# Local Identification of Subsets of Quantum states: A Stronger Quantum
Nonlocality

Pratik Ghosal  Department of Physics, Bose Institute, Unified Academic Campus,
EN 80, Sector V, Bidhan Nagar, Kolkata 700 091, India  Arkaprabha Ghosal
Department of Physics, Bose Institute, Unified Academic Campus, EN 80, Sector
V, Bidhan Nagar, Kolkata 700 091, India  Subhendu B. Ghosh  Physics and
Applied Mathematics Unit, Indian Statistical Institute, Kolkata, 203 B. T.
Road, Kolkata 700108, India  Amit Mukherjee  Department of Physics, Bose
Institute, Unified Academic Campus, EN 80, Sector V, Bidhan Nagar, Kolkata 700
091, India

###### Abstract

Nonolocality makes quantum theory nontrivially sacred and useful in the
paradigm of information theoretic tasks. Apart from Bell nonlocality, which
deals with measurement outcome statistics of spatially separated agents, there
is also another kind of quantum nonlocality, that is associated with perfect
distinguishability of quantum states by local operations and classical
communication (LOCC). We propose a distributed task: perfect identification of
subsets of a known set of multipartite orthogonal states by LOCC, namely,
local subset identification. Failure in accomplishing this task guarantees a
new notion of quantum nonlocality, viz., local subset unidentifiability. Here,
we show that both local distinguishability and local markability of quantum
states implies local subset identifiability, but the converse is not
necessarily true. This makes local subset unidentifiability a stronger quantum
nonlocal phenomenon than its predecessors – local indistinguishability and
local unmarkability. Moreover, we also present an even stronger version of
local subset unidentifiablity involving more than two spatially separated
parties namely, genuine local subset unidentifiability, where a given subset
becomes identifiable if and only if all the parties come together in a common
lab.

##  I Introduction

In any operational theory, if two or more spatially separated communicating
agents are able to perform a distributed task then it is trivially executable
when they come together and collaborate in the same lab. The converse
statement is true in classical world. Interestingly, in quantum regime, the
scenario gets more complex. There exist distributed quantum tasks that can not
be successfully accomplished locally, even though they are accomplishable when
the agents collaborate. This phenomenon exhibits a stark deviation of quantum
theory from its classical counterpart. Usually, this anomaly is termed as
quantum nonlocality. In general, there are two kinds of such non-classical
phenomena: one manifests nonlocality with the help of exotic structure and
topology of quantum state space solely, the other deals with the incapability
of explaining correlations by locality and realism. Customarily, the later one
was named after John Bell as Bell nonlocality [1]. However, both of them have
huge foundational and application-based significance in the current
development of quantum theory. In this work, we are especially interested in
the quantum nonlocality of the first kind. One such celebrated phenomenon is
the perfect indistinguishability of certain mutually orthogonal multipartite
quantum states via local operations and classical communication (LOCC). In the
past decades, a plethora of interesting results were developed along this line
[2, 3, 4, 5, 6, 7, 8, 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]. Very
recently, some other distributed quantum tasks have also been developed that
manifest a stronger form of quantum nonlocality, e.g., local irreducibility
[40], local unmarkability [41] etc. In this work, we present an even stronger
form of quantum nonlocality exhibited by certain sets of mutually orthogonal
entangled states. Here, the concerned distributed task, namely, local subset
identification (LSI), is to perfectly locally identify the elements of a
strict subset (having cardinality greater than one), chosen from a known set
of mutually orthogonal multipartite states by a number of spatially separated,
classically communicating agents. More precisely, from a known set of mutually
orthogonal states, more than one state will be chosen and shared among
multiple spatially separated agents. The winning condition / objective of the
task is to perfectly identify the shared states via LOCC. We show here that
the incapability of accomplishing this task for a set exhibits an even
stronger quantum nonlocal phenomenon than that discussed in the existing
literature.

The nearest related distributed task to the one proposed here is local state
marking (LSM) [41]. However, there are many obvious and crucial differences
between these two. For example, take a set of bipartite orthogonal states
$\\{\ket{\psi_{i}}\\}_{i=1}^{N}$. Here, the well known task of local state
discrimination (LSD) raise the question of perfectly locally distinguishing a
state that is chosen from the above set and shared among spatially separated
agents. However, LSM deals with a different question. For example, consider
that any two states from the set are shared between spatially separated
agents. At this point, local state marking asks to perfectly locally recognize
these two states as well as specify the order of the shared states, that is,
whether the shared states are $\ket{\psi_{j}}\otimes\ket{\psi_{k}}$ or
$\ket{\psi_{k}}\otimes\ket{\psi_{j}}$ for any $j\neq k$. If for a specific set
of states this distributed task can not be achieved perfectly locally, the set
is said to exhibit a stronger quantum nonlocality than the local
indistinguishability. This is denoted as local unmarkability [41]. However, in
case of LSI, the question is to perfectly locally recognize just the identity
of the states, not the order. For example, unlike the previous task, the
agents only need to perfectly identify which $\ket{\psi_{j}}$ and
$\ket{\psi_{k}}$ are provided from the original set
$\\{\ket{\psi_{i}}\\}_{i=1}^{N}$ via LOCC. Clearly, the complexity of this
proposed task is weaker than that of the LSM. Therefore, inability to
accomplish this task for a specific set demonstrates a form of quantum
nonlocality that is even stronger than local unmarkability as well as local
state indistinguishability. We termed this nonlocal phenomenon as local subset
unidentifiability. More importantly, this stronger nonlocality emerges due to
the presence of locally indistinguishable subspaces of rank not necessarily
restricted to one, whereas, local unmarkability appears because of the
existence of higher dimensional indistinguishable vectors.

It is quite evident that any exotic non-classical feature that evolves from
quantum mechanics gets more complex when multipartite scenario comes into the
picture. In case of LSI, we also explore the scenario involving more than two
spatially separated agents. Interestingly, we come up with a further stronger
version of the nonlocality we are introducing here. In a scenario involving
more than two agents, we present a set of tripartite states and a set of four
party states which are locally unidentifiable when all the agents are
spatially separated. Surprisingly, we also show that these sets are locally
unidentifiable in all possible bi-partitions. Therefore, to perfectly
accomplish the LSI task, all the agents need to come together or they must
resort to additional quantum resources. This certainly exhibits a stronger
form of nonlocality than its bipartite version. We term it as genuine local
subset unidentifiablity. It is important to mention here that, although the
LSM task has been formally defined for more than two spatially separated
agents, till now there are no examples in the existing literature, of genuine
local unmarkability, in the sense discussed above. By presenting our example
of genuine locally unidentifiable subsets of a particular set, we also provide
the first example of genuine locally unmarkable subsets. This is because, as
shown later in this paper, local unidentifiablility of a subset naturally
implies its local unmarkability.

Our manuscript is presented in the following way. In section II, we introduce
the concerned distributed task: LSI and present its elegance over LSM. In
section III, we discuss the stronger version local subset unidentifiablity in
multipartite framework. Followed by in section IV, we conclude with a
discussion and possible open directions.

##  II Local Subset Identification

We are now in a position to articulate the formal definition of our task –
local subset identification.

######  Definition 1
($(n,\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert})$-Local Subset
Identification).

Consider a set
$\mathcal{S}=\\{\ket{\psi_{1}},\ket{\psi_{2}},\ldots,\ket{\psi_{n}}\\}\subset\bigotimes_{k=1}^{N}\mathbb{C}^{d_{k}}$
of $n$ $N$-party orthogonal quantum states. A subset
$\mathcal{S}^{\prime}\subset\mathcal{S}$ containing
$1\leq\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert}<n$ quantum
states is randomly chosen and distributed among spatially separated agents
keeping the identities of the states hidden. The task of
$(n,\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert})$-local
identification is to perfectly determine the elements of the set
$\mathcal{S}^{\prime}$.

We call a set $\mathcal{S}$ to be
$(n,\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert})$-identifiable
if all possible subsets of $\mathcal{S}$, having the same cardinality as
$\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert}$, are locally
identifiable. It is evident that
$\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert}=n$ is a trivial
question in local state identification framework, as the elements of the
entire set $\mathcal{S}$ is already known to the agents. On the other extreme,
when $\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert}=1$, this
boils down to a well known problem:

######  Lemma 1.

For a given set $\mathcal{S}$ the task of $(n,1)$-LSI corresponds to the well-
known task of Local State Discrimination (LSD).

The reader may also wonder how LSI is related with the LSM task. But before
delving into that, let us first formally define the LSM task. Consider the
same set $\mathcal{S}$ of $n$ $N$-party orthogonal quantum states and a subset
$\mathcal{S}^{\prime}$ having
$\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert}$ quantum states is
randomly chosen from it and distributed among spatially separated agents
keeping the identities of the states hidden. The task of
$(n,\mathinner{\\!\left\lvert\mathcal{S}^{\prime}\right\rvert})$-LSM is to
perfectly determine the order of the elements of the set
$\mathcal{S}^{\prime}$ [41] as given to the $N$ agents. From definition of
these two tasks the next lemma follows directly.

######  Lemma 2 ($(n,m)$-LSM $\Rightarrow$ $(n,m)$-LSI).

If a set $\mathcal{S}$ is $(n,m)$-markable, then it readily follows that it is
also $(n,m)$-identifiable.

If the $N$ spatially separated agents can locally determine the order of the
states of the given subset $\mathcal{S^{\prime}}$, then the identity of the
states is trivially known to them. However, the converse of this statement is
not necessarily true.

######  Theorem 1.

If a set $\mathcal{S}$ is $(n,m)$-identifiable, then it is not necessarily
$(n,m)$-markable.

###### Proof.

The proof is constructive. Consider a set of three Bell states, say,
$\mathcal{S}=\\{\ket{B_{1}},\ket{B_{2}},\ket{B_{3}}\\}_{\mathtt{AB}}\subset\mathbb{C}_{\mathtt{A}}^{2}\otimes\mathbb{C}_{\mathtt{B}}^{2}$,
where
$\ket{B_{1}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}),\ket{B_{2}}=\frac{1}{\sqrt{2}}(\ket{00}-\ket{11}),\ket{B_{3}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$.
A subset containing two distinct Bell states from the set $\mathcal{S}$ can be
chosen in ${3\choose 2}=3$ ways, namely,
$\mathcal{S}^{\prime}_{1}=\\{\ket{B_{1}},\ket{B_{2}}\\},\mathcal{S}^{\prime}_{2}=\\{\ket{B_{2}},\ket{B_{3}}\\}$,
and $\mathcal{S}^{\prime}_{3}=\\{\ket{B_{3}},\ket{B_{1}}\\}$. Note that, the
order of the elements of the subsets is not important. One such subset is
randomly selected and the elements are distributed among two spatially
separated agents, Alice and Bob.

$(3,2)-$identifiability: Alice and Bob can successfully identify the shared
subset locally, using the following protocol. Each agent performs the
following measurement $\mathcal{M}_{\alpha}\mathrel{\mathop{\mathchar
58\relax}}=\\{P^{\alpha}_{i}|\sum_{i=1}^{4}P^{\alpha}_{i}=\mathbb{I}_{4}\\}$,
where $P^{\alpha}_{i}\mathrel{\mathop{\mathchar
58\relax}}=\ket{B_{i}}_{\alpha}\bra{B_{i}}$,
$\alpha\in\\{\mathtt{A}_{1}\mathtt{A}_{2},\mathtt{B}_{1}\mathtt{B}_{2}\\}$, on
their respective share of two qubits. Afterwards, the agents can know the
identity of the shared subset by tallying their measurement outcomes via
classical communication.

For example, if $\mathcal{S}^{\prime}_{1}$ is distributed among the agents
then their shared two states can be either of the following two orders—
$\ket{B_{1}}_{\mathtt{A}_{1}\mathtt{B}_{1}}\otimes\ket{B_{2}}_{\mathtt{A}_{2}\mathtt{B}_{2}}$
or,
$\ket{B_{2}}_{\mathtt{A}_{1}\mathtt{B}_{1}}\otimes\ket{B_{1}}_{\mathtt{A}_{2}\mathtt{B}_{2}}$.
However, we can write,

$\displaystyle\ket{B_{1}}_{\mathtt{A}_{1}\mathtt{B}_{1}}\ket{B_{2}}_{\mathtt{A}_{2}\mathtt{B}_{2}}=\ket{B_{1}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{2}}_{\mathtt{B}_{1}\mathtt{B}_{2}}+\ket{B_{2}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{1}}_{\mathtt{B}_{1}\mathtt{B}_{2}}$
$\displaystyle+\ket{B_{3}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{4}}_{\mathtt{B}_{1}\mathtt{B}_{2}}+\ket{B_{4}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{3}}_{\mathtt{B}_{1}\mathtt{B}_{2}},$

and,

$\displaystyle\ket{B_{2}}_{\mathtt{A}_{1}\mathtt{B}_{1}}\ket{B_{1}}_{\mathtt{A}_{2}\mathtt{B}_{2}}=\ket{B_{1}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{2}}_{\mathtt{B}_{1}\mathtt{B}_{2}}+\ket{B_{2}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{1}}_{\mathtt{B}_{1}\mathtt{B}_{2}}$
$\displaystyle-\ket{B_{3}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{4}}_{\mathtt{B}_{1}\mathtt{B}_{2}}-\ket{B_{4}}_{\mathtt{A}_{1}\mathtt{A}_{2}}\ket{B_{3}}_{\mathtt{B}_{1}\mathtt{B}_{2}}.$

It readily follows that if Alice and Bob perform
$\mathcal{M}_{\mathtt{A}_{1}\mathtt{A}_{2}}$ and
$\mathcal{M}_{\mathtt{B}_{1}\mathtt{B}_{2}}$ respectively, then their
corresponding outcomes will never be able to reveal the order of $\ket{B_{1}}$
and $\ket{B_{2}}$, but can distinguish between different
$\mathcal{S}^{\prime}_{i}$s (See Table 1).

$\mathcal{S}^{\prime}_{1}$ | $\mathcal{S}^{\prime}_{2}$ | $\mathcal{S}^{\prime}_{3}$
---|---|---
$(1,2)$ | $(1,4)$ | $(1,3)$
$(2,1)$ | $(2,3)$ | $(2,4)$
$(3,4)$ | $(3,2)$ | $(3,1)$
$(4,3)$ | $(4,1)$ | $(4,2)$
Table 1: Here, $(\clubsuit,\spadesuit)$ denotes that the outcome of Alice’s
measurement is $P_{\clubsuit}^{\mathtt{A}_{1}\mathtt{A}_{2}}$ and Bob’s is
$P_{\spadesuit}^{\mathtt{B}_{1}\mathtt{B}_{2}}$. The table states that for a
specific subset, say, $\mathcal{S}^{\prime}_{1}$, the outcomes at Alice’s and
Bob’s site can be any of $(1,2),(2,1),(3,4),(4,3)$.

A similar protocol can always be devised to identify two of any three Bell
states.

$(3,2)-$unmarkability: However, Alice and Bob can never be able to locally
mark (i.e., distinguish between the orders of) two states from any set of
three Bell states. This is because, this task tantamounts to distinguishing a
state from a set of six (${}^{3}P_{2}=6$)
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ maximally entangled states, which is
impossible according to Ref. [42]. ∎

The above example already establishes the fact that, the complexity of the LSI
task is essentially weaker than that of LSM. Nevertheless, LSI provides the
ground to demonstrate a hitherto unrevealed stronger form of quantum
nonlocality. If any set is incapable of accomplishing LSI then that must be an
example of stronger nonlocality than local unmarkability. Below we provide
some examples of sets which shows this strong form of nonlocality, namely,
local unidentifiability.

######  Theorem 2.

The set of four two-qubit maximally entangled states is (4,2)-unidentifiable.

###### Proof.

If a subset $\mathcal{S}^{\prime}_{i}$ containing two states from a set
$\mathcal{S}$ of four two-qubit Bell states is shared among two spatially
separated agents, Alice and Bob, they will never be able to locally
distinguish between the $\mathcal{S}^{\prime}_{i}$s, i.e., they will never be
able to know the identity of the two given Bell states, let alone their
ordering. Following is the detailed proof.

Consider that $\mathcal{S}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{1}},\ket{B_{2}},\ket{B_{3}},\ket{B_{4}}$}, where,
$\ket{B_{4}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$. A subset containing any
two Bell states can be chosen in $4\choose 2$, or six ways, namely,
$\mathcal{S}^{\prime}_{1}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{1}},\ket{B_{2}}\\}$,
$\mathcal{S}^{\prime}_{2}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{1}},\ket{B_{3}}\\}$,
$\mathcal{S}^{\prime}_{3}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{1}},\ket{B_{4}}\\}$,
$\mathcal{S}^{\prime}_{4}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{2}},\ket{B_{3}}\\}$,
$\mathcal{S}^{\prime}_{5}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{2}},\ket{B_{4}}\\}$,
$\mathcal{S}^{\prime}_{6}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{B_{3}},\ket{B_{4}}\\}$. Precisely, for example, if
$\mathcal{S}^{\prime}_{1}$ is distributed among the agents then they actually
share a mixed state,

$\begin{split}\rho_{1}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\ket{B_{1}}\bra{B_{1}}_{\mathtt{A_{1}B_{1}}}&\otimes\ket{B_{2}}\bra{B_{2}}_{\mathtt{A_{2}B_{2}}}\\\
&+\tfrac{1}{2}\ket{B_{2}}\bra{B_{2}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{1}}\bra{B_{1}}_{\mathtt{A_{2}B_{2}}}.\end{split}$

Similarly, other five states corresponding to the subsets are as follows:

$\displaystyle\begin{split}\rho_{2}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\ket{B_{1}}\bra{B_{1}}_{\mathtt{A_{1}B_{1}}}&\otimes\ket{B_{3}}\bra{B_{3}}_{\mathtt{A_{2}B_{2}}}\\\
&+\tfrac{1}{2}\ket{B_{3}}\bra{B_{3}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{1}}\bra{B_{1}}_{\mathtt{A_{2}B_{2}}},\\\
\rho_{3}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\ket{B_{1}}\bra{B_{1}}_{\mathtt{A_{1}B_{1}}}&\otimes\ket{B_{4}}\bra{B_{4}}_{\mathtt{A_{2}B_{2}}}\\\
&+\tfrac{1}{2}\ket{B_{4}}\bra{B_{4}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{1}}\bra{B_{1}}_{\mathtt{A_{2}B_{2}}},\\\
\rho_{4}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\ket{B_{2}}\bra{B_{2}}_{\mathtt{A_{1}B_{1}}}&\otimes\ket{B_{3}}\bra{B_{3}}_{\mathtt{A_{2}B_{2}}}\\\
&+\tfrac{1}{2}\ket{B_{3}}\bra{B_{3}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{2}}\bra{B_{2}}_{\mathtt{A_{2}B_{2}}},\\\
\rho_{5}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\ket{B_{2}}\bra{B_{2}}_{\mathtt{A_{1}B_{1}}}&\otimes\ket{B_{4}}\bra{B_{4}}_{\mathtt{A_{2}B_{2}}}\\\
&+\tfrac{1}{2}\ket{B_{4}}\bra{B_{4}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{2}}\bra{B_{2}}_{\mathtt{A_{2}B_{2}}},\\\
\rho_{6}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\ket{B_{3}}\bra{B_{3}}_{\mathtt{A_{1}B_{1}}}&\otimes\ket{B_{4}}\bra{B_{4}}_{\mathtt{A_{2}B_{2}}}\\\
&+\tfrac{1}{2}\ket{B_{4}}\bra{B_{4}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{3}}\bra{B_{3}}_{\mathtt{A_{2}B_{2}}}.\end{split}$

Clearly, the task of locally distinguishing the subsets
$\\{\mathcal{S}^{\prime}_{i}\\}_{i=1}^{6}$ is equivalent to distinguishing
between the mixed states $\\{\rho_{i}\\}_{i=1}^{6}$ by LOCC. We now prove our
claim by reductio ad absurdum.

Consider now that the states $\\{\rho_{i}\\}_{i=1}^{6}$ are locally
distinguishable. Consequently, their supports are also distinguishable via
LOCC [43]. Hence, any set of pure states $\\{\ket{\xi_{i}}\\}_{i=1}^{6}$,
where $\forall i$ $\ket{\xi_{i}}\in\text{Support}(\rho_{i})$, must also be
locally distinguishable. On the other hand, it is easy to see that the set
$\\{\ket{B_{j}}_{\mathtt{A_{1}B_{1}}}\otimes\ket{B_{k}}_{\mathtt{A_{2}B_{2}}}\\}_{j<k}$
for $j,k\in\\{1,2,3,4\\}$, is locally indistinguishable [42]. But, each
element of the above set of maximally entangled states belongs to the support
of six different $\rho_{i}$s. This readily contradicts our assumption that
$\\{\rho_{i}\\}_{i=1}^{6}$ are locally distinguishable and, thereby, concludes
our proof. ∎

We also come up with a number of sets containing orthogonal general maximally
entangled states which fails to accomplish the LSI task and thereby, shows
local subset unidentifiability. We discuss this in the following.

######  Theorem 3.

Consider a complete basis set $\mathcal{S}$ of maximally entangled states in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. There are $d^{2}\choose k$ possible
subsets, each containing $k$ distinct states from $\mathcal{S}$. The set
$\mathcal{S}$ is ($d^{2},k$)-unidentifiable, if ${d^{2}\choose k}>d^{k}$.
Moreover, the set of $D$ maximally entangled states ($D<d^{2}$) will also be
$(D,k)$-unidentifiable, provided ${D\choose k}>d^{k}$.

###### Proof.

Consider a maximally entangled state in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$
$\ket{\Gamma}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{\sqrt{d}}\sum_{j=1}^{d}\ket{jj}$. A complete basis of
orthonormal maximally entangled states can be generated from $\ket{\Gamma}$ as
$\mathcal{S}=\\{(\mathbb{U}_{l}\otimes\mathbb{I})\ket{\Gamma}\\}$, where the
set $\\{\mathbb{U}_{l}\\}_{l=1}^{d^{2}}$ forms a basis of Hilbert-Schmidt
orthogonal unitaries. A subset of $k$ distinct maximally entangled states can
be chosen from $\mathcal{S}$ in $\kappa\mathrel{\mathop{\mathchar
58\relax}}={d^{2}\choose k}$ ways. We denote the subsets as
$\mathcal{S}^{\prime(d^{2},k)}_{i}$, where $i=1$ to $\kappa$.

Now, the objective is to distinguish the subsets via LOCC. This actually
amounts to locally distinguishing corresponding mixed states
$\\{\rho_{i}^{m}\\}_{i=1}^{\kappa}$, where each state $\rho_{i}^{m}$ is an
equal classical mixture of all possible permutations of the composition of the
$k$ pure states from $\mathcal{S}^{\prime(d^{2},k)}_{i}$. These mixed states
are clearly orthogonal to each other because, by construction, each subset
contains at least one different element from the other. If the $\rho_{i}^{m}$s
are distinguishable, then any set of $\kappa$ vectors, each chosen from the
support of different $\rho_{i}^{m}$s in
$\mathbb{C}^{d^{k}}\otimes\mathbb{C}^{d^{k}}$, are also distinguishable.
However, any $\kappa$ pure maximally entangled states in
$\mathbb{C}^{d^{k}}\otimes\mathbb{C}^{d^{k}}$ are locally indistinguishable if
$\kappa>d^{k}$ [42]. Clearly, the set $\\{\rho_{i}^{m}\\}_{i=1}^{\kappa}$ is
locally indistinguishable, otherwise it leads to a contradiction. Evidently,
if we consider a set of $D$ ($<d^{2}$) maximally entangled states, it can be
easily shown following a similar approach as above, that the set will be
$(D,k)$-unidentifiable for ${D\choose k}>d^{k}$. This completes our proof. ∎

In the spirit of the theorem that any complete basis set of MES is locally
indistiguishable [42], the subsequent corollary follows.

######  Corollary 3.1.

Any two distinct states given from a complete basis set of MES of arbitrary
dimensions are locally unidentifiable.

###### Proof.

Consider $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ MES. Complete basis has $d^{2}$
such states. any two states given from it can occur in ${d^{2}\choose
2}=\frac{d^{2}(d^{2}-1)}{2}$ ways. This is always greater than $d^{2}$ for
$d\in\mathbb{Z^{+}}\backslash\\{1\\}$. Hence, locally unidentifiable. ∎

Figure 1: The bigger set corresponds to the set of all orthogonal set that
shows local unmarkability, whereas the small one contains all orthogonal sets
that shows local subset unidentifiability. The examples considered in Theorem
2 and 3 reside in the darkly shaded smaller set, whereas the example dealt in
Theorem 1 resides in the lightly shaded region outside the small one.

In the above, the Theorem 1 deals with an example of perfectly locally
identifiable set. It is obviously a nontrivial example as it is not locally
distinguishable or markable. There also exist other orthogonal sets which are
$(n,|\mathcal{S}^{\prime}|)$-identifiable, for example, consider a set of four
two-qubit Bell states. Now, if we explore the $(4,3)$-LSI task it is easy to
see that the set is $(4,3)$-identifiable. Here, each agent possesses three
qubits in their respective labs. They need to perform a joint projective
measurement in three-qubit GHZ basis on their respective three-qubit composite
subsystems. At the end, tallying of their measurement outcomes will perfectly
reveal the identity, but not the order, of the three shared Bell states. In
fact, the set of four two-qubit Bell states are $(4,3)$-unmarkable which
directly follows from its $(4,4)$-unmarkability [41]. There are also examples
of trivially identifiable sets. In general, any orthogonal set which is
locally distinguishable or even locally markable can be safely considered as
perfectly identifiable via LOCC.

At this point, the above results establish that our concerned task (LSI) is
distinct from LSM. Moreover, the incapability of accomplishing LSI designates
a stronger version of quantum nonlocality compared to the local unmarkability.
More precisely, a clear set inclusion relation can be established here (see
Fig. 1). All the orthogonal sets which are locally unindentifiable (Theorem 2
and 3) must also be locally unmarkable. However, the opposite is not
necessarily true (Theorem 1).

Hitherto, we have discussed examples in the bipartite framework. It is quite
an intriguing question to delve into the status of quantum nonlocality in
multipartite framework. Earlier works explored the notion of local
indistinguishability and local irreducibility in multipartite scenario [40,
44, 45]. Complex structure of the quantum state space in multiparty framework
empowers stronger notions of such quantum nonlocal phenomena. However, local
unmarkability in multiparty structure is not explored in existing literature.
We are now at a position to delve into the analysis of local subset
unidentifiablity in multipartite framework.

##  III Genuine Local Subset Unindetifiability

In this section, we analyze the status of LSI in multiparty scenario. We
provide an example of a class of tripartite orthogonal set that exhibits a
stronger form of local unidentifiability over its bipartite version. Moreover,
our example also serves as an stronger version of the existing notion of local
bipartite unmarkability.

We term this multipartite notion of unidentifiability as genuine because we
show that even when all but one agents come together in a same laboratory to
perfectly accomplish the LSI task, they fail. Of course, in this case, the
collaborating agents are allowed to perform the strongest possible operation
jointly on their composite subsystems. Nevertheless, this freedom does not
empower the agents enough to perfectly achieve their goal. Consider a set
$\mathcal{S}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{G_{\alpha}}\\}_{\alpha=1}^{8}$ of eight $3$-qubit GHZ
states, where

$\displaystyle\ket{G_{1,2}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{000}\pm\ket{111}\right)$ (1)
$\displaystyle\ket{G_{3,4}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{001}\pm\ket{110}\right)$ (2)
$\displaystyle\ket{G_{5,6}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{010}\pm\ket{101}\right)$ (3)
$\displaystyle\ket{G_{7,8}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{100}\pm\ket{011}\right).$ (4)

######  Theorem 4.

The set $\mathcal{S}$ is locally $(D,k)$-unidentifiable if ${D\choose
k}>2^{2k}$ (where $D\in(2,8]$ and $k\in(1,D)$) even when two out of three
parties come together in a same lab.

###### Proof.

Firstly, note that elements of the set $\mathcal{S}$ follow the relation
below,

$\displaystyle\ket{G_{\alpha}}=(\mathbb{I}\otimes\mathbb{U}_{\alpha\beta})\ket{G_{\beta}},~{}~{}\alpha,\beta\in\\{1,\ldots,8\\},$
(5)

where, $\mathbb{U}_{\alpha\beta}$ is a unitary acting on two qubits and
$\mathbb{I}$ is the identity operator acting on $\mathbb{C}^{2}$. Naturally,
any subset of $\mathcal{S}$ inherits the same property.

Let $\bar{\mathcal{S}}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{G_{\alpha}}\\}_{\alpha=1}^{D}$, $2<D\leq 8$. There are
$\kappa\mathrel{\mathop{\mathchar 58\relax}}={D\choose k}$ different ways in
which $k$ states can be chosen from $\bar{\mathcal{S}}$ to form subsets that
we label as $\mathcal{S}^{\prime}_{i}$, $i=1,\ldots,\kappa$. Locally
identifying $\mathcal{S}^{\prime}_{i}$ tantamounts to locally distinguishing
the set of mixed states $\\{\rho_{i}\\}_{i=1}^{\kappa}$ where

$\displaystyle\rho_{i}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{k!}\sum_{\mu=1}^{k!}$
$\displaystyle\ket{\Phi^{i}_{\mu}}\bra{\Phi^{i}_{\mu}},$ (6)
$\displaystyle\ket{\Phi^{i}_{\mu}}=\text{Perm}_{\mu}\ket{G_{\alpha_{1}}}$
$\displaystyle\ket{G_{\alpha_{2}}}\ldots\ket{G_{\alpha_{k}}},~{}\mu=1,\ldots,k!,$
$\displaystyle\alpha_{1}\neq\alpha_{2}\ldots\neq\alpha_{k}\in\\{1,\ldots,D\\},$
$\displaystyle\text{such that,}~{}~{}~{}~{}\ket{G_{\alpha_{1}}},$
$\displaystyle\ket{G_{\alpha_{2}}},\ldots,\ket{G_{\alpha_{k}}}\in\mathcal{S}^{\prime}_{i}.$

Here,
$\text{Perm}_{\mu}\ket{G_{\alpha_{1}}}\ket{G_{\alpha_{2}}}\ldots\ket{G_{\alpha_{k}}}$
means different possible permutations of the state
$\ket{G_{\alpha_{1}}}\ket{G_{\alpha_{2}}}\ldots\ket{G_{\alpha_{k}}}$. For
example,
$\text{Perm}_{1}\ket{G_{1}}\ket{G_{4}}\ket{G_{7}}=\ket{G_{1}}\ket{G_{4}}\ket{G_{7}}$,
$\text{Perm}_{2}\ket{G_{1}}\ket{G_{4}}\ket{G_{7}}=\ket{G_{7}}\ket{G_{4}}\ket{G_{1}}$,
$\text{Perm}_{3}\ket{G_{1}}\ket{G_{4}}\ket{G_{7}}=\ket{G_{4}}\ket{G_{7}}\ket{G_{1}}$,
etc. We examine the identifiability of $\mathcal{S}^{\prime}_{i}$s
(equivalently, distinguishability of the $\rho_{i}$s) when two out of the
three qubits of each of the $k$ $\ket{G_{\alpha}}$s are in the same lab. Under
such condition, it is possible to transform one permutation to any other by
implementing local unitaries on the two qubits of each of the $k$
$\ket{G_{\alpha}}$s. Therefore,

$\displaystyle\ket{G_{\alpha^{\prime}_{1}}}\ket{G_{\alpha^{\prime}_{2}}}\ldots\ket{G_{\alpha^{\prime}_{k}}}$
$\displaystyle=$
$\displaystyle(\mathbb{I}\otimes\mathbb{U}_{\alpha^{\prime}_{1}\alpha_{1}})\ket{G_{\alpha_{1}}}(\mathbb{I}\otimes\mathbb{U}_{\alpha^{\prime}_{2}\alpha_{2}})\ket{G_{\alpha_{2}}}\ldots(\mathbb{I}\otimes\mathbb{U}_{\alpha^{\prime}_{k}\alpha_{k}})\ket{G_{\alpha_{k}}}.$
(7)

Clearly, the set
$\\{\ket{\Phi^{i}_{\mu}}\\}_{i=1}^{\kappa}\subset\mathbb{C}^{2^{k}}\otimes\mathbb{C}^{2^{2k}}$
has the property:
$\ket{\Phi^{i}_{\mu}}=(\mathbb{I}\otimes\mathbb{U}_{ij,\mu\nu})\ket{\Phi^{j}_{\nu}}$
for $i,j=1,\ldots,\kappa$, $\mu,\nu=1,\ldots,k!$ where
$\mathbb{U}_{ij,\mu\nu}$s are $2k$-qubit unitaries. So, if $\kappa>2^{2k}$,
$\\{\ket{\Phi^{i}_{\mu}}\\}_{i=1}^{\kappa}$ is not perfectly distinguishable
[46].

On the other hand, if we assume that $\rho_{i}$s are distinguishable, then so
are their supports. This implies that the set
$\\{\ket{\Phi^{i}_{\mu}}\\}_{i=1}^{\kappa}$ is also distinguishable, as for
each $i$, $\ket{\Phi^{i}_{\mu}}\in\text{Support}(\rho_{i})$. But this leads to
contradiction for $\kappa>2^{2k}$. So, the $\mathcal{S}^{\prime}_{i}$s are
locally unidentifiable in each bipartition, provided ${D\choose k}>2^{2k}$,
that is, the sets $\mathcal{S}^{\prime}_{i}$s are genuinely unindentifiable. ∎

Note that unidentifiability of $\mathcal{S}^{\prime}_{i}$s in every
bipartition readily rules out the identifiability when the three parties are
separated. This is because when two parties collaborate they are able to
perform strongest possible joint operations on the qubits in their possession,
whereas in the later case, the parties are only allowed to perform a
relatively weak operation - LOCC.

While the last result exhibits genuine local unidentifibality in three-party
systems in every possible bi-partitions that are related by party
permutations, number of bi-partitions that are not related by party
permutations increases as we consider multipartite systems consisting of more
than three parties. This can be easily demonstrated by considering four-party
systems. In a four-party system, there are two classes of bi-partitions that
are not connected by party permutation viz., 1 party vs. 3 parties
($1\mathrel{\mathop{\mathchar 58\relax}}3$), and 2 parties vs. 2 parties
($2\mathrel{\mathop{\mathchar 58\relax}}2$), each class containing bi-
partitions that are related by party permutations. Sets of orthogonal four-
party states that exhibits local unidentifiability in all such possible bi-
partitions represent stronger nonlocality than that exhibited by three
parties. Consider the set $\mathcal{S}\mathrel{\mathop{\mathchar
58\relax}}=\\{\ket{G^{(4)}_{\alpha}}\\}_{\alpha=1}^{16}$, where

$\displaystyle\ket{G^{(4)}_{1,2}}$
$\displaystyle=\frac{1}{2}(\ket{0000}\pm\ket{0111}+\ket{1010}\pm\ket{1101}),$
$\displaystyle\ket{G^{(4)}_{3,4}}$
$\displaystyle=\frac{1}{2}(\ket{0000}\pm\ket{0111}-\ket{1010}\mp\ket{1101}),$
$\displaystyle\ket{G^{(4)}_{5,6}}$
$\displaystyle=\frac{1}{2}(\ket{0001}\pm\ket{0110}+\ket{1011}\pm\ket{1100}),$
$\displaystyle\ket{G^{(4)}_{7,8}}$
$\displaystyle=\frac{1}{2}(\ket{0001}\pm\ket{0110}-\ket{1011}\mp\ket{1100}),$
$\displaystyle\ket{G^{(4)}_{9,10}}$
$\displaystyle=\frac{1}{2}(\ket{0010}\pm\ket{0101}+\ket{1000}\pm\ket{1111}),$
$\displaystyle\ket{G^{(4)}_{11,12}}$
$\displaystyle=\frac{1}{2}(\ket{0010}\pm\ket{0101}-\ket{1000}\mp\ket{1111}),$
$\displaystyle\ket{G^{(4)}_{13,14}}$
$\displaystyle=\frac{1}{2}(\ket{0011}\pm\ket{0100}+\ket{1001}\pm\ket{1110}),$
$\displaystyle\ket{G^{(4)}_{15,16}}$
$\displaystyle=\frac{1}{2}(\ket{0011}\pm\ket{0100}-\ket{1001}\mp\ket{1110}).$
(8)

######  Theorem 5.

The set $\mathcal{S}$ is locally $(D,k)$-unidentifiable if ${D\choose
k}>2^{3k}$ (where $D\in(2,16]$ and $k\in(1,D)$) in all
$1\mathrel{\mathop{\mathchar 58\relax}}3$ as well $2\mathrel{\mathop{\mathchar
58\relax}}2$ bi-partitions.

###### Proof.

The set $\mathcal{S}$ has the property that

$\displaystyle\ket{G^{(4)}_{\alpha}}$
$\displaystyle=(\mathbb{I}\otimes\mathbb{U}_{\alpha\beta})\ket{G^{(4)}_{\beta}},$
$\displaystyle=(\mathbb{I}^{\prime}\otimes\mathbb{U}^{\prime}_{\alpha\beta})\ket{G^{(4)}_{\beta}},$
(9)

where, $\mathbb{U}_{\alpha\beta}$ is a unitary acting on three qubits,
$\mathbb{I}$ is the identity operator acting on $\mathbb{C}^{2}$,
$\mathbb{U}^{\prime}_{\alpha\beta}$ acts on two qubits and
$\mathbb{I}^{\prime}$ is the identity operator acting on $\mathbb{C}^{4}$.
Now, following the same arguments as given in the proof of Theorem 4, it can
be easily proved that a subset containing $k$ states, chosen from a set
$\bar{\mathcal{S}}$ containing known $D$ states of $\mathcal{S}$ is locally
unidentifiable in $2\mathrel{\mathop{\mathchar 58\relax}}2$ bi-partitions if
${D\choose k}>2^{2k}$ and $1\mathrel{\mathop{\mathchar 58\relax}}3$ bi-
partitions if ${D\choose k}>2^{3k}$. Thus, for ${D\choose k}>2^{3k}$ the
subset can be identified if and only if all four parties come in the same lab.
∎

Furthermore, Theorems 4 and 5 also present examples of the stronger form of
local unmarkability. The concerned sets are locally unmarkable in all possible
bipartitions.

##  IV Discussion

In summary, we have introduced a novel distributed task - local subset
identification. The task is to be played among multiple agents who are allowed
to perform LOCC only. The main aim of the task for spatially separated agents
is to perfectly identify the elements of a randomly chosen subset from a known
orthogonal set of quantum states, shared between them. We have proved that,
this task is distinct from the tasks / games explored earlier in the genre of
local state discrimination. The significance of this specific task is also
intriguing. Inability to perfectly accomplish the task certifies the presence
of a novel quantum nonlocal feature in the concerned orthogonal set, that is,
local subset unidentifiablity. Both in bipartite and multipartite scenario, we
have explored examples of such nonlocal phenomenon. Specifically, in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ bipartite system, subsets of cardinality
$k$ chosen from a set of orthogonal general maximally entangled states is
shown to be locally unidentifiable for some condition on $d$ and $k$, that is,
the set is $(d^{2},k)$-unidentifiable. Moreover, we have also established that
local subset unidentifiability is stronger than its nearest neighbouring
species - local unmarkability. This also makes local subset unidentifiability
the strongest one along this line as yet. Further, we have explored the
multipartite version of this new nonlocality and come up with an even stronger
form of it, where the set would remain locally unidentifiable in all possible
bipartitions. We denote it as genuine local subset unidentification. We have
also provided examples involving tripartite GHZ states in
${\mathbb{C}^{2}}^{\otimes 3}$ and four party genuine entangled states in
${\mathbb{C}^{2}}^{\otimes 4}$. As established in this article – locally
unidentifiable subsets are necessarily locally unmarkable, these also serves
as examples of genuine local unmarkability – strongest version of local
unmarkability as yet. Throughout the work, we have explored the local subset
unidentifiablity only in orthogonal entangled states. However, we believe that
it is not necessarily the exclusive feature of entanglement. It would be quite
interesting if the same feature can also be exhibited in sets of orthogonal
product states. Furthermore, the local subset identification task deals with
the problem of perfectly discriminating mutually orthogonal subspaces of rank
more than one, whereas the previous tasks in the state discrimination genre
e.g., local reducibility [40, 44], local markability [41] dealt only with the
rank one vectors. We, therefore, also believe that the local subset
unidetifiability will be a more prospective resource in various quantum
information processing protocols, such as, data hiding or secret sharing
protocols.

## Acknowledgments

We thank Guruprasad Kar, Somshubhro Bandyopadhyay, Sibasish Ghosh and Manik
Banik for stimulating discussions. PG would like to thank Department of
Science $\&$ Technology, Government of India for financial support. AG would
like to thank IMSc, Chennai for hospitality. AM acknowledges NASI for Research
Associate Fellowship.

## References

  * Bell [1964] J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika 1, 195 (1964).
  * 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).
  * Watrous [2005] J. Watrous, Bipartite subspaces having no bases distinguishable by local operations and classical communication, Phys. Rev. Lett. 95, 080505 (2005).
  * 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).
  * Ghosh  _et al._ [2022] S. B. Ghosh, T. Gupta, A. V. Ardra, A. Das Bhowmik, S. Saha, T. Guha, and A. Mukherjee, Activating strong nonlocality from local sets: An elimination paradigm, Phys. Rev. A 106, L010202 (2022).
  * Gupta  _et al._ [2022] T. Gupta, S. B. Ghosh, A. V. Ardra, A. D. Bhowmik, S. Saha, T. Guha, R. Rahaman, and A. Mukherjee, Genuine activation of quantum nonlocality: Stronger than local indistinguishability (2022).
  * Halder  _et al._ [2019] S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Strong quantum nonlocality without entanglement, Physical review letters 122, 040403 (2019).
  * Sen  _et al._ [2022] S. Sen, E. P. Lobo, S. G. Naik, R. K. Patra, T. Gupta, S. B. Ghosh, S. Saha, M. Alimuddin, T. Guha, S. S. Bhattacharya, _et al._ , Local quantum state marking, Physical Review A 105, 032407 (2022).
  * Hayashi  _et al._ [2006] M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication, Phys. Rev. Lett. 96, 040501 (2006).
  * Bandyopadhyay [2012] S. Bandyopadhyay, Entanglement, mixedness, and perfect local discrimination of orthogonal quantum states, Phys. Rev. A 85, 042319 (2012).
  * 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, Physical Review A 100, 032321 (2019).
  * Rout  _et al._ [2021] S. Rout, A. G. Maity, A. Mukherjee, S. Halder, and M. Banik, Multiparty orthogonal product states with minimal genuine nonlocality, Physical Review A 104, 052433 (2021).
  * Nathanson [2005] M. Nathanson, Distinguishing bipartitite orthogonal states using locc: Best and worst cases, Journal of Mathematical Physics 46, 062103 (2005), https://doi.org/10.1063/1.1914731 .