007] R. Duan, Y. Feng, Z. Ji, and M. Ying, Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication, Phys. Rev. Lett. 98, 230502 (2007).
  * Bandyopadhyay and Walgate [2009] S. Bandyopadhyay and J. Walgate, Local distinguishability of any three quantum states, J. Phys. A Math. Theor. 42, 072002 (2009).
  * Feng and Shi [2009] Y. Feng and Y. Shi, Characterizing locally indistinguishable orthogonal product states, IEEE Trans. Inf. Theory 55, 2799 (2009).
  * Duan  _et al._ [2010] R. Duan, Y. Xin, and M. Ying, Locally indistinguishable subspaces spanned by three-qubit unextendible product bases, Phys. Rev. A 81, 032329 (2010).
  * Yu  _et al._ [2012] N. Yu, R. Duan, and M. Ying, Four locally indistinguishable ququad-ququad orthogonal maximally entangled states, Phys. Rev. Lett. 109, 020506 (2012).
  * Yang  _et al._ [2013] Y.-H. Yang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Local distinguishability of orthogonal quantum states in a $2\bigotimes 2\bigotimes 2$ system, Phys. Rev. A 88, 024301 (2013).
  * Childs  _et al._ [2013] A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, A framework for bounding nonlocality of state discrimination, Commun. Math. Phys. 323, 1121 (2013).
  * Zhang  _et al._ [2014] Z.-C. Zhang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Nonlocality of orthogonal product basis quantum states, Phys. Rev. A 90, 022313 (2014).
  * Yu and Oh [2015] S. Yu and C. H. Oh, Detecting the local indistinguishability of maximally entangled states (2015), arXiv:1502.01274 [quant-ph] .
  * Zhang  _et al._ [2015] Z.-C. Zhang, F. Gao, S.-J. Qin, Y.-H. Yang, and Q.-Y. Wen, Nonlocality of orthogonal product states, Phys. Rev. A 92, 012332 (2015).
  * Wang  _et al._ [2015] Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, Nonlocality of orthogonal product-basis quantum states, Phys. Rev. A 92, 032313 (2015).
  * Chen and Johnston [2015] J. Chen and N. Johnston, The minimum size of unextendible product bases in the bipartite case (and some multipartite cases), Commun. Math. Phys. 333, 351 (2015).
  * Yang  _et al._ [2015] Y.-H. Yang, F. Gao, G.-B. Xu, H.-J. Zuo, Z.-C. Zhang, and Q.-Y. Wen, Characterizing unextendible product bases in qutrit-ququad system, Sci. Rep. 5, 11963 (2015).
  * Zhang  _et al._ [2016a] Z.-C. Zhang, F. Gao, Y. Cao, S.-J. Qin, and Q.-Y. Wen, Local indistinguishability of orthogonal product states, Phys. Rev. A 93, 012314 (2016a).
  * Xu  _et al._ [2016a] G.-B. Xu, Q.-Y. Wen, S.-J. Qin, Y.-H. Yang, and F. Gao, Quantum nonlocality of multipartite orthogonal product states, Phys. Rev. A 93, 032341 (2016a).
  * Zhang  _et al._ [2016b] X. Zhang, X. Tan, J. Weng, and Y. Li, LOCC indistinguishable orthogonal product quantum states, Sci. Rep. 6 (2016b).
  * Xu  _et al._ [2016b] G.-B. Xu, Y.-H. Yang, Q.-Y. Wen, S.-J. Qin, and F. Gao, Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system, Sci. Rep. 6 (2016b).
  * Wang  _et al._ [2017a] Y.-L. Wang, M.-S. Li, S.-M. Fei, and Z.-J. Zheng, Constructing unextendible product bases from the old ones (2017a), arXiv:1703.06542 [quant-ph] .
  * Zhang  _et al._ [2017a] Z.-C. Zhang, K.-J. Zhang, F. Gao, Q.-Y. Wen, and C. H. Oh, Construction of nonlocal multipartite quantum states, Phys. Rev. A 95, 052344 (2017a).
  * Xu  _et al._ [2017] G.-B. Xu, Q.-Y. Wen, F. Gao, S.-J. Qin, and H.-J. Zuo, Local indistinguishability of multipartite orthogonal product bases, Quantum Inf. Process. 16 (2017).
  * Wang  _et al._ [2017b] Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, The local indistinguishability of multipartite product states, Quantum Inf. Process. 16 (2017b).
  * Zhang  _et al._ [2017b] X. Zhang, J. Weng, X. Tan, and W. Luo, Indistinguishability of pure orthogonal product states by LOCC, Quantum Inf. Process. 16 (2017b).
  * Zhang  _et al._ [2017c] X. Zhang, C. Guo, W. Luo, and X. Tan, Local distinguishability of quantum states in bipartite systems (2017c), arXiv:1712.08830 [quant-ph] .
  * Zhang [2017] X. Zhang, Local distinguishability of quantum states in multipartite system (2017), arXiv:1712.08970 [quant-ph] .
  * Croke and Barnett [2017] S. Croke and S. M. Barnett, Difficulty of distinguishing product states locally, Phys. Rev. A 95, 012337 (2017).
  * Halder  _et al._ [2019a] S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Strong quantum nonlocality without entanglement, Phys. Rev. Lett. 122, 040403 (2019a).
  * Bennett  _et al._ [1999a] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality without entanglement, Phys. Rev. A 59, 1070 (1999a).
  * Rout  _et al._ [2019] S. Rout, A. G. Maity, A. Mukherjee, S. Halder, and M. Banik, Genuinely nonlocal product bases: Classification and entanglement-assisted discrimination, Phys. Rev. A 100, 032321 (2019).
  * Agrawal  _et al._ [2019] S. Agrawal, S. Halder, and M. Banik, Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition, Phys. Rev. A 99, 032335 (2019).
  * Halder  _et al._ [2019b] S. Halder, M. Banik, and S. Ghosh, Family of bound entangled states on the boundary of the peres set, Phys. Rev. A 99, 062329 (2019b).
  * Halder and Sengupta [2020] S. Halder and R. Sengupta, Distinguishability classes, resource sharing, and bound entanglement distribution, Phys. Rev. A 101, 012311 (2020).
  * Bhattacharya  _et al._ [2020] S. S. Bhattacharya, S. Saha, T. Guha, and M. Banik, Nonlocality without entanglement: Quantum theory and beyond, Phys. Rev. Research 2, 012068 (2020).
  * Halder and Srivastava [2020] S. Halder and C. Srivastava, Locally distinguishing quantum states with limited classical communication, Phys. Rev. A 101, 052313 (2020).
  * Rout  _et al._ [2021] S. Rout, A. G. Maity, A. Mukherjee, S. Halder, and M. Banik, Multiparty orthogonal product states with minimal genuine nonlocality, Phys. Rev. A 104, 052433 (2021).
  * Zhang  _et al._ [2020] Z.-C. Zhang, X. Wu, and X. Zhang, Locally distinguishing unextendible product bases by using entanglement efficiently, Phys. Rev. A 101, 022306 (2020).
  * Shi  _et al._ [2020] F. Shi, M. Hu, L. Chen, and X. Zhang, Strong quantum nonlocality with entanglement, Phys. Rev. A 102, 042202 (2020).
  * Jiang and Xu [2020] D.-H. Jiang and G.-B. Xu, Nonlocal sets of orthogonal product states in an arbitrary multipartite quantum system, Phys. Rev. A 102, 032211 (2020).
  * Zhang and Zhang [2019] Z.-C. Zhang and X. Zhang, Strong quantum nonlocality in multipartite quantum systems, Phys. Rev. A 99, 062108 (2019).
  * Yuan  _et al._ [2020] P. Yuan, G. Tian, and X. Sun, Strong quantum nonlocality without entanglement in multipartite quantum systems, Phys. Rev. A 102, 042228 (2020).
  * Terhal  _et al._ [2001] B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Hiding bits in bell states, Phys. Rev. Lett. 86, 5807 (2001).
  * DiVincenzo  _et al._ [2002] D. DiVincenzo, D. Leung, and B. Terhal, Quantum data hiding, IEEE Transactions on Information Theory 48, 580 (2002).
  * Eggeling and Werner [2002] T. Eggeling and R. F. Werner, Hiding classical data in multipartite quantum states, Phys. Rev. Lett. 89, 097905 (2002).
  * Matthews  _et al._ [2009] W. Matthews, S. Wehner, and A. Winter, Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding, Commun. Math. Phys. 291, 813 (2009).
  * Markham and Sanders [2008] D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Rev. A 78, 042309 (2008).
  * Bandyopadhyay and Halder [2021] S. Bandyopadhyay and S. Halder, Genuine activation of nonlocality: From locally available to locally hidden information, Phys. Rev. A 104, L050201 (2021).
  * Forster  _et al._ [2009] M. Forster, S. Winkler, and S. Wolf, Distilling nonlocality, Phys. Rev. Lett. 102, 120401 (2009).
  * Brito  _et al._ [2019] S. G. A. Brito, M. G. M. Moreno, A. Rai, and R. Chaves, Nonlocality distillation and quantum voids, Phys. Rev. A 100, 012102 (2019).
  * Li and Zheng [2022] M.-S. Li and Z.-J. Zheng, Genuine hidden nonlocality without entanglement: from the perspective of local discrimination (2022), arXiv:2111.02891 [quant-ph] .
  * Walgate and Hardy [2002] J. Walgate and L. Hardy, Nonlocality, asymmetry, and distinguishing bipartite states, Phys. Rev. Lett. 89, 147901 (2002).
  * Bennett  _et al._ [1999b] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product bases and bound entanglement, Phys. Rev. Lett. 82, 5385 (1999b).
  * DiVincenzo  _et al._ [2003] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product bases, uncompletable product bases and bound entanglement, Commun. Math. Phys. 238, 379 (2003).

## Proof of Proposition 3

###### Proof.

We will first prove that the above set of states are free from local
redundancy through a detailed case-wise analysis:

Case-I: More than one players discard their qutrits: Observe that, such
operations lead to $27$ tripartite states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ (when all the
qutrits have been traced out) or in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{6}$ or its party
permutations (when two of the qutrits have been traced out). Such reduced sets
can not be orthogonal in any way.

Case-II: Any one player discards their qutrit: Without loss of generality, we
can analyze the case when Charlie has traced out the qutrit system. Same
argument also works for other two players.

Note that under such an operation (discarding the qutrit) only possibility of
non-orthogonality arise from any of the twelve pairs of
$\\{\ket{\xi_{i}^{\pm}}\\}$.

Again, from the symmetric structure of these states, the non-orthogonality of
all those $12$ pairs can be sufficiently guaranteed by checking only
$\\{\ket{\xi_{1}^{\pm}}\\}$ and $\\{\ket{\xi_{6}^{\pm}}\\}$. Checking the
first one is sufficient for $i\in\\{1,2,3,4\\}$ pairs and the second one is
sufficient for $i\in\\{6,7,8,9,11,12,13,14\\}$ pairs. From now on, we will use
the party annotation for clarity. We will denote each party’s qubit system
with $1$ in subscript and qutrit with $2$ in subscript. For example,
$\ket{\xi_{1}^{\pm}}_{{a_{1}a_{2}b_{1}b_{2}c_{1}c_{2}}}$ implies that Alice,
Bob and Charlies’ qubits are denoted by the subsystems $a_{1},b_{1}$ and
$c_{1}$, whereas their qutrit subsystems are denoted by $a_{2},b_{2}$ and
$c_{2}$.

Now, tracing out Charlie’s qutrit from
$\ket{\xi_{1}^{\pm}}_{{a_{1}a_{2}b_{1}b_{2}c_{1}c_{2}}}$, we will obtain,
$\rho_{{a_{1}a_{2}b_{1}b_{2}c_{1}}}^{1\pm}=\text{Tr}_{c_{2}}(\ket{\xi_{1}^{\pm}}\bra{\xi_{1}^{\pm}}_{{a_{1}a_{2}b_{1}b_{2}c_{1}c_{2}}})=\ket{\mathbf{0-4}}\bra{\mathbf{0-4}}_{a_{1}a_{2}}\otimes\ket{\mathbf{1-5}}\bra{\mathbf{1-5}}_{b_{1}b_{2}}\otimes(\frac{1}{2}\ket{\alpha_{\pm}}\bra{\alpha_{\pm}}+\frac{1}{2}\times\frac{\mathbb{I}}{2})_{c_{1}}$,
where $\ket{\alpha_{\pm}}=\frac{\ket{0}\pm\ket{1}}{\sqrt{2}}$ and the
superscript of $\rho_{{a_{1}a_{2}b_{1}b_{2}c_{1}}}^{1\pm}$ implies that it is
a reduced state of $\ket{\xi_{1}^{\pm}}$. It trivially follows that the states
$\rho_{{a_{1}a_{2}b_{1}b_{2}c_{1}}}^{1\pm}$ are not orthogonal to each other.

Case-III: Any one player discards their qubit: It is sufficient to analyze the
case $\mathbb{C}^{6}\otimes\mathbb{C}^{6}\otimes\mathbb{C}^{3}$. This does not
lose any generality.

Consider the states $\ket{\xi_{1}^{\pm}}_{{a_{1}a_{2}b_{1}b_{2}c_{1}c_{2}}}$.
Discarding Charlie’s qubit from this state, we get
$\rho_{{a_{1}a_{2}b_{1}b_{2}c_{2}}}^{1\pm}={Tr}_{c_{1}}(\ket{\xi_{1}^{\pm}}\bra{\xi_{1}^{\pm}}_{{a_{1}a_{2}b_{1}b_{2}c_{1}c_{2}}})=\ket{\mathbf{0-4}}\bra{\mathbf{0-4}}_{a_{1}a_{2}}\otimes\ket{\mathbf{1-5}}\bra{\mathbf{1-5}}_{b_{1}b_{2}}\otimes(\frac{1}{2}\ket{\alpha_{\pm}}\bra{\alpha_{\pm}}+\frac{1}{2}\ket{\beta_{\pm}}\bra{\beta_{\pm}})_{c_{2}}$,
where $\ket{\alpha_{\pm}}=\frac{\ket{0}\pm\ket{1}}{\sqrt{2}}$ and
$\ket{\beta_{\pm}}=\frac{\ket{1}\pm\ket{2}}{\sqrt{2}}$. Note that,
$\ket{\alpha_{\pm}}$ is not orthogonal to $\ket{\beta_{\pm}}$, it is, thus,
straightforward to assure the non-orthogonality of
$\rho_{{a_{1}a_{2}b_{1}b_{2}c_{2}}}^{1\pm}$.

Case-IV: More than one players discard their qubits: The non-orthogonality
depicted in Case-III, guarantees that tracing out more systems from
$\rho_{{a_{1}a_{2}b_{1}b_{2}c_{2}}}^{1\pm}$ can not make them orthogonal.

Figure 1: Local discrimination protocol for the set $\mathcal{G}_{3}$. The
green boxes stand for the case where the state has been distinguished by
clicking of a particular projector and the blue boxes refer to the ultimate
measurement which has to be performed on a set of two states to perfectly
distinguish them.

Case-V: Any one player discards their qubits and any other discards their
qutrit: This case is also a subset of Case-III. It is evident that discarding
any qutrits from Alice’s or Bob’s system does not make the states
$\rho_{{a_{1}a_{2}b_{1}b_{2}c_{2}}}^{1\pm}$ orthogonal. This completes our
proof that $\mathcal{G}_{3}$ does not have local redundancy.

We now move to the proof of local distinguishability of the set
$\mathcal{G}_{3}$. We first define some important measurements which will be
required to perform by the players at different steps of the protocol:

$\displaystyle\mathcal{M}_{1}$
$\displaystyle\equiv\\{P_{1}:=P[\ket{\mathbf{0-4}}],P_{2}:={P}[\ket{\mathbf{1-5}}],P_{3}:={P}[\ket{\mathbf{2-3}}],P_{4}:=\mathbb{I}-({P}[\ket{\mathbf{0-4}}]+{P}[\ket{\mathbf{1-5}}]+{P}[\ket{\mathbf{2-3}}])\\}$
$\displaystyle\mathcal{M}_{2}$
$\displaystyle\equiv\\{Q_{1}:=P[\ket{\mathbf{0+1+4+5}}],Q_{2}:=P[\ket{\mathbf{0-1+4-5}}],Q_{3}:=\mathbb{I}-(P[\ket{\mathbf{0+1+4+5}}]+[\ket{\mathbf{0-1+4-5}}])\\}$
$\displaystyle\mathcal{M}_{3}$
$\displaystyle\equiv\\{R_{1}:=P[\ket{\mathbf{0+2+4+3}}],R_{2}:=P[\ket{\mathbf{0-2+4-3}}],R_{3}:=\mathbb{I}-(P[\ket{\mathbf{0+2+4+3}}]+{P}[\ket{\mathbf{0-2+4-3}}])\\}.$

For rest of the proof, we use $\mathcal{M}_{k}^{i}$ to mean that the
party-$i\in\\{A,B,C\\}$ has performed the measurement $k\in\\{1,2,3\\}$. In a
similar fashion, $P_{k}^{i}$ implies that $P_{k}$ projector is clicked when
measurement $\mathcal{M}_{1}^{i}$ is performed on $i$th party’s subsystem. Let
us now move to the distinguishability protocol. First, Alice performs the
local measurement $\mathcal{M}_{1}^{A}$. Consequently, depending upon the
given state, any four of the projectors can click. Whatever outcome Alice
gets, Bob will perform measurement $\mathcal{M}_{1}^{A}$ in the second step
and so on. The detailed step by step analysis is pictorially presented in
Fig.1. Note that it is evident that any two pure orthogonal multipartite
states can be perfectly locally distinguished always. We thus described our
protocol up to the point when the players sole task boils down to locally
distinguish two pure orthogonal states. It is also important to mention that
the measurements $\mathcal{M}_{2}$ and $\mathcal{M}_{3}$ will be mainly
required to distinguish two orthogonal pure states at the ultimate steps of
the protocol. This completes the proof.

∎

## Proof of Proposition 4

###### Proof.

The proof that the set $\mathcal{G}_{4}$ is free from local redundancy is
quite evident from the proof of Proposition 3. Thus, here, we will describe
only the local distinguishability protocol of $\mathcal{G}_{4}$.

Step-1: Charlie performs a measurement:

$\displaystyle\mathcal{M}^{C}_{1}$
$\displaystyle\equiv\\{P_{1}^{C}:=P[\ket{\mathbf{0-4}}_{C}],P_{2}^{C}:={P}[\ket{\mathbf{1-5}}_{C}],P_{3}^{C}:={P}[\ket{\mathbf{2-3}}_{C}],P_{4}^{C}:=\mathbb{I}-({P}[\ket{\mathbf{0-4}}_{C}]+{P}[\ket{\mathbf{1-5}}_{C}]+{P}[\ket{\mathbf{2-3}}_{C}])\\}$

When $P_{1}^{C}$ clicks, the given state must be any of
$\ket{\zeta_{6}^{\pm}},\ket{\zeta_{7}^{\pm}}$ or $\ket{\zeta_{5}}$. If
$P_{2}^{C}$ clicks, we can say that the state is one of
$\ket{\zeta_{8}^{\pm}},\ket{\zeta_{11}^{\pm}},\ket{\zeta_{14}^{\pm}}$ or
$\ket{\zeta_{10}}$. For the third outcome: click of $P_{3}^{C}$ implies that
the state can be one of
$\ket{\zeta_{9}^{\pm}},\ket{\zeta_{12}^{\pm}},\ket{\zeta_{13}^{\pm}}$ or
$\ket{\zeta_{15}}$. Lastly, if $P_{4}^{C}$ clicks, the state can be any of
$\ket{\zeta_{1}^{\pm}},\ket{\zeta_{2}^{\pm}},\ket{\zeta_{3}^{\pm}},\ket{\zeta_{4}^{\pm}}$.

Step-2: For different outcomes of $\mathcal{M}_{1}^{C}$, our protocol splits
into different branches. For each outcome of $\mathcal{M}_{1}^{C}$, we will
describe the next step of our protocol below.

When the outcome $P_{1}^{C}$ clicks on measuring $\mathcal{M}_{1}^{C}$, three
parties are now to discriminate between
$\ket{\zeta_{6}^{\pm}},\ket{\zeta_{7}^{\pm}}$ and $\ket{\zeta_{5}}$ states.
Now, Alice will perform a measurement

$\displaystyle\mathcal{M}^{A}_{1}$
$\displaystyle\equiv\\{P_{1}^{A}:=P[\ket{0}_{A}],P_{2}^{A}:={P}[\ket{{1}}_{A}],P_{3}^{A}:={P}[\ket{{2}}_{A}]\\}$

In case when $P_{1}^{A}$ clicks, the players will be sure that the given state
is $\ket{\zeta_{5}}$. Otherwise, when $P_{2}^{A}$ or $P_{3}^{A}$ clicks, in
each case the players are to distinguish between only two states
$\ket{\zeta_{6}^{\pm}}$ or $\ket{\zeta_{7}^{\pm}}$ respectively and this is
always possible [11, 70].

Consider now that after the measurement $\mathcal{M}_{1}^{C}$, $P_{2}^{C}$
clicks. Therefore, the task is now to distinguish among
$\ket{\zeta_{8}^{\pm}},\ket{\zeta_{11}^{\pm}},\ket{\zeta_{14}^{\pm}}$ and
$\ket{\zeta_{10}}$. At this point, Bob will perform a measurement,

$\displaystyle\mathcal{M}^{B}_{1}$
$\displaystyle\equiv\\{P_{1}^{B}:=P[(\ket{0},\ket{1})_{B}],P_{2}^{B}:={P}[\ket{{2}}_{B}]\\}$

Now, if $P_{2}^{B}$ clicks the given state must be either of
$\ket{\zeta_{14}^{\pm}}$ which can be easy distinguished by a measurement on
Alice’s side [11, 70]. When $P_{1}^{B}$ clicks the given state can be any of
$\ket{\zeta_{8}^{\pm}},\ket{\zeta_{11}^{\pm}}$ and $\ket{\zeta_{10}}$. At this
point, Alice will perform a measurement:

$\displaystyle\mathcal{M}^{A}_{2}$
$\displaystyle\equiv\\{Q_{1}^{A}:=P[(\ket{0},\ket{1})_{A}],Q_{2}^{A}:={P}[\ket{{2}}_{A}]\\}$

If $Q_{2}^{A}$ clicks the given state must be either of
$\ket{\zeta_{8}^{\pm}}$ which can again be distinguished perfectly [11, 70].
When $Q_{1}^{A}$ clicks, the players are to distinguish among
$\ket{\zeta_{11}^{\pm}}$ and $\ket{\zeta_{10}}$. Now, Bob will perform a
measurement:
$\mathcal{M}^{B}_{2}\equiv\\{Q_{1}^{B}:=P[\ket{0}_{B}],Q_{2}^{B}:={P}[\ket{{1}}_{B},Q_{3}^{B}:={P}[\ket{{2}}_{B}]$.
If $Q_{2}^{B}$ clicks, then the given state is certainly $\ket{\zeta_{10}}$.
Otherwise, if $Q_{1}^{B}$ clicks the state is either of
$\ket{\zeta_{11}^{\pm}}$ which can be perfectly locally discriminated [11,
70].

Now, consider that on measuring $\mathcal{M}_{1}^{C}$, $P_{3}^{C}$ clicks.
Consequently, the players are now to distinguish between
$\ket{\zeta_{9}^{\pm}},\ket{\zeta_{12}^{\pm}},\ket{\zeta_{13}^{\pm}}$ or
$\ket{\zeta_{15}}$. Now, Bob will perform a measurement:
$\mathcal{M}^{B}_{3}\equiv\\{T_{1}^{B}:=P[(\ket{0},\ket{2})_{B}],T_{2}^{B}:={P}[\ket{{1}}_{B}]\\}$.
If $T_{2}^{B}$ clicks then the given state must be either of
$\ket{\zeta_{13}^{\pm}}$ which can be perfectly locally distinguished [11,
70]. On the other hand, if $T_{1}^{B}$ clicks, the state is certainly any of
$\ket{\zeta_{9}^{\pm}},\ket{\zeta_{12}^{\pm}},\ket{\zeta_{15}}$. Now, Alice
performs a measurement:
$\mathcal{M}^{A}_{2}\equiv\\{R_{1}^{A}:=P[(\ket{0},\ket{2})_{A}],R_{2}^{A}:={P}[\ket{{1}}_{A}]\\}$.
If $R_{2}^{A}$ clicks, the given state is either of $\ket{\zeta_{9}^{\pm}}$
which can be perfectly distinguished via LOCC. Otherwise, for the click
$R_{1}^{A}$ the state can be any of $\ket{\zeta_{12}^{\pm}},\ket{\zeta_{15}}$.
At this point, Bob will again perform a measurement,
$\mathcal{M}^{B}_{4}\equiv\\{N_{1}^{B}:=P[\ket{0}_{B}],N_{2}^{B}:={P}[\ket{{1}}_{B}],N_{3}^{B}:={P}[\ket{{2}}_{B}]\\}$.
If $N_{1}^{B}$ clicks the state must be either of $\ket{\zeta_{12}^{\pm}}$
which can be locally distinguishable [11, 70]. Otherwise, if $N_{3}^{B}$
clicks the state must be $\ket{\zeta_{15}}$.

Ultimately, consider the case when $P_{4}^{C}$ clicks when
$\mathcal{M}_{1}^{C}$ is measured. Here, the players are to distinguish
between
$\ket{\zeta_{1}^{\pm}},\ket{\zeta_{2}^{\pm}},\ket{\zeta_{3}^{\pm}},\ket{\zeta_{4}^{\pm}}$.
Now, Alice will perform a measurement:
$\mathcal{M}^{A}_{3}\equiv\\{T_{1}^{A}:=P[\ket{0}_{A}],T_{2}^{A}:={P}[\ket{{1}}_{A}],T_{3}^{A}:={P}[\ket{{2}}_{A}]\\}$.
If $T_{2}^{A}$ clicks, the given state must be either of
$\ket{\zeta_{3}^{\pm}}$ which can be locally distinguished [11, 70]. If
$T_{3}^{A}$ clicks, the given state is any of $\ket{\zeta_{4}^{\pm}}$ which
can again be locally distinguished [11, 70]. Otherwise, if $T_{1}^{A}$ clicks
then the given state can be any of
$\ket{\zeta_{1}^{\pm}},\ket{\zeta_{2}^{\pm}}$. Bob will now perform a
measurement:
$\mathcal{M}^{B}_{5}\equiv\\{L_{1}^{B}:=P[\ket{0}_{B}],L_{2}^{B}:={P}[\ket{{1}}_{B}],L_{3}^{B}:={P}[\ket{{2}}_{B}]\\}$.
If $L_{2}^{B}$ clicks the given state can be either of $\ket{\zeta_{1}^{\pm}}$
and if $L_{3}^{B}$ clicks the given state is any of $\ket{\zeta_{2}^{\pm}}$.
Further, $\ket{\zeta_{1}^{\pm}}$ (or, $\ket{\zeta_{2}^{\pm}}$) can be
perfectly locally distinguished [11, 70]. This concludes the local
distinguishability protocol of $\mathcal{G}_{4}$. ∎