# Genuine hidden nonlocality without entanglement: from the perspective of
local discrimination

Mao-Sheng Li<EMAIL_ADDRESS>School of Mathematics, South China
University of Technology, Guangzhou 510641, China  Zhu-Jun Zheng
<EMAIL_ADDRESS>School of Mathematics, South China University of
Technology, Guangzhou 510641, China

###### Abstract

Quantum nonlocality without entanglement is a fantastic phenomenon in quantum
theory. This kind of quantum nonlocality is based on the task of local
discrimination of quantum states. Recently, Bandyopadhyay and Halder [Phys.
Rev. A 104, L050201 (2021)] studied the problem: is there any set of
orthogonal states which can be locally distinguishable, but under some
orthogonality preserving local measurement, each outcome will lead to a
locally indistinguishable set. We say that the set with such property has
hidden nonlocality. Moreover, if such phenomenon can not arise from discarding
subsystems which is termed as local irredundancy, we call it genuine hidden
nonlocality. There, they presented several sets of entangled states with
genuine hidden nonlocality. However, they doubted the existence of a set
without entanglement but with genuine hidden nonlocality. In this paper, we
eliminate this doubt by constructing a series of sets without entanglement but
whose nonlocality can be genuinely activated. We derive a method to tackle
with the local irredundancy problem which is a key tricky for the systems
whose local dimensions are composite numbers. Unexpectedly, the constructions
of genuine hidden nonlocal sets without entanglement seems to be easier than
that with entanglement. Therefore, this kind of nonlocality is rather
different from the Bell nonlocality.

##  I Introduction

The predictions of quantum theory are incompatible with those based on the
concept of locality (Bell-1964, ; Brunner-et-al-2014, ). Such phenomenon is
known as quantum nonlocality. Quantum nonlocality is usually revealed by
violating Bell type inequalities (CHSH-1969, ; Freedman-Causer-1972, ;
Aspect+-1981, ; Aspect+1982, ; Hensen+2015, ; handsteiner+2017, ; Rosenfeld+,
; BIG-BELL, ) which can only arise from entangled states. The nonlocality
arising from this way is termed as Bell nonlocality (Brunner-et-al-2014, ;
Bell-1964, ; CHSH-1969, ). It is well-known that entanglement Entanglement-
horodecki can not be generated from separable states under local operations.
However, there exists some states which admit a local hidden variable model
(hence is local) but violating Bell type inequalities after judicious local
filters are applied. This kind of states is said to have hidden nonlocality
Popescu-1995 ; Gisin-1996 ; navascues+-2011 ; Palazuelos-2012 ; Klobus+2012 ;
Hirsch+2013 .

Except the Bell nonlocality, there are other forms of nonlocality that have
attracted our attention. In fact, the local indistinguishability of some set
of orthogonal quantum states has been widely used to illustrate the phenomenon
of quantum nonlocality. In this setting, two or more observers share the parts
of a composite quantum system prepared in a state from a known orthogonal set.
The task is to identify the state by performing local operations and classical
communication (LOCC). Note that we can always identify the state correctly if
global measurements are allowed as the set of states are assumed to be
orthogonal. If such task can be accomplished perfectly under LOCC, we say that
the set is _locally distinguishable_ , otherwise, _locally indistinguishable_.
Bennett et al. ben99 presented the first example of orthogonal product states
that are locally indistinguishable and named such a phenomenon as quantum
nonlocality without entanglement. The nonlocality here is in the sense that
there exists some quantum information that could be inferred from global
measurement but cannot be read from local correlations of the subsystems.
Since then, this kind of nonlocality has been studied extensively (see Refs.
ben99 ; ben99u ; Walgate-2000 ; Ghosh-2001 ; grois01 ; walgate-2002 ; divin03
; HSSH ; Ghosh-2004 ; rin04 ; Watrous-2005 ; fan-2005 ; Nathanson-2005 ;
Wootters-2006 ; Hayashi-etal-2006 ; nis06 ; Duan2007 ; Duan-2009 ; feng09 ;
Bandyo-2011 ; BGK-2011 ; Yu-Duan-2012 ; BN-2013 ; Cosentino-2013 ; Yus15 ;
B-IQC-2015 ; childs13 ; Cosentino-Russo-2014 ; Ha21 ; B-IQC-2015 ; Li15 ;
Yang13 ; zhang14 ; wang15 ; zhang15 ; Yang15 ; xu16-1 ; Xu-16-2 ; zhang16 ;
zhang16-1 ; Xu-17 ; Wang-2017-Qinfoprocess ; Zhang-Oh-2017 ; zhang17-1 ;
halder ; Li20 ; Li18 ; Halder1909 ; Xu20b ; Xu20a ; Halder20c for an
incomplete list). Most studies are focused on identifying sets of orthogonal
product states or sets of the maximally entangled states. In addition, a
stronger manifestation of nonlocality in multipartite systems has been studied
which is based on the notion of local irreducibility in all biparitions of
subsystems Halder19 ; Zhang1906 ; Shi20S ; Tian20 ; Wang21 ; Shi21 ; Shi21b ;
Banik21 . An orthogonal set of pure states in multipartite quantum system is
called _locally irreducible_ if it is not possible to eliminate one or more
states from the set by orthogonality-preserving local measurements.

The local indistinguishability of quantum states has been practically applied
in quantum cryptography primitives such as data hiding Terhal01 ; DiVincenzo02
and secret sharing Markham08 ; Rahaman15 ; WangJ17 . Therefore, local
indistinguishability of quantum states can be considered as a resource in
quantum information processing. If there are only locally distinguishable sets
at hand, how can we transfer them into resources that have applications in
data hiding? This is what Bandyopadhyay and Halder Bandyopadhyay21 recently
studied. In fact, they studied the problem: is there any set of orthogonal
states which can be locally distinguishable, but under some orthogonality
preserving local measurement, each outcome will lead to a locally
indistinguishable set. As there are some trivial sets with this property, they
introduced the concept of local irredundancy. An orthogonal set is said to be
locally redundant if it remains orthogonal after discarding one or more
subsystems. Otherwise, it is called to be locally irredundant. If a locally
irredundant set satisfies the aforementioned property, then we said its
nonlocality can be activated genuinely. There, they give several examples of
such sets with entanglement. However, deeper research on this property remains
to be explored. For example, is there any locally distinguishable set without
entanglement whose nonlocality can be genuinely activated? Which state spaces
have sets with this property? In this paper, we tend to solve the two problems
partially.

The rest of this article is organized as follows. In Sec. II, we review the
concept of genuine activation of nonlocality (genuine hidden nonlocality).
Then we study the constructions of sets with genuine hidden nonlocality with
respect to two cases: this first is cardinality preserving while the second is
cardinality decreasing. In Sec. III, we present examples for the first case
and discuss its constructions in multipartite systems. In Sec. IV, we give a
example for the second case and also and discuss its constructions in
multipartite systems. Finally, we draw a conclusion in section V.

##  II Preliminary

It is well known that a set of quantum states is perfectly distinguished by
global measurement if and only if the given set of states are mutually
orthogonal. Therefore, every LOCC protocol that distinguishes a set of
orthogonal states is a sequence of orthogonality-preserving-local-measurements
(OPLM). In this paper, we consider this specific class of LOCC measurements,
i.e., OPLM.

Now we give a brief review of the main problem studied in Ref. Bandyopadhyay21
. Suppose that $\mathcal{S}$ is an orthogonal set of multipartite states which
is locally distinguishable. The participants aim to find some OPLM
$\mathbb{M}$ such that for each outcome $\mu$ of this measurement, they end up
to a new orthogonal set $\mathcal{S}_{\mu}^{\prime}$ which is locally
indistinguishable. If we could find such measurement, we call that the
nonlocality of $\mathcal{S}$ can be _activated_ (Compared with setting in the
Bell nonlocality , we also call that $\mathcal{S}$ has _hidden nonlocality_
here). However, the authors in Ref. Bandyopadhyay21 have pointed out that
there are some trivial cases which arise from taking partial trace. Such cases
would happen if the original set is local redundancy, i.e., the set remains
orthogonal if we discard one or more subsystems. If the set $\mathcal{S}$ is
locally irredundant and satisfies the aforementioned property, we can call
that the nonlocality of $\mathcal{S}$ can be _genuinely activated_ (And we
also call that $\mathcal{S}$ has _genuine hidden nonlocality_ here). Note that
the cardinality of $\mathcal{S}_{\mu}^{\prime}$ always satisfies
$\left|\mathcal{S}_{\mu}^{\prime}\right|\leq\left|\mathcal{S}\right|.$
According to the differences of the cardinality, we separate the genuine
hidden nonlocality into two types.

  1. (a)

Genuine hidden nonlocality of type I: $\mathcal{S}$ is locally distinguishable
and locally irredundant. And there is some OPLM $\mathbb{M}$ such that for
each outcome $\mu$ of this measurement, the set of post-measurement states
$\mathcal{S}_{\mu}^{\prime}$ is locally indistinguishable and
$|\mathcal{S}_{\mu}^{\prime}|=|\mathcal{S}|$. Moreover, if for each outcome
$\mu$ of this measurement, the set of post-measurement states
$\mathcal{S}_{\mu}^{\prime}$ is locally irreducible, we call it has genuine
hidden strong form nonlocality of type I.

  2. (b)

Genuine hidden nonlocality of type II: $\mathcal{S}$ is locally
distinguishable and locally irredundant. And there is some OPLM $\mathbb{M}$
such that for each outcome $\mu$ of this measurement, the set of post-
measurement states $\mathcal{S}_{\mu}^{\prime}$ is locally indistinguishable.
In addition, there is some outcome $\mu$ such that
$|\mathcal{S}_{\mu}^{\prime}|<|\mathcal{S}|.$

In this paper, we will study how to construct sets without entanglement but
with genuine hidden nonlocality of type I or II. Throughout this paper, we
will use the following notations. Let $d\geq 2$ be an integer. Considering a
quantum system $\mathcal{H}$ of dimension $d$, we usually denote
$\\{|0\rangle,|1\rangle,\cdots,|d-1\rangle\\}$ as its a computational basis.
We also use the notation $|i+j\rangle$ (w.r.t. $|i-j\rangle$) which presents
$|i\rangle+|j\rangle$ (w.r.t. $|i\rangle-|j\rangle$ ). Moreover, we use
$|+_{n}\rangle\equiv\sum_{i=0}^{n}|i\rangle,$ and
$|{}_{m}+_{n}\rangle\equiv\sum_{i=m}^{n}|i\rangle$ where $m<n$. We also use
$\mathbb{I}_{d}$ represent the identity operator on this system, i.e.,
$\mathbb{I}_{d}=\sum_{i=0}^{d-1}|i\rangle\langle i|.$ And the states
throughout this paper may be unnormalized.

##  III Genuine hidden nonlocality of type I

In this section, we study how to construct sets without entanglement but have
genuine hidden nonlocality of type I. We will use the local
indistinguishability of the sets without entanglement as follows.

Figure 1: This shows two examples of the structures of the product states
constructed in Theorem 1 in $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ and
$\mathbb{C}^{6}\otimes\mathbb{C}^{6}$. The squares indicated by the same label
represent a unique state. For examples, there are two squares (that is,
$(2,0),$ and $(2,2)$) with label ‘$h_{2}$’ in the left hand side graph, they
correspond to the state
$|2\rangle|0-2\rangle\in\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ and there are two
squares (that is, $(0,5),$ and $(4,5)$) with label ‘$V_{5}$’ in the right hand
side graph, they correspond to the state
$|0-4\rangle|5\rangle\in\mathbb{C}^{6}\otimes\mathbb{C}^{6}.$

######  Theorem 1 (Yus15 ).

In $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ system with $d\geq 3$, the following
$2d-1$ orthogonal pure product states are locally indistinguishable:

$\\{|n\rangle|\delta_{n}\rangle\\}_{n=1}^{d-1}\cup\\{|\delta_{n}\rangle|n_{+}\rangle\\}_{n=1}^{d-1}\cup\\{|+_{(d-1)}\rangle|+_{(d-1)}\rangle\\}$

where $|\delta_{n}\rangle\equiv|0-n\rangle$ and $n_{+}=n+1$ for $1\leq n\leq
d-2$ while $n_{+}=1$ for $n=d-1$.

Now let’s start with the following example, the construction of whose states
is inspired from the above Theorem for the cases
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ and
$\mathbb{C}^{6}\otimes\mathbb{C}^{6}$ (see Fig. 1 for their structures).

Figure 2: This shows the states structure of set of product states in Example
1 . The squares indicated by the same label represent a unique state. For
examples, there are four squares (that is, $(1,0),(1,1),(1,9),$ and $(1,10)$)
with label ‘$h_{1}$’, they correspond to the state
$|\psi_{1}\rangle:=|1\rangle|0-1+9-10\rangle,$ and there are four squares
(that is, $(5,2),(5,8),$ $(7,2)$, and $(7,8)$) with label ‘$V_{3}$’, they
correspond to the state $|\phi_{8}\rangle:=|5-7\rangle|8-2\rangle.$ The first
five rows are axisymmetric about the middle column, and the first two columns
and the last two columns of the last six rows are also axisymmetric about the
middle column.

######  Example 1.

Let $\mathcal{S}$ be the set of product states in
$\mathbb{C}^{11}\otimes\mathbb{C}^{11}$ whose elements are listed as below
(see Fig. 2)):

$\begin{array}[]{rclrcl}\left|\psi_{1}\right\rangle&\equiv&|1\rangle|0-1+9-10\rangle,&\left|\psi_{5}\right\rangle&\equiv&|0-4\rangle|1-9\rangle,\\\
\left|\psi_{2}\right\rangle&\equiv&|2\rangle|0-2+8-10\rangle,&\left|\psi_{6}\right\rangle&\equiv&|0-1\rangle|2-8\rangle,\\\
\left|\psi_{3}\right\rangle&\equiv&|3\rangle|0-3+7-10\rangle,&\left|\psi_{7}\right\rangle&\equiv&|0-2\rangle|3-7\rangle,\\\
\left|\psi_{4}\right\rangle&\equiv&|4\rangle|0-4+6-10\rangle,&\left|\psi_{8}\right\rangle&\equiv&|0-3\rangle|4-6\rangle,\\\
\left|\phi_{1}\right\rangle&\equiv&|6\rangle|5-6+2-3\rangle,&\left|\phi_{6}\right\rangle&\equiv&|5-10\rangle|6-3\rangle,\\\
\left|\phi_{2}\right\rangle&\equiv&|7\rangle|5-7+3-4\rangle,&\left|\phi_{7}\right\rangle&\equiv&|5-6\rangle|7-4\rangle,\\\
\left|\phi_{3}\right\rangle&\equiv&|8\rangle|5-8+2-4\rangle,&\left|\phi_{8}\right\rangle&\equiv&|5-7\rangle|8-2\rangle,\\\
\left|\phi_{4}\right\rangle&\equiv&|9\rangle|5-9+1-4\rangle,&\left|\phi_{9}\right\rangle&\equiv&|5-8\rangle|9-1\rangle,\\\
\left|\phi_{5}\right\rangle&\equiv&|10\rangle|5-10+0-4\rangle,&\left|\phi_{10}\right\rangle&\equiv&|5-9\rangle|10-0\rangle,\end{array}$

and $|\psi_{9}\rangle\equiv|+_{4}\rangle|+_{10}\rangle,$ and
$|\phi_{11}\rangle\equiv|{}_{5}+_{10}\rangle|+_{10}\rangle.$ Then the set
$\mathcal{S}$ has genuine hidden nonlocality of type I.

Now we show that $\mathcal{S}$ is locally distinguishable. For each
$|\Theta\rangle=|\theta_{1}\rangle|\theta_{2}\rangle\in\mathcal{S}$, we denote

$\mathcal{C}(|\Theta\rangle):=\\{(i,j)\in\mathbb{Z}_{11}\times\mathbb{Z}_{11}\big{|}\langle
i|\theta_{1}\rangle\langle j|\theta_{2}\rangle\neq 0\\}.$

We call it the coordinates of the state $|\Theta\rangle$. Note that except the
states $|\psi_{9}\rangle$ and $|\phi_{11}\rangle$, the coordinates of other
states in $\mathcal{S}$ are all of cardinality $4$ and these states can be
formed into two types: their coordinates lie on the same line and are labeled
with $h_{i}$ or $H_{i}$ in Fig. 2 (For examples,
$\mathcal{C}(|\psi_{1}\rangle)=\\{(1,0),(1,1),(1,9),(1,10)\\}$) and their
coordinates lie on the four corners of a rectangle and are labeled with
$v_{j}$ or $V_{j}$ in Fig. 2 (For example,
$\mathcal{C}(|\phi_{6}\rangle)=\\{(5,3),(5,6),(10,3),(10,6)\\}$). Moreover,
for different states
$|\Theta_{1}\rangle,|\Theta_{2}\rangle\in\mathcal{S}\setminus\\{|\psi_{9}\rangle,|\phi_{11}\rangle\\}$,
their coordinates are disjoint, i.e.,
$\mathcal{C}(|\Theta_{1}\rangle)\cap\mathcal{C}(|\Theta_{2}\rangle)=\emptyset.$

First, Alice perform the measurement
$\mathbb{M}_{1}^{A}:=\\{\pi_{i}^{A}=|i\rangle\langle i|\ \big{|}\
i\in\mathbb{Z}_{11}\\}.$ If the outcome of $\mathbb{M}_{1}^{A}$ is ‘$i$’, by
the construction of the set $\mathcal{S}$ and the above observation, the
postmeasurement states of Bob’s part are mutually orthogonal and hence can be
distinguished. For example, if the outcome of $\mathbb{M}_{1}^{A}$ is ‘$5$’,
the state must be one of $\\{|\phi_{7,8,9,10,11}\rangle\\}$ whose Bob’s part
are the following five orthognal states
$|7-4\rangle,|8-2\rangle,|9-1\rangle,|10-0\rangle,|+_{10}\rangle$
respectively.

Now we prove that the set $\mathcal{S}$ satisfies the second property. Suppose
Bob perform the measurement
$\mathbb{M}^{B}:=\\{\pi_{1}^{B}=\sum_{i=0}^{4}|i\rangle\langle i|,\
\pi_{2}^{B}=\sum_{j=5}^{10}|j\rangle\langle j|\\}.$

If the outcome of $\mathbb{M}^{B}$ is ‘1’, the states are transferred to

$\begin{array}[]{rclrcl}\left|\tilde{\psi}_{1}\right\rangle&=&|1\rangle|0-1\rangle,&\left|\tilde{\psi}_{5}\right\rangle&=&|0-4\rangle|1\rangle,\\\
\left|\tilde{\psi}_{2}\right\rangle&=&|2\rangle|0-2\rangle,&\left|\tilde{\psi}_{6}\right\rangle&=&|0-1\rangle|2\rangle,\\\
\left|\tilde{\psi}_{3}\right\rangle&=&|3\rangle|0-3\rangle,&\left|\tilde{\psi}_{7}\right\rangle&=&|0-2\rangle|3\rangle,\\\
\left|\tilde{\psi}_{4}\right\rangle&=&|4\rangle|0-4\rangle,&\left|\tilde{\psi}_{8}\right\rangle&=&|0-3\rangle|4\rangle,\\\
\left|\tilde{\phi}_{1}\right\rangle&=&|6\rangle|2-3\rangle,&\left|\tilde{\phi}_{6}\right\rangle&=&|5-10\rangle|3\rangle,\\\
\left|\tilde{\phi}_{2}\right\rangle&=&|7\rangle|3-4\rangle,&\left|\tilde{\phi}_{7}\right\rangle&=&|5-6\rangle|4\rangle,\\\
\left|\tilde{\phi}_{3}\right\rangle&=&|8\rangle|2-4\rangle,&\left|\tilde{\phi}_{8}\right\rangle&=&|5-7\rangle|2\rangle,\\\
\left|\tilde{\phi}_{4}\right\rangle&=&|9\rangle|1-4\rangle,&\left|\tilde{\phi}_{9}\right\rangle&=&|5-8\rangle|1\rangle,\\\
\left|\tilde{\phi}_{5}\right\rangle&=&|10\rangle|0-4\rangle,&\left|\tilde{\phi}_{10}\right\rangle&=&|5-9\rangle|0\rangle,\\\
\left|\tilde{\psi}_{9}\right\rangle&=&|+_{4}\rangle|+_{4}\rangle,&\left|\tilde{\phi}_{11}\right\rangle&=&|{}_{5}+_{10}\rangle|+_{4}\rangle,\end{array}$

which are mutually orthogonal and contain $9$ states
$\\{|\tilde{\psi}_{i}\rangle\\}_{i=1}^{9}$ that is known to be locally
indistinguishable in $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$.

If the outcome of $\mathbb{M}^{B}$ is ‘2’, the states are transferred to

$\begin{array}[]{rclrcl}\left|\tilde{\psi}_{1}\right\rangle&=&|1\rangle|9-10\rangle,&\left|\tilde{\psi}_{5}\right\rangle&=&|0-4\rangle|9\rangle,\\\
\left|\tilde{\psi}_{2}\right\rangle&=&|2\rangle|8-10\rangle,&\left|\tilde{\psi}_{6}\right\rangle&=&|0-1\rangle|8\rangle,\\\
\left|\tilde{\psi}_{3}\right\rangle&=&|3\rangle|7-10\rangle,&\left|\tilde{\psi}_{7}\right\rangle&=&|0-2\rangle|7\rangle,\\\
\left|\tilde{\psi}_{4}\right\rangle&=&|4\rangle|6-10\rangle,&\left|\tilde{\psi}_{8}\right\rangle&=&|0-3\rangle|6\rangle,\\\
\left|\tilde{\phi}_{1}\right\rangle&=&|6\rangle|5-6\rangle,&\left|\tilde{\phi}_{6}\right\rangle&=&|5-10\rangle|6\rangle,\\\
\left|\tilde{\phi}_{2}\right\rangle&=&|7\rangle|5-7\rangle,&\left|\tilde{\phi}_{7}\right\rangle&=&|5-6\rangle|7\rangle,\\\
\left|\tilde{\phi}_{3}\right\rangle&=&|8\rangle|5-8\rangle,&\left|\tilde{\phi}_{8}\right\rangle&=&|5-7\rangle|8\rangle,\\\
\left|\tilde{\phi}_{4}\right\rangle&=&|9\rangle|5-9\rangle,&\left|\tilde{\phi}_{9}\right\rangle&=&|5-8\rangle|9\rangle,\\\
\left|\tilde{\phi}_{5}\right\rangle&=&|10\rangle|5-10\rangle,&\left|\tilde{\phi}_{10}\right\rangle&=&|5-9\rangle|10\rangle,\\\
\left|\tilde{\psi}_{9}\right\rangle&=&|+_{4}\rangle|{}_{5}+_{10}\rangle,&\left|\tilde{\phi}_{11}\right\rangle&=&|{}_{5}+_{10}\rangle|{}_{5}+_{10}\rangle,\end{array}$

which are mutually orthogonal and contain $11$ states
$\\{|\tilde{\phi}_{i}\rangle\\}_{i=1}^{11}$ that is known to be locally
indistinguishable in $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$. ∎

The above construction and argument can be easily extend to
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ for any prime number $d\geq 11$. Here we
give one more example without proof.

Figure 3: This shows the states structure of set of product states in Example
2 . The first six rows are axisymmetric about the middle column, and the first
three columns and the last three columns of the last seven rows are also
axisymmetric about the middle column.

######  Example 2.

Let $\mathcal{S}$ be the set of product states in
$\mathbb{C}^{13}\otimes\mathbb{C}^{13}$ whose elements are listed as below
(see Fig. 3)):

$\begin{array}[]{rclrcl}\left|\psi_{1}\right\rangle&\equiv&|1\rangle|0-1+11-12\rangle,&\left|\psi_{6}\right\rangle&\equiv&|0-5\rangle|1-11\rangle,\\\
\left|\psi_{2}\right\rangle&\equiv&|2\rangle|0-2+10-12\rangle,&\left|\psi_{7}\right\rangle&\equiv&|0-1\rangle|2-10\rangle,\\\
\left|\psi_{3}\right\rangle&\equiv&|3\rangle|0-3+9-12\rangle,&\left|\psi_{8}\right\rangle&\equiv&|0-2\rangle|3-9\rangle,\\\
\left|\psi_{4}\right\rangle&\equiv&|4\rangle|0-4+8-12\rangle,&\left|\psi_{9}\right\rangle&\equiv&|0-3\rangle|4-8\rangle,\\\
\left|\psi_{5}\right\rangle&\equiv&|5\rangle|0-5+7-12\rangle,&\left|\psi_{10}\right\rangle&\equiv&|0-4\rangle|5-7\rangle,\\\
\left|\phi_{1}\right\rangle&\equiv&|7\rangle|6-7+3-4\rangle,&\left|\phi_{7}\right\rangle&\equiv&|6-12\rangle|7-4\rangle,\\\
\left|\phi_{2}\right\rangle&\equiv&|8\rangle|6-8+4-5\rangle,&\left|\phi_{8}\right\rangle&\equiv&|6-7\rangle|8-5\rangle,\\\
\left|\phi_{3}\right\rangle&\equiv&|9\rangle|6-9+3-5\rangle,&\left|\phi_{9}\right\rangle&\equiv&|6-8\rangle|9-3\rangle,\\\
\left|\phi_{4}\right\rangle&\equiv&|10\rangle|6-10+2-5\rangle,&\left|\phi_{10}\right\rangle&\equiv&|6-9\rangle|10-2\rangle,\\\
\left|\phi_{5}\right\rangle&\equiv&|11\rangle|6-11+1-5\rangle,&\left|\phi_{11}\right\rangle&\equiv&|6-10\rangle|11-1\rangle,\\\
\left|\phi_{6}\right\rangle&\equiv&|12\rangle|6-12+0-5\rangle,&\left|\phi_{12}\right\rangle&\equiv&|6-11\rangle|12-0\rangle,\end{array}$

and $|\psi_{11}\rangle\equiv|+_{5}\rangle|+_{12}\rangle,$ and
$|\phi_{13}\rangle\equiv|{}_{6}+_{12}\rangle|+_{12}\rangle.$ Then the set
$\mathcal{S}$ has genuine hidden nonlocality of type I.

For further discussion in the composite dimension, we need the following
lemma.

######  Lemma 1.

Let $\mathcal{N}$ be a quantum channel from system $\mathcal{H}_{A}$ to system
$\mathcal{H}_{B}$. Then $\mathcal{N}$ preserves non-orthogonality. That is,
for any density operators $\rho,\sigma$ of systems $\mathcal{H}_{A}$,

$\langle\rho,\sigma\rangle_{A}\neq 0,\ \Rightarrow\
\langle\mathcal{N}(\rho),\mathcal{N}(\sigma)\rangle_{B}\neq 0.$

Here, the inner product
$\langle\rho,\sigma\rangle_{A}:=\mathrm{Tr}[\rho^{\dagger}\sigma].$
Particularly, if we choose $\mathcal{N}$ to be the partial trace operation,
then we have that the partial trace operation preserves non-orthogonality.

Lemma 1 can be understood as a corollary of the well known result: a set of
quantum states is perfect distinguishable under global measurement if and only
if the given set of states are mutually orthogonal. For the sake of
completeness, we give a direct proof in the APPENDIX.

Method for checking local irredundancy: Now we return to discuss the genuine
activation of nonlocality in general dimension. Suppose $d=2k+1\geq 11$, we
can construct the set $\mathcal{S}$ to be the union of
$\\{|\psi_{i}\rangle\\}_{i=1}^{2k-1}$ and
$\\{|\phi_{i}\rangle\\}_{j=1}^{2k+1}$ similarly to Example 1 and Example 2. In
the following, we show that even for the case of composite number $d$, the
above set of states is locally irredundant whenever $d\geq 11$. Let
$d=p_{1}p_{2}\cdots p_{n}$, where $p_{i}|d$ is some prime divisor of $d$ (here
we assume that $3\leq p_{1}\leq p_{2}\leq\cdots\leq p_{n}$ and $n\geq 2$).
Suppose that $\mathcal{H}_{X}=\mathbb{C}^{d}$, it can be factored into an
$n$-particles systems $\otimes_{i=1}^{n}\mathcal{H}_{X_{i}}$ where
$\mathcal{H}_{X_{i}}=\mathbb{C}^{p_{i}}$ and $X\in\\{A,B\\}$. In fact, we will
show that Alice and Bob can not preserve the orthogonality of the given states
by discarding one or more of their subsystems. Written
$|\psi_{i}\rangle=|\psi_{i}\rangle_{A}|\psi_{i}\rangle_{B}$ for
$i=1,2,\cdots,2(k-1)$, one observes that

$\langle\psi_{i}|\psi_{j}\rangle_{B}\neq 0,\
\langle\psi_{k-1+i}|\psi_{k-1+j}\rangle_{A}\neq 0$ (1)

for different $i,j\in\\{0,1,\cdots,k-1\\}.$ If the set is local redundancy,
there must exist not all empty sets
$\mathcal{S}_{A}\subseteq\\{A_{1},A_{2},\cdots,A_{n}\\}$,
$\mathcal{S}_{B}\subseteq\\{B_{1},B_{2},\cdots,B_{n}\\}$ and some unitaries
$U_{A},U_{B}\in U(d),$ such that

$\\{\mathrm{Tr}_{\mathcal{S}_{A}\cup\mathcal{S}_{B}}[(U_{A}\otimes
U_{B})|\psi_{i}\rangle\langle\psi_{i}|(U_{A}^{\dagger}\otimes
U_{B}^{\dagger})]\\}_{i=1}^{2k-2}$

are mutually orthogonal. And this is equivalent to the mutual orthogonality of
the following set

$\\{\mathrm{Tr}_{\mathcal{S}_{A}}[U_{A}|\psi_{i}\rangle_{A}\langle\psi_{i}|U_{A}^{\dagger}]\otimes\mathrm{Tr}_{\mathcal{S}_{B}}[U_{B}|\psi_{i}\rangle_{B}\langle\psi_{i}|U_{B}^{\dagger}]\\}_{i=1}^{2k-2}.$

To ensure the satisfaction of these orthogonality relations, by Eq. (1) and
Lemma 1, both
$\\{\mathrm{Tr}_{\mathcal{S}_{A}}[U_{A}|\psi_{i}\rangle_{A}\langle\psi_{i}|U_{A}^{\dagger}]\\}_{i=1}^{k-1}$
and
$\\{\mathrm{Tr}_{\mathcal{S}_{B}}[U_{B}|\psi_{k-1+i}\rangle_{B}\langle\psi_{k-1+i}|U_{B}^{\dagger}]\\}_{i=1}^{k-1}$
are mutually orthogonal. By assumption, at least one of $\mathcal{S}_{A}$ and
$\mathcal{S}_{B}$ is nonempty. Without loss of generality, we assume that
$\mathcal{S}_{A}\neq\emptyset.$ Therefore, the resulting states
$\\{\mathrm{Tr}_{\mathcal{S}_{A}}[U_{A}|\psi_{i}\rangle_{A}\langle\psi_{i}|U_{A}^{\dagger}]\\}_{i=1}^{k-1}$
are $(k-1)$ orthogonal states in the systems corresponding to
$\\{A_{1},\cdots,A_{n}\\}\setminus\mathcal{S}_{A}$ whose dimension is at most
$d/p_{1}\leq d/3<k-1$. This deduces a contradiction as there are at most $N$
orthogonal positive semidefinite matrices in system with dimensional $N$.
Therefore, our constructing set is locally irredundant.

######  Theorem 2.

Let $d\geq 11$ be an odd integer. Then there exists some orthogonal set
without entanglement in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ which has
genuine hidden nonlocality of type I .

One finds that the above results can be also use to construct sets with this
property in multipartite systems. For example, let
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}\otimes\mathcal{H}_{D}\otimes\mathcal{H}_{E}\otimes\mathcal{H}_{F}$
where

$\dim_{\mathbb{C}}(\mathcal{H}_{A})=\dim_{\mathbb{C}}(\mathcal{H}_{B})=\dim_{\mathbb{C}}(\mathcal{H}_{C})=\dim_{\mathbb{C}}(\mathcal{H}_{D})=11$

and
$\dim_{\mathbb{C}}(\mathcal{H}_{E})=\dim_{\mathbb{C}}(\mathcal{H}_{F})=13.$
Set $\mathbb{M}^{B}:=\\{\pi_{1}^{B}=\sum_{i=0}^{4}|i\rangle\langle i|,\
\pi_{2}^{B}=\sum_{j=5}^{10}|j\rangle\langle j|\\}.$ Denote
$\\{|\Phi_{i}\rangle_{AB}\\}_{i=1}^{20}$ be the set constructed in Example 1.
Finding four unfilled squares (with labels $h_{i},H_{i},v_{j},V_{j}$) in Fig.
2 such that they come from exactly four different colored blocks and form a
rectangle. For example, $\\{(3,2),(3,8),(9,2),(9,8)\\}.$ Using the four
cubics, we can define a state $|\Phi\rangle_{AB}:=|3-9\rangle|2-8\rangle$ with
the following properties: (1). The set
$\\{|\Phi_{i}\rangle_{AB}\\}_{i=1}^{20}\cup\\{|\Phi\rangle_{AB}\\}$ is an
orthogonal set which is still locally distinguishable (whose proof is similar
to the proof showed in Example 1); (2). $\pi_{k}^{B}|\Phi\rangle_{AB}$ is
nonzero for each $k=1,2$; (3). $\pi_{k}^{B}|\Phi\rangle_{AB}$ is orthogonal to
each of $\\{\pi_{k}^{B}|\Phi_{i}\rangle_{AB}\\}_{i=1}^{20}$for $k=1,2$. Denote
$\\{|\Psi_{j}\rangle_{EF}\\}_{j=1}^{24}$ be the set constructed in Example 2.
And we choose $|\Psi\rangle_{EF}$ to be $|4-10\rangle|3-9\rangle$ which has
similar properties with $|\Phi\rangle_{AB}$. One can check that the union of
the following set

$\begin{array}[]{l}\\{|\Phi_{i}\rangle_{AB}|\Phi\rangle_{CD}|\Psi\rangle_{EF}\\}_{i=1}^{20},\\\\[5.69054pt]
\\{|\Phi\rangle_{AB}|\Phi_{j}\rangle_{CD}|\Psi\rangle_{EF}\\}_{j=1}^{20},\\\\[5.69054pt]
\\{|\Phi\rangle_{AB}|\Phi\rangle_{CD}|\Psi_{k}\rangle_{EF}\\}_{k=1}^{24},\end{array}$

has genuine hidden nonlocality of type I in
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}\otimes\mathcal{H}_{D}\otimes\mathcal{H}_{E}\otimes\mathcal{H}_{F}$.
Firstly, the set is locally distinguishable as
$\\{|\Phi_{i}\rangle_{AB}\\}_{i=1}^{20}\cup\\{|\Phi\rangle_{AB}\\}$,
$\\{|\Phi_{i}\rangle_{CD}\\}_{j=1}^{20}\cup\\{|\Phi\rangle_{CD}\\}$ and
$\\{|\Psi_{i}\rangle_{EF}\\}_{k=1}^{24}\cup\\{|\Psi\rangle_{EF}\\}$ are all
locally distinguishable. Now if the $B,D,F$ parts make the following local
measurements

$\begin{array}[]{l}\mathbb{M}^{B}:=\\{\pi_{1}^{B}=\sum_{i=0}^{4}|i\rangle\langle
i|,\ \pi_{2}^{B}=\sum_{j=5}^{10}|j\rangle\langle j|\\},\\\\[2.84526pt]
\mathbb{M}^{D}:=\\{\pi_{1}^{D}=\sum_{i=0}^{4}|i\rangle\langle i|,\
\pi_{2}^{D}=\sum_{j=5}^{10}|j\rangle\langle j|\\},\\\\[2.84526pt]
\mathbb{M}^{F}:=\\{\pi_{1}^{F}=\sum_{i=0}^{5}|i\rangle\langle i|,\
\pi_{2}^{F}=\sum_{j=6}^{12}|j\rangle\langle j|\\}\end{array}$

there are eight outcomes. One can check that the post-measurement states of
each outcome form an orthogonal set with $64$ elements which is locally
indistinguishable. Generally, this result can be extended to
$\bigotimes_{i=1}^{N}(\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{B_{i}})$ where
each
$d_{i}=\mathrm{dim}_{\mathbb{C}}(\mathcal{H}_{A_{i}})=\mathrm{dim}_{\mathbb{C}}(\mathcal{H}_{B_{i}})$
is an odd integer greater than 10 for $i\in\\{1,2,\cdots,N\\}$.

Note that locally irreducible (see Halder19 ) is a stronger form of
nonlocality than locally indistinguishable. Therefore, it is interesting to
find that whether can we genuinely activate this stronger form of nonlocality
under OPLM without decreasing the cardinality of the given set. The examples
with three elements provided by Bandyopadhyay21 do satisfy this property. In
the following, we present an example of such set without entanglement.

Figure 4: This shows the states structure of set of product states in Example
3 . The notation here is similar with that in Fig. 2. The four squares
($(1,0),(1,9),(6,0),(6,9)$ labeled with ‘$m$’ correspond to the states
$|\mathcal{M}\rangle=|1-6\rangle|0-9\rangle$.

######  Example 3.

Let $\mathcal{S}$ be the set of product states in
$\mathbb{C}^{11}\otimes\mathbb{C}^{11}$ whose elements are listed as below
(see Fig. 4)):

$\begin{array}[]{rclrcl}\left|\psi_{1}\right\rangle&\equiv&|1\rangle|3-4+5-6\rangle,&\left|\psi_{5}\right\rangle&\equiv&|0-4\rangle|3-6\rangle,\\\
\left|\psi_{2}\right\rangle&\equiv&|2\rangle|2-4+5-7\rangle,&\left|\psi_{6}\right\rangle&\equiv&|0-1\rangle|2-7\rangle,\\\
\left|\psi_{3}\right\rangle&\equiv&|3\rangle|1-4+5-8\rangle,&\left|\psi_{7}\right\rangle&\equiv&|0-2\rangle|1-8\rangle,\\\
\left|\psi_{4}\right\rangle&\equiv&|4\rangle|0-4+5-9\rangle,&\left|\psi_{8}\right\rangle&\equiv&|0-3\rangle|0-9\rangle,\\\
\left|\phi_{1}\right\rangle&\equiv&|6\rangle|5-6+2-3\rangle,&\left|\phi_{6}\right\rangle&\equiv&|5-10\rangle|6-3\rangle,\\\
\left|\phi_{2}\right\rangle&\equiv&|7\rangle|5-7+3-4\rangle,&\left|\phi_{7}\right\rangle&\equiv&|5-6\rangle|7-4\rangle,\\\
\left|\phi_{3}\right\rangle&\equiv&|8\rangle|5-8+2-4\rangle,&\left|\phi_{8}\right\rangle&\equiv&|5-7\rangle|8-2\rangle,\\\
\left|\phi_{4}\right\rangle&\equiv&|9\rangle|5-9+1-4\rangle,&\left|\phi_{9}\right\rangle&\equiv&|5-8\rangle|9-1\rangle,\\\
\left|\phi_{5}\right\rangle&\equiv&|10\rangle|5-10+0-4\rangle,&\left|\phi_{10}\right\rangle&\equiv&|5-9\rangle|10-0\rangle,\par\end{array}$

and $|S\rangle\equiv|+_{10}\rangle|+_{10}\rangle,$ and
$|\mathcal{M}\rangle\equiv|1-6\rangle|0-9\rangle.$ Then the set $\mathcal{S}$
has genuine hidden strong form nonlocality of type I.

Similar with Example 1, we can show that $\mathcal{S}$ is locally
distinguishable. Suppose Bob perform the measurement
$\mathbb{M}^{B}:=\\{\pi_{1}^{B}=\sum_{i=0}^{4}|i\rangle\langle i|,\
\pi_{2}^{B}=\sum_{j=5}^{10}|j\rangle\langle j|\\}.$ Note that, for both
outcomes of the measurement, the cardinaility of the possible states are
unchanged. In appendix, we show that for both outcomes of the measurement the
postmeasurement states are locally irreducible.

##  IV Genuine hidden nonlocality of type II

In this section, we study how to construct sets without entanglement but have
genuine hidden nonlocality of type II. We will use the local
indistinguishability of the sets without entanglement as follows.

######  Theorem 3 (Xu20a ).

In $\mathbb{C}^{m}\otimes\mathbb{C}^{n}$ system with $m,n\geq 3$, there exists
a set with $2(m+n)-4$ orthogonal pure product states which is locally
indistinguishable.

Figure 5: This shows the examples of the four types of states structure in
Theorem 3 in $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$,
$\mathbb{C}^{4}\otimes\mathbb{C}^{5}$, $\mathbb{C}^{5}\otimes\mathbb{C}^{8}$,
and $\mathbb{C}^{6}\otimes\mathbb{C}^{4}$ respectively. The squares indicated
by the same label of cardinality $2,3$ represent $2,3$ states respectively.
For example, there are three squares (that is, $(0,3),(1,3),$ and $(2,3)$)
with label ‘$v_{1}$’ at the right down corner figure, they correspond to
$|0+w1+w^{2}2\rangle|3\rangle\in\mathbb{C}^{6}\otimes\mathbb{C}^{4}$ where
$w\in\\{1,e^{\frac{2\pi\sqrt{-1}}{3}},e^{\frac{4\pi\sqrt{-1}}{3}}\\}$).

According to the parity of $m$ and $n$, the structure of the set of states can
be divided into four types (see Fig. 5). For more details on the sets of
quantum states, the authors may see the reference Xu20a . Now let’s start with
the following example.

######  Example 4.

Let $\mathcal{S}$ be the set of product states in
$\mathbb{C}^{7}\otimes\mathbb{C}^{8}$ whose elements are listed as below:

$\begin{array}[]{rclrcl}\left|\psi_{1,2}\right\rangle&\equiv&|0\rangle|0\pm
1\rangle,&\left|\psi_{8,9}\right\rangle&\equiv&|3\rangle|3\pm 4+5\pm
6\rangle,\\\ \left|\psi_{3,4}\right\rangle&\equiv&|0\rangle|2\pm
3\rangle,&\left|\psi_{10,11}\right\rangle&\equiv&|3\rangle|1\pm 2\rangle,\\\
\left|\psi_{5,6,7}\right\rangle&\equiv&|0+w1+w^{2}2\rangle|4\rangle,&\left|\psi_{12,13,14}\right\rangle&\equiv&|1+w2+w^{2}3\rangle|0\rangle,\\\
\left|\phi_{1,2}\right\rangle&\equiv&|4\rangle|5\pm 6+3\pm
4\rangle,&\left|\phi_{5,6}\right\rangle&\equiv&|6\rangle|6\pm 7\rangle,\\\
\left|\phi_{3,4}\right\rangle&\equiv&|4\pm 5\rangle|7\rangle,&\
\left|\phi_{7,8}\right\rangle&\equiv&|5\pm 6\rangle|5\rangle,\end{array}$

where $w\in\\{1,e^{\frac{2\pi\sqrt{-1}}{3}},e^{\frac{4\pi\sqrt{-1}}{3}}\\}$
which is any cubic root of unit. Then the set $\mathcal{S}$ has genuine hidden
nonlocality of type II.

Figure 6: This shows the states structure of set of product states in
$\mathbb{C}^{7}\otimes\mathbb{C}^{8}$ of Example 4. The squares indicated by
the same label of cardinality $2,3,4$ represent $2,3,2$ states respectively.
For examples, there are two squares (that is, $(0,2),(0,3)$) with label
‘$h_{2}$’, they correspond to two states $|\psi_{3,4}\rangle:=|0\rangle|2\pm
3\rangle,$ and there are four squares (that is, $(3,3),(3,4),$ $(3,5)$, and
$(3,6)$) with label ‘$h_{3}$’, they correspond to two states
$|\phi_{8,9}\rangle:=|3\rangle|3\pm 4+5\pm 6\rangle$ (More exactly,
$|\phi_{8}\rangle:=|3\rangle|3+4+5+6\rangle$ and
$|\phi_{9}\rangle:=|3\rangle|3-4+5-6\rangle$).

Now we show that $\mathcal{S}$ is locally distinguishable. First, Alice
perform the measurement
$\mathbb{M}_{1}^{A}:=\\{\pi_{1}^{A}=\sum_{i=0}^{3}|i\rangle\langle i|,\
\pi_{2}^{A}=\sum_{j=4}^{6}|j\rangle\langle j|\\}.$

If the outcome of $\mathbb{M}_{1}^{A}$ is ‘1’, the state must be one of
$\\{|\psi_{i}\rangle\\}_{i=1}^{14}.$ And the states do not change under this
measurement. Now Bob perform the measurement:

$\mathbb{M}_{1}^{B}:=\\{\pi_{1}^{B}=|4\rangle\langle 4|+|5\rangle\langle 5|,\
\pi_{2}^{B}=\mathbb{I}_{8}-\pi_{i}^{B}\\}.$

If the outcome of $\mathbb{M}_{1}^{B}$ is ‘1’, the state must be one of
$\\{|\psi_{5,6,7,8,9}\rangle\\}$ which may change to be

$\displaystyle\left|\psi_{5,6,7}\right\rangle=|0+w1+w^{2}2\rangle|4\rangle,\
\left|\tilde{\psi}_{8,9}\right\rangle=|3\rangle|5\pm 4\rangle.$

It is easy to derive a LOCC protocal to distinguish the states of above set.
If the outcome of $\mathbb{M}_{1}^{B}$ is ‘2’, the state must be one of
$\\{|\psi_{i}\rangle\\}_{i=1}^{14}\setminus\\{|\psi_{5,6,7}\rangle\\}$
(without the states corresponding to the label $v_{1}$ in Fig. 6) which may
change to

$\begin{array}[]{rclrcl}\left|\psi_{1,2}\right\rangle&=&|0\rangle|0\pm
1\rangle,&\left|\tilde{\psi}_{8,9}\right\rangle&=&|3\rangle|3\pm 6\rangle,\\\
\left|\psi_{3,4}\right\rangle&=&|0\rangle|2\pm
3\rangle,&\left|\psi_{10,11}\right\rangle&=&|3\rangle|1\pm 2\rangle,\\\
&&&\left|\psi_{12,13,14}\right\rangle&=&|1+w2+w^{2}3\rangle|0\rangle.\end{array}$

It is also easy to derive a LOCC protocal to distinguish the states of above
set.

If the outcome of $\mathbb{M}_{1}^{A}$ is ‘2’, the state must be one of
$\\{|\phi_{j}\rangle\\}_{j=1}^{8}.$ And the states do not change under this
measurement. Now Bob perform the measurement:

$\mathbb{M}_{1}^{B}:=\\{\pi_{1}^{B}=|4\rangle\langle 4|+|5\rangle\langle 5|,\
\pi_{2}^{B}=\mathbb{I}_{8}-\pi_{i}^{B}\\}.$

If the outcome of $\mathbb{M}_{1}^{B}$ is ‘1’, the state must be one of
$\\{|\phi_{1,2,7,8}\rangle\\}$ which may change to be

$\displaystyle\left|\tilde{\phi}_{1,2}\right\rangle=|4\rangle|5\pm 4\rangle,\
\left|\phi_{7,8}\right\rangle=|5\pm 6\rangle|5\rangle.$

It is easy to derive a LOCC protocal to distinguish these four states. If the
outcome of $\mathbb{M}_{1}^{B}$ is ‘2’, the state must be one of
$\\{|\phi_{i}\rangle\\}_{i=1}^{8}\setminus\\{|\phi_{7,8}\rangle\\}$ (without
the states corresponding to the label $V_{2}$ in Fig. 6) which may change to
be

$\begin{array}[]{rclrcl}\left|\phi_{1,2}\right\rangle&=&|4\rangle|3\pm
6\rangle,&\left|\phi_{5,6}\right\rangle&=&|6\rangle|6\pm 7\rangle,\\\
\left|\phi_{3,4}\right\rangle&=&|4\pm 5\rangle|7\rangle.&&&\end{array}$

It is also easy to derive a LOCC protocal to distinguish the states of above
set.

Now we prove the second property. Suppose Bob perform the measurement
$\mathbb{M}^{B}:=\\{\pi_{1}^{B}=\sum_{i=0}^{4}|i\rangle\langle i|,\
\pi_{2}^{B}=\sum_{j=5}^{7}|j\rangle\langle j|\\}.$

If the outcome of $\mathbb{M}^{B}$ is ‘1’, the states are transferred to

$\begin{array}[]{rclrcl}\left|\psi_{1,2}\right\rangle&=&|0\rangle|0\pm
1\rangle,&\left|\tilde{\psi}_{8,9}\right\rangle&=&|3\rangle|3\pm 4\rangle,\\\
\left|\psi_{3,4}\right\rangle&=&|0\rangle|2\pm
3\rangle,&\left|\psi_{10,11}\right\rangle&=&|3\rangle|1\pm 2\rangle,\\\
\left|\psi_{5,6,7}\right\rangle&=&|0+w1+w^{2}2\rangle|4\rangle,&\left|\tilde{\psi}_{12,13,14}\right\rangle&=&|1+w2+w^{2}3\rangle|0\rangle,\\\
\left|\phi_{1,2}\right\rangle&=&|4\rangle|3\pm 4\rangle,&&&\end{array}$

which are mutually orthogonal and contains $14$ states that is known to be
locally indistinguishable in $\mathbb{C}^{4}\otimes\mathbb{C}^{5}$.

If the outcome of $\mathbb{M}^{B}$ is ‘2’, the states are transferred to

$\begin{array}[]{rclrcl}&&&\left|\tilde{\psi}_{8,9}\right\rangle&\equiv&|3\rangle|5\pm
6\rangle,\\\ \left|\tilde{\phi}_{1,2}\right\rangle&\equiv&|4\rangle|5\pm
6\rangle,&\left|\phi_{5,6}\right\rangle&\equiv&|6\rangle|6\pm 7\rangle,\\\
\left|\phi_{3,4}\right\rangle&\equiv&|4\pm 5\rangle|7\rangle,&\
\left|\phi_{7,8}\right\rangle&\equiv&|5\pm 6\rangle|5\rangle,\end{array}$

which are mutually orthogonal and contains $8$ states that is known to be
locally indistinguishable in $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$.

Now we prove that the set is locally irredundant. Written
$|\psi_{i}\rangle=|\psi_{i}\rangle_{A}|\psi_{i}\rangle_{B}$ for
$i=1,2,\cdots,14.$ One may observe that

$\langle\psi_{i}|\psi_{j}\rangle_{A}\neq 0,\text{ and
}\langle\psi_{k}|\psi_{l}\rangle_{B}\neq 0$

for different $i,j\in\\{1,2,3,4,5\\}$ and different $k,l\in\\{1,12,13,14\\}.$
If the set $\mathcal{S}$ is local redundancy, there must exist
$\mathcal{S}_{B}\subseteq\\{B_{1},B_{2},B_{3}\\}$ and some unitary $U_{B}\in
U(d),$ such that
$\\{\mathrm{Tr}_{\mathcal{S}_{B}}[U_{B}|\psi_{i}\rangle_{B}\langle\psi_{i}|U_{B}^{\dagger}]\\}_{i=1}^{5}$
are mutually orthogonal. However, the dimension of the system corresponding to
$\\{B_{1},B_{2},B_{3}\\}\setminus\mathcal{S}_{B}$ is at most $4$ which is less
than $5$. ∎

The above argument can be easily extend to
$\mathbb{C}^{m}\otimes\mathbb{C}^{n}$ provided the conditions $m,n\geq 7$.
Therefore, we arrive at the following theorem.

######  Theorem 4.

Let $m,n\geq 7$ be an integer. Then there exists some orthogonal set without
entanglement in $\mathbb{C}^{m}\otimes\mathbb{C}^{n}$ which has genuine hidden
nonlocality of type II .

For an integer $m\geq 7$ define $f_{m}=\lfloor\frac{m-1}{2}\rfloor$ and
$c_{m}=m-f_{m}$. Similarly, one finds that $c_{m}$ is the smallest integer
which is larger than $m/2$. Similarly, one can construct a set with $2(m+n)-8$
states which are inspired by Xu et al.’s results Xu20a in
$\mathbb{C}^{c_{m}}\otimes\mathbb{C}^{c_{n}}$ and
$\mathbb{C}^{f_{m}}\otimes\mathbb{C}^{f_{n}}.$ The local irredundancy can be
obtained similarly as the above argument since there are at least
$c_{n}>\frac{n}{2}$ states for which Alice’s part are mutually non-orthogonal
and $c_{m}>\frac{m}{2}$ states for which Bob’s part are mutually non-
orthogonal. Similar with the discussion in the end of Sec. III, we can extend
the construction to multipartite systems whose local dimensions are greater
than $6$.

##  V Conclusions and Discussions

We studied genuine hidden nonlocality without entanglement in the settings of
local discrimination of quantum states. There are two cases: the first one is
cardinality preserving under OPLM and the second one is cardinality decreasing
under OPLM. For both cases, we provided examples to demonstrate the idea how
to construct set of product states with genuine hidden nonlocality. For the
first case, the construction can even be extended to $2N$-parties systems
$\bigotimes_{i=1}^{N}(\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{B_{i}})$ whenever
each
$d_{i}=\mathrm{dim}_{\mathbb{C}}(\mathcal{H}_{A_{i}})=\mathrm{dim}_{\mathbb{C}}(\mathcal{H}_{B_{i}})$
is an odd integer greater than 10 for $i\in\\{1,2,\cdots,N\\}.$ And for the
second case, we can extend the idea to any $N$-parties systems whose local
dimensions are greater than $6$. Moreover, we provided a method to deal with
the local redundancy problem which might be helpful to tackling this problem.
Compared with results with those in Bandyopadhyay21 , it is a bit surprise as
sets without entanglement seems can be more easier to be genuinely activated
than sets with entanglement. Therefore, it is interesting find more sets with
entanglement (but with more complicated structure) and genuine hidden
nonlocality. In this case, finding a way to check the local irredundancy is a
tricky problem.

###### Acknowledgements.

We thank Shi Fei for pointing out several mistakes of our orginal maniscript.
This work is supported by National Natural Science Foundation of China under
Grant No. 12005092, the China Postdoctoral Science Foundation (2020M681996),
the Key Research and Development Project of Guangdong province under Grant No.
2020B0303300001, the Guangdong Basic and Applied Research Foundation under
Grant No. 2020B1515310016.

APPENDIX

Proof of Lemma 1. Suppose not, we have
$\langle\mathcal{N}(\rho),\mathcal{N}(\sigma)\rangle_{B}=0.$ Let
$\rho=\sum_{i=1}^{m}\lambda_{i}|x_{i}\rangle\langle
x_{i}|,\sigma=\sum_{j=1}^{n}\mu_{j}|y_{j}\rangle\langle y_{j}|$ be the
spectral decomposition of $\rho$ and $\sigma$ respectively. By the linearity
of $\mathcal{N}$ and the inner product,

$\langle\mathcal{N}(\rho),\mathcal{N}(\sigma)\rangle_{B}=\sum_{i=1}^{m}\sum_{j=1}^{n}\lambda_{i}\mu_{j}\langle\mathcal{N}(|x_{i}\rangle\langle
x_{i}|),\mathcal{N}(|y_{j}\rangle\langle y_{j}|)\rangle_{B}.$ (2)

It is well known that $\mathrm{Tr}[MN]\geq 0$ if $M$ and $N$ are positive
semidefinite matrices. Therefore, the zero of
$\langle\mathcal{N}(\rho),\mathcal{N}(\sigma)\rangle_{B}$ implies all the
terms on the right hand side of Eq. (2) are zeros, i.e.,

$\langle\mathcal{N}(|x_{i}\rangle\langle
x_{i}|),\mathcal{N}(|y_{j}\rangle\langle y_{j}|)\rangle_{B}=0$

for all $i,j$. For each $i,j$, we compute the
$\langle\mathcal{N}(|x_{i}\rangle\langle
x_{i}|),\mathcal{N}(|y_{j}\rangle\langle y_{j}|)\rangle_{B}$ by using the
Kraus representation of $\mathcal{N}$ which is given by

$\mathcal{N}(\tau)=\sum_{k=1}^{K}A_{k}\tau A_{k}^{\dagger},\ \text{ where
}\sum_{k=1}^{K}A_{k}^{\dagger}A_{k}=\mathbb{I}_{A}.$

Therefore, we have

$\begin{array}[]{rcl}\mathcal{N}(|x_{i}\rangle\langle
x_{i}|),\mathcal{N}(|y_{j}\rangle\langle
y_{j}|)\rangle_{B}&=&\displaystyle\sum_{k=1}^{K}\sum_{l=1}^{K}\mathrm{Tr}[A_{k}|x_{i}\rangle\langle
x_{i}|A_{k}^{\dagger}A_{l}|y_{j}\rangle\langle
y_{j}|A_{l}^{\dagger}]\\\\[5.69054pt]
&=&\displaystyle\sum_{k=1}^{K}\sum_{l=1}^{K}\mathrm{Tr}[\langle
x_{i}|A_{k}^{\dagger}A_{l}|y_{j}\rangle\langle
y_{j}|A_{l}^{\dagger}A_{k}|x_{i}\rangle]\\\\[5.69054pt]
&=&\displaystyle\sum_{k=1}^{K}\sum_{l=1}^{K}|\langle
x_{i}|A_{k}^{\dagger}A_{l}|y_{j}\rangle|^{2}.\end{array}$

As $\langle\mathcal{N}(|x_{i}\rangle\langle
x_{i}|),\mathcal{N}(|y_{j}\rangle\langle y_{j}|)\rangle_{B}=0$, we have
$|\langle x_{i}|A_{k}^{\dagger}A_{l}|y_{j}\rangle|^{2}=0$ for all $1\leq
k,l\leq K$. Particularly, $\langle x_{i}|A_{k}^{\dagger}A_{k}|y_{j}\rangle=0$
for $1\leq k\leq K$ which imply that $\langle x_{i}|y_{j}\rangle=0$ as
$\sum_{k=1}^{K}A_{k}^{\dagger}A_{k}=\mathbb{I}_{A}.$ Using these results and
the spectral decomposition of $\rho$ and $\sigma$, we obtain that
$\rho\sigma=\mathbf{0}$ which is a zero matrix. This is contradicted with the
non-orthogonality of $\rho$ and $\sigma$. ∎

Proof of the local irreducibility of postmeasurement states in Example 3. If
the outcome of $\mathbb{M}^{B}$ is ‘1’, the states are transferred to
$\tilde{\mathcal{S}}_{1}$

$\begin{array}[]{rclrcl}\left|\tilde{\psi}_{1}\right\rangle&=&|1\rangle|3-4\rangle,&\left|\tilde{\psi}_{5}\right\rangle&=&|0-4\rangle|3\rangle,\\\
\left|\tilde{\psi}_{2}\right\rangle&=&|2\rangle|2-4\rangle,&\left|\tilde{\psi}_{6}\right\rangle&=&|0-1\rangle|2\rangle,\\\
\left|\tilde{\psi}_{3}\right\rangle&=&|3\rangle|1-4\rangle,&\left|\tilde{\psi}_{7}\right\rangle&=&|0-2\rangle|1\rangle,\\\
\left|\tilde{\psi}_{4}\right\rangle&=&|4\rangle|0-4\rangle,&\left|\tilde{\psi}_{8}\right\rangle&=&|0-3\rangle|0\rangle,\\\
\left|\tilde{\phi}_{1}\right\rangle&=&|6\rangle|2-3\rangle,&\left|\tilde{\phi}_{6}\right\rangle&=&|5-10\rangle|3\rangle,\\\
\left|\tilde{\phi}_{2}\right\rangle&=&|7\rangle|3-4\rangle,&\left|\tilde{\phi}_{7}\right\rangle&=&|5-6\rangle|4\rangle,\\\
\left|\tilde{\phi}_{3}\right\rangle&=&|8\rangle|2-4\rangle,&\left|\tilde{\phi}_{8}\right\rangle&=&|5-7\rangle|2\rangle,\\\
\left|\tilde{\phi}_{4}\right\rangle&=&|9\rangle|1-4\rangle,&\left|\tilde{\phi}_{9}\right\rangle&=&|5-8\rangle|1\rangle,\\\
\left|\tilde{\phi}_{5}\right\rangle&=&|10\rangle|0-4\rangle,&\left|\tilde{\phi}_{10}\right\rangle&=&|5-9\rangle|0\rangle,\\\
\left|\tilde{S}\right\rangle&=&|+_{10}\rangle|+_{4}\rangle,&\left|\tilde{\mathcal{M}}\right\rangle&=&|1-6\rangle|0\rangle,\end{array}$

which can be seen as states in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}:=\mathbb{C}^{11}\otimes\mathbb{C}^{5}$
with computation basis $\\{|i\rangle|j\rangle\ \big{|}\
i\in\mathbb{Z}_{11},j\in\mathbb{Z}_{5}\\}.$ We only need to show that $A$ and
$B_{1}$ can only start with trivial OPLMs. In fact, to preserve the
orthogonality of
$\\{|\tilde{\psi}_{i}\rangle\\}_{i=1}^{8}\cup\\{|\tilde{S}\rangle\\},$ $B_{1}$
can only start with trivial measurement. Now it is sufficient to prove that if
$E=(a_{i,j})_{i,j\in\mathbb{Z}_{11}}$ is an $11\times 11$ Hermitian matrix and

$\langle\Theta_{1}|E\otimes\mathbb{I}_{B_{1}}|\Theta_{2}\rangle=0,\ \
|\Theta_{1}\rangle\neq|\Theta_{2}\rangle\in\tilde{\mathcal{S}}_{1},$ (3)

then $E\propto\mathbb{I}_{A}.$ As the $B_{1}$ part of the states
$\\{|\tilde{\phi}_{i}\rangle\\}_{i=1}^{4}\cup\\{|\tilde{\psi}_{j}\rangle\\}_{j=1}^{5}$,
applying Eq. (4) to this subset, we obtain

$a_{k,l}=0,\ \ \forall\ k,l\in\mathbb{Z}_{11}\setminus\\{0,5\\},\text{ and
}k\neq l.$

Substitute $|\Theta_{1}\rangle$ and $|\Theta_{2}\rangle$ in Eq. (3) by one of
$\\{|\tilde{\psi}_{i}\rangle\\}_{i=1}^{4}\cup\\{|\tilde{\phi}_{j}\rangle\\}_{j=2}^{5}$
and $|\tilde{\phi}_{7}\rangle$ respectively, we obtain
$a_{x,5}=a_{x,6}=0\text{ for }x\in\\{1,2,3,4,7,8,9,10\\}.$ Now substitute
$|\Theta_{1}\rangle$ and $|\Theta_{2}\rangle$ in Eq. (3) by
$|\tilde{\phi}_{1}\rangle$ and $|\tilde{\phi}_{6}\rangle$, we obtain that
$a_{6,5}=a_{6,10}=0.$ Now applying Eq, (3) to the following pairs of states
$(|\tilde{\psi}_{5}\rangle,|\tilde{\psi}_{1}\rangle),$
$(|\tilde{\psi}_{6}\rangle,|\tilde{\psi}_{2}\rangle),$
$(|\tilde{\psi}_{7}\rangle,|\tilde{\psi}_{3}\rangle),$
$(|\tilde{\psi}_{8}\rangle,|\tilde{\psi}_{4}\rangle),$
$(|\tilde{\psi}_{5}\rangle,|\tilde{\phi}_{6}\rangle),$
$(|\tilde{\psi}_{5}\rangle,|\tilde{\phi}_{2}\rangle),$
$(|\tilde{\psi}_{6}\rangle,|\tilde{\phi}_{3}\rangle),$
$(|\tilde{\psi}_{7}\rangle,|\tilde{\phi}_{4}\rangle),$
$(|\tilde{\psi}_{8}\rangle,|\tilde{\phi}_{5}\rangle),$ and
$(|\tilde{\psi}_{5}\rangle,|\tilde{\phi}_{6}\rangle)$, we get $a_{0,x}=0$ for
$x\in\mathbb{Z}_{11}\setminus\\{0\\}.$ Therefore, $E$ is a diagonal matrix. At
last, applying Eq.(3) to one of
$\\{|\tilde{\psi}_{i}\rangle\\}_{i=5}^{8}\cup\\{|\tilde{\phi}_{j}\rangle\\}_{j=6}^{10}\cup\\{|\tilde{\mathcal{M}}\rangle\\}$
and $|\tilde{S}\rangle,$ we have $E\propto\mathbb{I}_{A}$.

If the outcome of $\mathbb{M}^{B}$ is ‘2’, the states are transferred to
$\tilde{\mathcal{S}}_{2}$

$\begin{array}[]{rclrcl}\left|\tilde{\psi}_{1}\right\rangle&=&|1\rangle|5-6\rangle,&\left|\tilde{\psi}_{5}\right\rangle&=&|0-4\rangle|6\rangle,\\\
\left|\tilde{\psi}_{2}\right\rangle&=&|2\rangle|5-7\rangle,&\left|\tilde{\psi}_{6}\right\rangle&=&|0-1\rangle|7\rangle,\\\
\left|\tilde{\psi}_{3}\right\rangle&=&|3\rangle|5-8\rangle,&\left|\tilde{\psi}_{7}\right\rangle&=&|0-2\rangle|8\rangle,\\\
\left|\tilde{\psi}_{4}\right\rangle&=&|4\rangle|5-9\rangle,&\left|\tilde{\psi}_{8}\right\rangle&=&|0-3\rangle|9\rangle,\\\
\left|\tilde{\phi}_{1}\right\rangle&=&|6\rangle|5-6\rangle,&\left|\tilde{\phi}_{6}\right\rangle&=&|5-10\rangle|6\rangle,\\\
\left|\tilde{\phi}_{2}\right\rangle&=&|7\rangle|5-7\rangle,&\left|\tilde{\phi}_{7}\right\rangle&=&|5-6\rangle|7\rangle,\\\
\left|\tilde{\phi}_{3}\right\rangle&=&|8\rangle|5-8\rangle,&\left|\tilde{\phi}_{8}\right\rangle&=&|5-7\rangle|8\rangle,\\\
\left|\tilde{\phi}_{4}\right\rangle&=&|9\rangle|5-9\rangle,&\left|\tilde{\phi}_{9}\right\rangle&=&|5-8\rangle|9\rangle,\\\
\left|\tilde{\phi}_{5}\right\rangle&=&|10\rangle|5-10\rangle,&\left|\tilde{\phi}_{10}\right\rangle&=&|5-9\rangle|10\rangle,\\\
\left|\tilde{S}\right\rangle&=&|+_{10}\rangle|{}_{5}+_{10}\rangle,&\left|\tilde{\mathcal{M}}\right\rangle&=&|1-6\rangle|9\rangle,\end{array}$

which can be seen as states in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B_{2}}:=\mathbb{C}^{11}\otimes\mathbb{C}^{6}$
with computation basis $\\{|i\rangle|j\rangle\ \big{|}\
i\in\mathbb{Z}_{11},j\in\\{5,6,\cdots,10\\}\\}.$ We only need to show that $A$
and $B_{2}$ can only start with trivial OPLMs. In fact, to preserve the
orthogonality of
$\\{|\tilde{\phi}_{i}\rangle\\}_{i=1}^{10}\cup\\{|\tilde{S}\rangle\\},$
$B_{2}$ can only start with trivial measurement. Now it is sufficient to prove
that if $E=(a_{i,j})_{i,j\in\mathbb{Z}_{11}}$ is an $11\times 11$ Hermitian
matrix and

$\langle\Theta_{1}|E\otimes\mathbb{I}_{B_{2}}|\Theta_{2}\rangle=0,\ \
|\Theta_{1}\rangle\neq|\Theta_{2}\rangle\in\tilde{\mathcal{S}}_{2},$ (4)

then $E\propto\mathbb{I}_{A}.$ Written
$E=\left(\begin{array}[]{cc}E_{11}&E_{12}\\\ E_{21}&E_{22}\end{array}\right)$
where $E_{11},E_{12}=E_{21}^{\dagger},E_{22}$ are $5\times 5,5\times 6,$ and
$6\times 6$ matrices respectively. Following similar proof with that in Ref.
Yus15 , we can prove that $E_{11}=\alpha_{1}\sum_{i=0}^{4}|i\rangle\langle i|$
and $E_{22}=\alpha_{2}\sum_{j=5}^{10}|j\rangle\langle j|$ for some
$\alpha_{1},\alpha_{2}\geq 0$ by using the orthogonality relations of Eq. (4)
from the sets
$\\{|\tilde{\psi}_{i}\rangle\\}_{i=1}^{8}\cup\\{|\tilde{S}\rangle\\}$ and
$\\{|\tilde{\phi}_{j}\rangle\\}_{j=1}^{10}\cup\\{|\tilde{S}\rangle\\}$
respectively. In the following, we show that $E_{12}$ is a zero matrix. Note
that the $B_{2}$ part of $|\tilde{\psi}_{i}\rangle$ and
$|\tilde{\phi}_{j}\rangle$ are pairwise nonorthogonal for $i=1,2,3,4$ and
$j=1,2,3,4,5.$ Substitute $|\Theta_{1}\rangle$ and $|\Theta_{2}\rangle$ in Eq.
(4) by $|\tilde{\psi}_{i}\rangle$ and $|\tilde{\phi}_{j}\rangle$ respectively,
we obtain

$a_{i,j+5}=\langle i|E|j+5\rangle=0$

for $i=1,2,3,4$ and $j=1,2,3,4,5.$ Now substitute $|\Theta_{1}\rangle$ and
$|\Theta_{2}\rangle$ in Eq. (4) by $|\tilde{\psi}_{i+4}\rangle$ and
$|\tilde{\phi}_{i}\rangle$ ($i=1,2,3,4$), we obtain

$\begin{array}[]{cccc}a_{0,6}=a_{4,6}=0,&a_{0,7}=a_{1,7}=0,&a_{0,8}=a_{2,8}=0,&a_{0,9}=a_{3,9}=0.\end{array}$

Now substitute $|\Theta_{1}\rangle$ and $|\Theta_{2}\rangle$ in Eq. (4) by
$|\tilde{\psi}_{i}\rangle$ and $|\tilde{\phi}_{i+1}\rangle$ ($i=5,6$), we
obtain $a_{0,5}-a_{0,10}+a_{4,5}-a_{4,10}=0,$ and
$a_{0,5}-a_{0,6}+a_{1,5}-a_{1,6}=0.$ By the previous result, we get
$a_{0,5}=a_{0,6}=0.$ Therefore, the matrix $E_{12}$ is a zero matrix. Hence
$E$ is a diagonal matrix. At last, consider that
$\langle\tilde{\mathcal{M}}|E\otimes\mathbb{I}_{B_{2}}|\tilde{S}\rangle=0,$ we
have $a_{1,1}=a_{6,6}$. That is, $\alpha_{1}=\alpha_{2}.$ So we conclude that
$E\propto\mathbb{I}_{A}.$ ∎

## References

  * (1) J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195 (1964).
  * (2) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014); Erratum, Rev. Mod. Phys. 86, 839 (2014).
  * (3) J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  * (4) S.J. Freedman and J.F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972).
  * (5) A. Aspect, P. Grangier, and G. Roger, Experimental Tests of Realistic Local Theories via Bell’s Theorem, Phys. Rev. Lett. 47, 460 (1981).
  * (6) A. Aspect, J. Dalibard, and G. Roger, Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982).
  * (7) B. Hensen _et al.,_ Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015).
  * (8) J. Handsteiner _et al_., Cosmic Bell Test: Measurement Settings from Milky Way Stars, Phys. Rev. Lett. 118, 060401 (2017).
  * (9) W. Rosenfeld _et al_., Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes, Phys. Rev. Lett. 119, 010402 (2017).
  * (10) BIG Bell Test Collaboration, Challenging local realism with human choices, Nature 557, 212 (2018).
  * (11) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * (12) S. Popescu, Bell’s Inequalities and Density Matrices: Revealing “Hidden” Nonlocality, Phys. Rev. Lett. 74, 2619 (1995).
  * (13) N. Gisin, Hidden quantum nonlocality revealed by local filters, Phys. Lett. A 210, 151 (1996).
  * (14) M. Navascués, and T. Vértesi, Activation of Nonlocal Quantum Resources, Phys. Rev. Lett. 106, 060403 (2011).
  * (15) C. Palazuelos, Superactivation of Quantum Nonlocality, Phys. Rev. Lett. 109, 190401 (2012).
  * (16) W. Klobus, W. Laskowski, M. Markiewicz, and A. Grudka, Nonlocality activation in entanglement-swapping chains, Phys. Rev. A 86, 020302(R) (2012).
  * (17) Flavien Hirsch, Marco Tu´lio Quintino, Joseph Bowles, and Nicolas Brunner, Genuine hidden quantum nonlocality, Phys. Rev. Lett. 111, 160402 (2013).
  * (18) 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 (1999).
  * (19) 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 (1999).
  * (20) J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local distinguishability of multipartite orthogonal quantum states, Phys. Rev. Lett. 85, 4972 (2000).
  * (21) S. Ghosh, G. Kar, A. Roy, A. Sen (De), and U. Sen, Distinguishability of Bell states, Phys. Rev. Lett. 87, 277902 (2001).
  * (22) B. Groisman and L. Vaidman, Nonlocal variables with product state eigenstates,  J. Phys. A: Math. Gen. 34, 6881 (2001).
  * (23) J. Walgate and L. Hardy, Nonlocality, asymmetry, and distinguishing bipartite states, Phys. Rev. Lett. 89, 147901 (2002).
  * (24) M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Local indistinguishability: more nonlocality with less entanglement, Phys. Rev. Lett. 90, 047902 (2003).
  * (25) 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).
  * (26) S. De Rinaldis, Distinguishability of complete and unextendible product bases, Phys. Rev. A 70, 022309 (2004).
  * (27) S. Ghosh, G. Kar, A. Roy, and D. Sarkar, Distinguishability of maximally entangled states, Phys. Rev. A 70, 022304 (2004).
  * (28) H. Fan, Distinguishability and indistinguishability by local operations and classical communication, Phys. Rev. Lett. 92, 177905 (2004).
  * (29) M. Nathanson, Distinguishing bipartite orthogonal states by LOCC: best and worst cases, Journal of Mathematical Physics 46, 062103 (2005).
  * (30) J. Watrous, Bipartite subspaces having no bases distinguishable by local operations and classical communication, Phys. Rev. Lett. 95, 080505 (2005).
  * (31) M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Bounds on entangled orthogonal state discrimination using local operations and classical communication, Phys. Rev. Lett. 96, 040501 (2006).
  * (32) W. K. Wootters, Distinguishing unentangled states with an unentangled measurement, Int. J. Quantum Inf. 4, 219 (2006).
  * (33) J. Niset and N. J. Cerf, Multipartite nonlocality without entanglement in many dimensions, Phys. Rev. A 74, 052103 (2006).
  * (34) R. Y. Duan, Y. Feng, Z. F. Ji, and M. S. Ying, Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication, Phys. Rev. Lett. 98, 230502 (2007).
  * (35) Y. Feng and Y.-Y. Shi, Characterizing locally indistinguishable orthogonal product states, IEEE Trans. Inf. Theory 55, 2799 (2009).
  * (36) R. Y. Duan, Y. Feng, Y. Xin, and M. S. Ying, Distinguishability of quantum states by separable operations, IEEE Trans. Inform. Theory 55, 1320 (2009).
  * (37) S. Bandyopadhyay, S. Ghosh and G. Kar, LOCC distinguishability of unilaterally transformable quantum states, New J. Phys. 13, 123013 (2011).
  * (38) S. Bandyopadhyay, More nonlocality with less purity, Phys. Rev. Lett. 106, 210402 (2011).
  * (39) N. Yu, R. Duan, and M. Ying, Four Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled States, Phys. Rev. Lett. 109, 020506 (2012).
  * (40) A. Cosentino, Positive-partial-transpose-indistinguishable states via semidefinite programming, Phys. Rev. A 87, 012321 (2013).
  * (41) M.-S. Li, Y.-L. Wang, S.-M. Fei and Z.-J. Zheng, $d$ locally indistinguishable maximally entangled states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, Phys. Rev. A 91, 042318 (2015).
  * (42) S. X. Yu and C. H. Oh, Detecting the local indistinguishability of maximally entangled states. arXiv:1502.01274v1.
  * (43) M.-S. Li, S.-M. Fei, Z.-X. Xiong, and Y.-L. Wang, Twist-teleportation-based local discrimination of maximally entangled states, SCIENCE CHINA Physics, Mechnics $\&$ Astronomy 63 8, 280312 (2020).
  * (44) D. Ha and Y. Kwon, Quantum nonlocality without entanglement: explicit dependence on prior probabilities of nonorthogonal mirror-symmetric states, npj Quantum Inf. 7, 81 (2021).
  * (45) S. Bandyopadhyay, M. Nathanson, Tight bounds on the distinguishability of quantum states under separable measurements, Phys. Rev. A 88, 052313 (2013).
  * (46) A. Cosentino and V. Russo, Small sets of locally indistinguishable orthogonal maximally entangled states, Quantum Information and Computation 14, 1098 (2014).
  * (47) S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous, and N. Yu, Limitations on separable measurements by convex optimization, IEEE Transactions on Information Theory, 61, 3593 (2015).
  * (48) A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, A framework for bounding nonlocality of state discrimination, Commun. Math. Phys. 323, 1121 (2013).
  * (49) Y.-H. Yang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Local distinguishability of orthogonal quantum states in a $2\otimes 2\otimes 2$ system, Phys. Rev. A 88, 024301 (2013).
  * (50) 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).
  * (51) 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).
  * (52) 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).
  * (53) 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).
  * (54) 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, 31048 (2016).
  * (55) 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 (2016).
  * (56) 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 (2016).
  * (57) X. Zhang, X. Tan, J. Weng, and Y. Li, LOCC indistinguishable orthogonal product quantum states, Sci. Rep. 6, 28864 (2016).
  * (58) G.-B. Xu, Q.-Y. Wen, F. Gao, S.-J. Qin, and H.-J. Zuo, Local indistinguishability of multipartite orthogonal product bases, Quantum Inf. Processing 16, 276 (2017).
  * (59) Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, The local indistinguishability of multipartite product states, Quantum Inf. Processing 16, 5 (2017).
  * (60) 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 (2017).
  * (61) X. Zhang, J. Weng, X. Tan, and W. Luo, Indistinguishability of pure orthogonal product states by LOCC, Quantu Inf. Processing 16, 168 (2017).
  * (62) S. Halder, Several nonlocal sets of multipartite pure orthogonal product states, Phys. Rev. A 98, 022303 (2018).
  * (63) M.-S. Li and Y.-L. Wang, Alternative method for deriving nonlocal multipartite product states, Phys. Rev. A 98, 052352 (2018).
  * (64) D.-H. Jiang, and G.-B. Xu, Nonlocal sets of orthogonal product states in arbitrary multipartite quantum system, Phys. Rev. A 102, 032211 (2020).
  * (65) G.-B. Xu, and D.-H. Jiang, Novel methods to construct nonlocal sets of orthogonal product states in any bipartite high-dimensional system,Quantum Inf. Process. 20, 128 (2021).
  * (66) S. Halder, and C. Srivastava, Locally distinguishing quantum states with limited classical communication, Phys. Rev. A 101, 052313 (2020).
  * (67) S. Halder, R. Sengupta, Distinguishability classes, resource sharing, and bound entanglement distribution, Phys. Rev. A 101, 012311 (2020).
  * (68) S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Strong Quantum Nonlocality without Entanglement, Phys. Rev. Lett. 122, 040403 (2019).
  * (69) Z.-C. Zhang and X. Zhang, Strong quantum nonlocality in multipartite quantum systems, Phys. Rev. A 99, 062108 (2019).
  * (70) F. Shi, M. Hu, L. Chen, and X. Zhang, Strong quantum nonlocality with entanglement, Phys. Rev. A 102, 042202 (2020).
  * (71) P. Yuan, G. J. Tian, and X. M. Sun, Strong quantum nonlocality without entanglement in multipartite quantum systems, Phys. Rev. A 102, 042228 (2020).
  * (72) Y.-L. Wang, M.-S. Li, and M.-H. Yung, Graph-connectivity-based strong quantum nonlocality with genuine entanglement, Phys. Rev. A 104, 012424 (2021).
  * (73) F. Shi, M.-S. Li, M. Hu, L. Chen, M.-H. Yung, Y.-L. Wang and X. Zhang, Strongly nonlocal unextendible product bases do exist, arXiv:2101.00735.
  * (74) F. Shi, M.-S. Li, M. Hu, L. Chen, M.-H. Yung, Y.-L. Wang and X. Zhang, Strong quantum nonlocality from hypercubes, arXiv:2110.08461.
  * (75) M. Banik, T. Guha, M. Alimuddin, G. Kar, S. Halder, and S. S. Bhattacharya, Multicopy Adaptive Local Discrimination: Strongest Possible Two-Qubit Nonlocal Bases, Phys. Rev. Lett. 126 210505 (2021).
  * (76) B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Hiding Bits in Bell States, Phys. Rev. Lett. 86, 5807 (2001).
  * (77) D. P. DiVincenzo, D.W. Leung and B.M. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48, 580 (2002).
  * (78) D. Markham and B. C. Sanders, Graph States for Quantum Secret Sharing, Phys. Rev. A 78, 042309 (2008).
  * (79) R. Rahaman and M. G. Parker, Quantum scheme for secret sharing based on local distinguishability, Phys. Rev. A 91, 022330 (2015).
  * (80) J. Wang, L. Li, H. Peng, and Y. Yang, Quantum-secret-sharing scheme based on local distinguishability of orthogonal multiqudit entangled states, Phys. Rev. A 95, 022320 (2017).
  * (81) S. Bandyopadhyay, S. Halder, Genuine activation of nonlocality: From locally available to locally hidden information, Phys. Rev. A 104, L050201 (2021).