#  Local distinguishability based genuinely quantum nonlocality without
entanglement

Mao-Sheng Li  Department of Physics, Southern University of Science and
Technology, Shenzhen, 518055, China  Department of Physics, University of
Science and Technology of China, Hefei, 230026, China  Yan-Ling Wang
<EMAIL_ADDRESS>School of Computer Science and Techonology, Dongguan
University of Technology, Dongguan, 523808, China  Fei Shi
<EMAIL_ADDRESS>School of Cyber Security, University of Science and
Technology of China, Hefei, 230026, China.  Man-Hong Yung<EMAIL_ADDRESS>Department of Physics, Southern University of Science and Technology,
Shenzhen, 518055, China  Institute for Quantum Science and Engineering, and
Department of Physics, Southern University of Science and Technology,
Shenzhen, 518055, China

###### Abstract

Recently, Halder _et al._ [Phys. Rev. Lett. 122, 040403 (2019)] proposed the
concept strong nonlocality without entanglement: an orthogonal set of fully
product states in multipartite quantum systems that is locally irreducible for
every bipartition of the subsystems. As the difficulty of the problem, most of
the results are restricted to tripartite systems. Here we consider a weaker
form of nonlocality called local distinguishability based genuine nonlocality.
A set of orthogonal multipartite quantum states is said to be genuinely
nonlocal if it is locally indistinguishable for every bipartition of the
subsystems. In this work, we tend to study the latter form of nonlocality.
First, we present an elegant set of product states in bipartite systems that
is locally indistinguishable. After that, based on a simple observation, we
present a general method to construct genuinely nonlocal sets of multipartite
product states by using those sets that are genuinely nonlocal but with less
parties. As a consequence, we obtain that genuinely nonlocal sets of fully
product states exist for all possible multipartite quantum systems.

##  I Introduction

Quantum states discrimination plays a fundamental role in quantum information
processing. It is well known that a set of quantum states can be perfectly
distinguished by positive operation value measurement (POVM) if and only if
these states are pairwise orthogonal nils . In a multipartite setting, due to
the physical obstacles, sometimes we can not take a global measurement but
only can use local operations with classical communication (LOCC). Bennett et
al. Ben99  presented examples of orthogonal product states that are
indistinguishable under LOCC 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. A
set of orthogonal states which is indistinguishable under LOCC is also called
as being locally indistinguishable or nonlocal. The local indistinguishability
has been practically applied in quantum cryptography primitives such as data
hiding Terhal01 ; DiVincenzo02  and secret sharing Markham08 ; Rahaman15 ;
WangJ17 .

Since the work of Bennett et al. Ben99 , the problem of local discrimination
of quantum states has attracted much attention. The maximally entangled states
and the product states, as being two extreme sets among the pure states, their
local distinguishability is the most attractive. Here we present an incomplete
list of the results about the local distinguishability of maximally entangled
states Gho01 ; Wal00 ; Wal02 ; Fan04 ; Nathanson05 ; Cohen07 ; Bandyopadhyay11
; Li15 ; Fan07 ; Yu12 ; Cos13 ; Yu115 ; Wang19 ; Xiong19 ; Li20  and product
states Ben99 ; Ran04 ; Hor03 ; Ben99b ; DiVincenzo03 ; Zhang14 ; Zhang15 ;
Zhang16 ; Xu16b ; Xu16m ; Zhang16b ; Wang15 ; Wang17 ; Feng09 ; Yang13 ;
Zhang17 ; Zhangj17 ; Halder18 ; Li18 ; Halder1909 ; Xu20a ; Xu20b . Another
direction of related research is to study how much resource of entanglement
are needed in order to distinguish quantum states which are locally
indistinguishable Cohen08 ; Bandyopadhyay16 ; Zhang16E ; Bandyopadhyay18 ;
Lilv19 . Another important sets which are known to be locally
indistinguishable are those unextendible product bases (UPB), sets of
incomplete orthonormal product states whose complementary space has no product
states Ben99b ; DiVincenzo03 ; Feng06 ; J14 ; CJ15 .

Recently, Halder _et al._ Halder19  introduced a stronger form of local
indistinguishability, i.e., local irreducibility. A set of multipartite
orthogonal quantum states is said to be locally irreducible if it is not
possible to locally eliminate one or more states from the set while preserving
orthogonality of the postmeasurement states. Under this setting, they proposed
the concept strong nonlocality without entanglement. A set of orthogonal
multipartite product states is called to be strong nonlocality if it is
locally irreducible for every bipartition of the systems. They provided the
first two examples of strongly nonlocal sets of product states in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ and
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ and raised the
questions of how to extend their results to multipartite quantum systems and
sets of unextendible product bases Halder19 . Quite recently, Zhang _et al._
Zhang1906  extended the concept of strong nonlocality to more general
settings. However, there are only a few sets which have been proven to be
strongly nonlocal Halder19 ; Zhang1906 ; Rout1909 ; Rout1910 ; Tian20 ; Shi20S
. Most of the known results are in the tripartite quantum settings. Here we
propose a form of nonlocality called genuine nonlocality whose nonlocality is
lying between the local distinguishablity based nonlocality and the local
irreducibility based strong nonlocality (the definition here is slightly
different from that defined by Rout _et al._ in Ref. Rout1909 ). A set of
orthogonal multipartite quantum states is said to be genuinely nonlocal if it
is locally indistinguishable for every bipartition of the systems. A nature
question arises: are there genuinely nonlocal set of fully product states for
any possible multipartite quantum systems? In this paper, we tend to address
this problem.

The rest of this article is organized as follows. In Sec. II, we give some
necessary notation, definitions and some basic result of local nonlocality of
bipartite product basis. In Sec. III, we present a general method to obtaining
genuinely nonlocal set of multipartite product states. Finally, we draw a
conclusion and present some interesting problems in section IV.

##  II Locally indistinguishable set of bipartite product states

For any integer $n\geq 2$, we denote $U(n)$ to be the set of all unitary
matrices of dimensional $n$. And throughout this paper, we use the following
subset of unitary matrices

$U_{FL}(n):=\\{(h_{ij})_{i,j}^{n}\in U(n)|h_{1k},h_{k1}\neq
0,k=1,\cdots,n\\}.$

That is, the set of $n$ dimensional unitary matrices whose elements on the
first and last rows are all nonzero.

Let $n\geq 3$ be an integer and $\mathcal{H}$ be a Hilbert space of
dimensional $n$. Assume that $\mathcal{B}=(|1\rangle,\cdots,|n\rangle)$ is an
$n$-tuple of vectors in $\mathcal{H}$ and these $n$ vectors are consisting of
an orthonormal basis of $\mathcal{H}$. And we call $\mathcal{B}$ an ordered
orthonormal basis of $\mathcal{H}$. For any $H=(h_{jk})_{j,k=1}^{n-1}\in
U_{FL}(n-1)$, we define two operations on $\mathcal{H}$ with respect to
$\mathcal{B}$

$H_{\mathcal{B}}^{(U)}:=\displaystyle\sum_{j=1}^{n-1}\sum_{k=1}^{n-1}h_{jk}|j\rangle\langle
k|,H_{\mathcal{B}}^{(D)}:=\displaystyle\sum_{j=1}^{n-1}\sum_{k=1}^{n-1}h_{jk}|j+1\rangle\langle
k+1|.$

That is, under the computational basis $\\{|1\rangle,\cdots,|n\rangle\\}$,
their matrix representations are as follows

$\begin{array}[]{c}H_{\mathcal{B}}^{(U)}=\left[\begin{array}[]{ll}H&\mathbf{0}_{(n-1)\times
1}\\\ \mathbf{0}_{1\times(n-1)}&0\end{array}\right],\\\\[8.53581pt]
H_{\mathcal{B}}^{(D)}=\left[\begin{array}[]{ll}0&\mathbf{0}_{1\times(n-1)}\\\
\mathbf{0}_{(n-1)\times 1}&H\end{array}\right].\end{array}$

We call them the up extension and down extension of $H$ with respect to
$\mathcal{B}$ respectively.

Figure 1: States structure corresponding to $\\{|\psi\rangle\\}$ in Theorem 1
(or $\\{|\Psi\rangle\\}$ in Theorem 3).

Motivated by the constructions of nonlocal sets of product states in Ref.
Xu20a , the following theorem is a generalized version of their results but
the proof is more elegant.

###### Theorem 1

Let $x,y\geq 3$ be integers and $X\in U_{FL}(x-1)$, $Y\in U_{FL}(y-1)$. Let
$\mathcal{H}_{A}$($\mathcal{H}_{B}$) be a Hilbert space of dimension $x$($y$)
with an ordered orthonormal basis
$\mathcal{A}=(|1\rangle_{A},\cdots,|x\rangle_{A})$($\mathcal{B}=(|1\rangle_{B},\cdots,|y\rangle_{B})$).
The following $2(x+y)-4$ product states in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ are locally indistinguishable (See
Fig. 1)

$\begin{array}[]{l}|\psi_{i}\rangle:=|1\rangle_{A}\otimes(Y_{\mathcal{B}}^{(U)}|i\rangle_{B}),\\\
|\psi_{y-1+j}\rangle:=(X_{\mathcal{A}}^{(U)}|j\rangle_{A})\otimes|y\rangle_{B},\\\
|\psi_{x+y-3+k}\rangle:=|x\rangle_{A}\otimes(Y_{\mathcal{B}}^{(D)}|k\rangle_{B}),\\\
|\psi_{x+2y-4+l}\rangle:=(X_{\mathcal{A}}^{(D)}|l\rangle_{A})\otimes|1\rangle_{B},\\\
\end{array}$

where $1\leq i\leq y-1,1\leq j\leq x-1,2\leq k\leq y,2\leq l\leq x.$

_Proof_. Suppose Alice starts with a measurement
$\\{M_{a}^{\dagger}M_{a}\\}_{a=1}^{S}$. The postmeasurement states should be
orthogonal to each other, i.e.

$\langle\psi_{i}|M_{a}^{\dagger}M_{a}\otimes\mathbb{I}_{B}|\psi_{j}\rangle=0,\text{
for }i\neq j.$

Let $M:=M_{a}^{\dagger}M_{a}$. Suppose its matrix representation under the
ordered basis $\mathcal{A}$ is $(m_{ij})_{i,j=1}^{x}$. Then one finds

$M=\displaystyle\sum_{i=1}^{x}\sum_{j=1}^{x}m_{ij}|i\rangle_{A}\langle j|.$

Figure 2:  The left hand side draw the boundary states corresponding to the
ordered bases
$\mathcal{A}:=(|1\rangle_{A},|2\rangle_{A},|3\rangle_{A},|4\rangle_{A},|5\rangle_{A},|6\rangle_{A})$
and
$\mathcal{B}:=(|1\rangle_{B},|2\rangle_{B},|3\rangle_{B},|4\rangle_{B},|5\rangle_{B},|6\rangle_{B},|7\rangle_{B},|8\rangle_{B},|9\rangle_{B})$.
The right hand side presents the boundary states corresponding to the ordered
bases
$\mathcal{A}^{\prime}:=(|2\rangle_{A},|1\rangle_{A},|3\rangle_{A},|4\rangle_{A},|5\rangle_{A},|6\rangle_{A})$
and
$\mathcal{B}^{\prime}:=(|2\rangle_{B},|1\rangle_{B},|3\rangle_{B},|4\rangle_{B},|5\rangle_{B},|6\rangle_{B},|7\rangle_{B},|9\rangle_{B},|8\rangle_{B})$

but draws them under the ordered bases $\mathcal{A}$ and $\mathcal{B}.$

Because $M_{a}\otimes\mathbb{I}_{y}|\psi_{1}\rangle$ is orthogonal to the set
of states
$\\{M_{a}\otimes\mathbb{I}_{y}|\psi_{x+2y-4+l}\rangle\big{|}l=2,3,\cdots,x\\}$,
we have the following equations

${}_{A}\langle 1|MX_{\mathcal{A}}^{(D)}|l\rangle_{A}=0,l=2,3,\cdots,x.$

These equalities are equivalent to the matrix equality
$[m_{12},m_{13},\cdots,m_{1x}]X=[0,0,\cdots,0]$. As $X$ is invertible, we have
$[m_{12},m_{13},\cdots,m_{1x}]=[0,0,\cdots,0]$. As $M$ is Hermitian, we also
have $[m_{21},m_{31},\cdots,m_{x1}]=[0,0,\cdots,0]$.

Because $M_{a}\otimes\mathbb{I}_{y}|\psi_{x+y-3+2}\rangle$ is orthogonal to
the set of states
$\\{M_{a}\otimes\mathbb{I}_{y}|\psi_{y-1+j}\rangle\big{|}j=1,2,\cdots,x-1\\}$,
we have the following equations

${}_{A}\langle x|MX_{\mathcal{A}}^{(U)}|j\rangle_{A}=0,j=1,2,\cdots,y-1.$

These equalities are equivalent to the matrix equality
$[m_{x1},m_{x2},\cdots,m_{x(x-1)}]X=[0,0,\cdots,0]$. As $X$ is invertible, we
have $[m_{x1},m_{x2},\cdots,m_{x(x-1)}]=[0,0,\cdots,0]$. As $M$ is Hermitian,
we also have $[m_{1x},m_{2x},\cdots,m_{(x-1)x}]=[0,0,\cdots,0]$.

Because the set of states
$\\{M_{a}\otimes\mathbb{I}_{y}|\psi_{y-1+j}\rangle\big{|}j=1,2,\cdots,x-1\\}$
are pairwise orthogonal to each other, we have the following equations

${}_{A}\langle
j_{1}|{X_{\mathcal{A}}^{(U)}}^{\dagger}MX_{\mathcal{A}}^{(U)}|j_{2}\rangle_{A}=0,\text{
for }1\leq j_{1}\neq j_{2}\leq x-1.$

If we define $M^{(u)}:=(m_{ij})_{i,j=1}^{x-1}$, the above equalities are
equivalent to
$X^{\dagger}M^{(u)}X=\text{diag}(\alpha_{1},\cdots,\alpha_{x-1})$. Since $X$
is a unitary matrix, we have

$M^{(u)}X=X\text{diag}(\alpha_{1},\cdots,\alpha_{x-1}).$

In the following, we will compare the first row of the matrices at both hand
sides of the above equality. Suppose that $X=(X_{ij})_{i,j=1}^{x-1}$. Then the
first row of $M^{(u)}X$ is
$[m_{11}X_{11},m_{11}X_{12},\cdots,m_{11}X_{1(x-1)}]$. Meanwhile, the first
row of $X\text{diag}(\alpha_{1},\cdots,\alpha_{x-1})$ is
$[\alpha_{1}X_{11},\alpha_{2}X_{12},\cdots,\alpha_{x-1}X_{1(x-1)}].$ Comparing
these two vectors, one can derive
$[\alpha_{1},\alpha_{2},\cdots,\alpha_{x-1}]=[m_{11},m_{11},\cdots,m_{11}]$ as
$X_{11},X_{12},\cdots,X_{1(x-1)}$ are all nonzero. Hence

$M^{(u)}=X\text{diag}(\alpha_{1},\cdots,\alpha_{x-1})X^{\dagger}=m_{11}\mathbb{I}_{x-1}.$

Using the orthogonal relations among the states in
$\\{M_{a}\otimes\mathbb{I}_{y}|\psi_{x+2y-4+l}\rangle\big{|}l=2,3,\cdots,x\\}$,
we have

${}_{A}\langle
l_{1}|{X_{\mathcal{A}}^{(D)}}^{\dagger}MX_{\mathcal{A}}^{(D)}|l_{2}\rangle_{A}=0,\text{
for }2\leq l_{1}\neq l_{2}\leq x.$

If we define $M^{(d)}:=(m_{ij})_{i,j=2}^{x}$, the above equalities are
equivalent to $X^{\dagger}M^{(d)}X=\text{diag}(\beta_{2},\cdots,\beta_{x})$.
In a similar way (but here we should consider the last row instead of the
first row), we can get $M^{(d)}=m_{xx}\mathbb{I}_{x-1}.$

Therefore, the Hermitian matrix $M$ is of the form $m_{11}\mathbb{I}_{x}$.
That is, Alice can only start with trivial measurement. By the symmetry of the
constructed states, Bob can only start with trivial measurement.

Remark: Notice that the product states we constructed in Theorem 1 are spanned
by the “boundary states” with respect to the ordered bases $\mathcal{A}$ and
$\mathcal{B}$ (the outermost layer of a rectangle under the ordered bases
$\mathcal{A}$ and $\mathcal{B}$). One finds that
$\text{span}_{\mathbb{C}}\\{|\psi_{1}\rangle,\cdots,|\psi_{2x+2y-4}\rangle\\}$
is equal to

$\text{span}_{\mathbb{C}}\\{\mathcal{A}_{i}\otimes\mathcal{B}_{j}|i\in\\{1,x\\}\text{
or }j\in\\{1,y\\}\\}$

where $\mathcal{A}_{i}$($\mathcal{B}_{j}$) is the $i$-th ($j$-th) element of
the $x$($y$)-tuples $\mathcal{A}$ ($\mathcal{B}$) (see Fig. 2).

##  III Constructing genuinely nonlocal set from known ones

As any set of orthogonal product states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{d}$ is locally distinguishable DiVincenzo03
, a necessary condition for an orthogonal set of fully product states in
$\bigotimes_{i=1}^{L}\mathbb{C}^{d_{i}}$ to be genuinely nonlocal is
$d_{i}\geq 3$ for all $i$. In this section, we show that there always exists
some genuinely nonlocal set of fully product states in
$\bigotimes_{i=1}^{L}\mathbb{C}^{d_{i}}$ if the previous necessary condition
is fulfilled.

###### Theorem 2

Let $L\geq 3$ and $d_{i}\geq 3\ (i=1,2,\cdots,L)$ be integers. Then there
always exists an orthogonal set of fully product states in
$\otimes_{i=1}^{L}\mathbb{C}^{d_{i}}$ that is genuinely nonlocal.

This conclusion can be derived from Theorem 1, Theorem 3, Proposition 1 of
this paper and the genuinely nonlocal set of
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ constructed in Ref.
Rout1909 .

Notice that if $S=\\{|\phi_{k}\rangle_{A}|\theta_{k}\rangle_{B}\\}_{k=1}^{N}$
is $A|B$ locally indistinguishable, then
$\mathcal{S}=\\{|\phi_{k}\rangle_{A}|\theta_{k}\rangle_{B}|\varphi\rangle_{A_{1}}|\vartheta\rangle_{B_{1}}\\}_{k=1}^{N}$
is also $AA_{1}|BB_{1}$ locally indistinguishable. Otherwise, in the first
setting, Alice and Bob can prepare the ancillar qudit states as
$|\varphi\rangle_{A_{1}}$, $|\vartheta\rangle_{B_{1}}$ respectively on
themselves side and using the latter distinguish strategy to locally
distinguish the set of states in $S$. Moreover, we have the following
observation(see also in Ref. Rout1910 )

###### Observation 1

Let $S=\\{|\Psi_{k}\rangle_{AB}\\}_{k=1}^{N}$ be a nonlocal product set shared
between Alice and Bob. Consider the set
$\mathcal{S}:=\\{|\Psi_{k}\rangle_{AB}\otimes|\Phi_{0}\rangle_{A_{1}\cdots
A_{m}}\otimes|\Theta_{0}\rangle_{B_{1}\cdots B_{n}}\\}_{k=1}^{N}$, where
$|\Phi_{0}\rangle_{A_{1}\cdots A_{m}}$ and $|\Theta_{0}\rangle_{B_{1}\cdots
B_{n}}$ are some fully product states with some of the subsystems
$\\{A_{i}\\}_{i=1}^{m}$ and $\\{B_{j}\\}_{j=1}^{n}$ being in possession of
some parties. The resulting set $\mathcal{S}$ is also nonlocal between
$\mathcal{H}_{A}\otimes(\otimes_{i=1}^{m}\mathcal{H}_{A_{i}})$ and
$\mathcal{H}_{B}\otimes(\otimes_{j=1}^{n}\mathcal{H}_{B_{j}})$.

With this observation, we show how to use the special structure of nonlocal
sets in Theorem 1 to construct genuinely nonlocal set of product states in
tripartite systems.

Figure 3: States structure corresponding to $\\{|\Phi\rangle\\}$ in the
Theorem 3.

###### Theorem 3

Let $x,z\geq 3,y\geq 4$ be integers and $X$, $Y$, $Z$ belong to $U_{FL}(x-1)$,
$U_{FL}(y-1)$ and $U_{FL}(z-1)$ respectively. Let $\mathcal{H}_{A}$,
$\mathcal{H}_{B}$, $\mathcal{H}_{C}$ be Hilbert spaces of dimension $x,y,z$
respectively. Suppose that $\mathcal{A}=(|1\rangle_{A},\cdots,|x\rangle_{A})$,
$\mathcal{B}=(|1\rangle_{B},\cdots,|y\rangle_{B})$, and
$\mathcal{C}=(|1\rangle_{C},\cdots,|z\rangle_{C})$ are ordered orthonormal
bases with respect to $\mathcal{H}_{A}$, $\mathcal{H}_{B}$, $\mathcal{H}_{C}$.
The following $2x+4y+2z-8$ product states in
$\mathbb{C}^{x}\otimes\mathbb{C}^{y}\otimes\mathbb{C}^{z}$ are pairwise
orthogonal and they form a set of product states which is genuinely nonlocal
(see Fig. 1)

$\begin{array}[]{l}|\Psi_{i}\rangle:=|1\rangle_{A}\otimes(Y_{\mathcal{B}}^{(U)}|i\rangle_{B})\otimes|1\rangle_{C},\\\
|\Psi_{y-1+j}\rangle:=(X_{\mathcal{A}}^{(U)}|j\rangle_{A})\otimes|y\rangle_{B}\otimes|1\rangle_{C},\\\
|\Psi_{x+y-3+k}\rangle:=|x\rangle_{A}\otimes(Y_{\mathcal{B}}^{(D)}|k\rangle_{B})\otimes|1\rangle_{C},\\\
|\Psi_{x+2y-4+l}\rangle:=(X_{\mathcal{A}}^{(D)}|l\rangle_{A})\otimes|1\rangle_{B}\otimes|1\rangle_{C},\\\
\end{array}$

where $1\leq i\leq y-1,1\leq j\leq x-1,2\leq k\leq y,2\leq l\leq x$ and (see
Fig. 3)

$\begin{array}[]{l}|\Phi_{i}\rangle:=|2\rangle_{A}\otimes|1^{\prime}\rangle_{B}\otimes(Z_{\mathcal{C}^{\prime}}^{(U)}|i^{\prime}\rangle_{C}),\\\
|\Phi_{z-1+j}\rangle:=|2\rangle_{A}\otimes(Y_{\mathcal{B}^{\prime}}^{(U)})|j^{\prime}\rangle_{B}\otimes|z^{\prime}\rangle_{C},\\\
|\Phi_{y+z-3+k}\rangle:=|2\rangle_{A}\otimes|y^{\prime}\rangle_{B}\otimes(Z_{\mathcal{C}^{\prime}}^{(D)}|k^{\prime}\rangle_{C}),\\\
|\Phi_{y+2z-4+l}\rangle:=|2\rangle_{A}\otimes(Y_{\mathcal{B}^{\prime}}^{(D)}|l^{\prime}\rangle_{B})\otimes|1^{\prime}\rangle_{C}.\\\
\end{array}$

where $1\leq i\leq z-1,1\leq j\leq y-1,2\leq k\leq z,2\leq l\leq y$. Here the
ordered bases $\mathcal{B}^{\prime}$ and $\mathcal{C}^{\prime}$ are defined as
follows

$\begin{array}[]{l}\mathcal{B}^{\prime}:=(|1^{\prime}\rangle_{B},|2^{\prime}\rangle_{B},\cdots,|(y-1)^{\prime}\rangle_{B},|y^{\prime}\rangle_{B}),\\\
\mathcal{C}^{\prime}:=(|1^{\prime}\rangle_{C},|2^{\prime}\rangle_{C},\cdots,|z^{\prime}\rangle_{C})\end{array}$

where
$|1^{\prime}\rangle_{B}=|2\rangle_{B}$,$|2^{\prime}\rangle_{B}=|1\rangle_{B}$,
$|y-1^{\prime}\rangle_{B}=|y\rangle_{B}$,
$|y^{\prime}\rangle_{B}=|y-1\rangle_{B}$,
$|j^{\prime}\rangle_{B}=|j\rangle_{B}$ for $3\leq j\leq y-1$ and
$|1^{\prime}\rangle_{C}=|2\rangle_{C}$,$|2^{\prime}\rangle_{C}=|1\rangle_{C}$,
$|k^{\prime}\rangle_{C}=|k\rangle_{C}$ for $3\leq j\leq z$.

Figure 4: This is a schematic diagram for the states in Theorem 3.

_Proof._ We notice that
$\text{span}_{\mathbb{C}}\\{|\Psi_{1}\rangle,|\Psi_{2}\rangle\cdots,|\Psi_{2x+2y-4}\rangle\\}$
is equal to the linear space spanned by

$\mathcal{S}_{\Psi}:=\\{|i\rangle_{A}|j\rangle_{B}|1\rangle_{C}\big{|}\ \
i\in\\{1,x\\}\text{ or }j\in\\{1,y\\}\\}$

while
$\text{span}_{\mathbb{C}}\\{|\Phi_{1}\rangle,|\Phi_{2}\rangle,\cdots,|\Phi_{2y+2z-4}\rangle\\}$
is equal to the linear space spanned by

$\begin{array}[]{l}\mathcal{S}_{\Phi}:=\\{|2\rangle_{A}|j^{\prime}\rangle_{B}|k^{\prime}\rangle_{C}\big{|}\
\ j\in\\{1,y\\}\text{ or }k\in\\{1,z\\}\\}\\\
=\\{|2\rangle_{A}|j\rangle_{B}|k\rangle_{C}\big{|}\ \ j\in\\{2,y-1\\}\text{ or
}k\in\\{2,z\\}\\}.\end{array}$

As $\mathcal{S}_{\Psi}\cap\mathcal{S}_{\Phi}=\emptyset$ and
$\mathcal{S}_{\Psi},\mathcal{S}_{\Phi}\subseteq\\{|i\rangle_{A}|j\rangle_{B}|k\rangle_{C}\big{|}\
1\leq i\leq x,1\leq j\leq y,1\leq k\leq z\\}$ which is a orthonormal basis of
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$, we have
$\langle\Psi_{u}|\Phi_{v}\rangle=0$ for integers $u,v$ with $1\leq u\leq
2x+2y-4,1\leq v\leq 2y+2z-4.$ Therefore, the states in
$\\{|\Psi_{u}\rangle\\}_{u=1}^{2x+2y-4}\cup\\{|\Phi_{v}\rangle\\}_{v=1}^{2y+2z-4}$
are pairwise orthogonal ( Fig. 4 is more intuitive for the orthogonality).

To prove that the set of states we construct are genuinely nonlocal. There are
only three ways to separate $ABC$ into two sets. That is, $A|BC,B|CA,C|AB$. By
Theorem 1 and Observation 1, the set $\\{|\Psi_{u}\rangle\\}_{u=1}^{2x+2y-4}$
is locally indistinguishable as the partitions $A|BC$ and $B|CA$. And the set
$\\{|\Phi_{v}\rangle\\}_{v=1}^{2y+2z-4}$ is locally indistinguishable as the
partitions $B|CA$ and $C|AB$. Hence the given set is genuinely nonlocal.

In the following, we begin to strengthen the results in Zhang17  where they
constructed locally indistinguishable multipartite product states from known
bipartite ones. The following two propositions enhance their results to
genuine nonlocality settings.

Figure 5: States structure corresponding to the Proposition 1.

###### Proposition 1

Let $L\geq 4$ be an integer and $d_{i}\geq 3$ for all $1\leq i\leq L$. Let
$S_{i}=\\{|\psi_{j}^{(i)}\rangle|\phi_{j}^{(i)}\rangle\\}_{j=1}^{n_{i}}\subseteq\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{i+1}}$
be sets of product states that are locally indistinguishable for
$i=1,2,\cdots,L-1$. Then the union of the following sets (See Fig. 5)

$\begin{array}[]{lllll}\mathcal{S}_{1}=\\{|\psi_{j}^{(1)}\rangle|\phi_{j}^{(1)}\rangle|1\rangle|1\rangle|1\rangle\cdots|2\rangle\\}_{j=1}^{n_{1}},\\\\[5.69054pt]
\mathcal{S}_{2}=\\{|\psi_{j}^{(2)}\rangle|2\rangle|\phi_{j}^{(2)}\rangle|1\rangle|1\rangle\cdots|1\rangle\\}_{j=1}^{n_{2}},\\\\[5.69054pt]
\mathcal{S}_{3}=\\{|\psi_{j}^{(3)}\rangle|1\rangle|2\rangle|\phi_{j}^{(3)}\rangle|1\rangle\cdots|1\rangle\\}_{j=1}^{n_{3}},\\\\[5.69054pt]
\ \ \ \ \ \ \ \ \ \ \ \vdots\\\\[5.69054pt]
\mathcal{S}_{L-1}=\\{|\psi_{j}^{(L-1)}\rangle|1\rangle|1\rangle\cdots|1\rangle|2\rangle|\phi_{j}^{(L-1)}\rangle\\}_{j=1}^{n_{L-1}}\end{array}$
(1)

is also a genuinely nonlocal set of product states in
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_{L}}$.

_Proof._ To distinguish the states of $\mathcal{S}_{1}$, by Observation 1, the
first two parties must come together and perform a global measurement.
Similarly, to distinguish the states of $\mathcal{S}_{i}$, the $1$-th and
($i+1$)-th parties must come together and perform a global measurement.

Therefore, all the parties must come together to distinguish all the states in
Eq. (1). Hence such a set of states is genuinely nonlocal.

Figure 6: States structure corresponding to the Proposition 2.

###### Proposition 2

Let $L\geq 5$ be an integer and $d_{i}\geq 3$ for all $1\leq i\leq L$. Let
$S_{i}:=\\{|\psi_{j}^{(i)}\rangle|\phi_{j}^{(i)}\rangle\\}_{j=1}^{n_{i}}\subseteq\mathbb{C}^{d_{i}}\otimes\mathbb{C}^{d_{i+1}}$
be sets of product states that are locally indistinguishable for
$i=1,2,\cdots,L-1$. Then the union of the following sets (See Fig. 6)

$\begin{array}[]{lllll}\mathcal{S}_{1}=\\{|\psi_{j}^{(1)}\rangle|\phi_{j}^{(1)}\rangle|2\rangle|1\rangle|1\rangle|1\rangle\cdots|1\rangle\\}_{j=1}^{n_{1}},\\\\[5.69054pt]
\mathcal{S}_{2}=\\{|1\rangle|\psi_{j}^{(2)}\rangle|\phi_{j}^{(2)}\rangle|2\rangle|1\rangle|1\rangle\cdots|1\rangle\\}_{j=1}^{n_{2}},\\\\[5.69054pt]
\mathcal{S}_{3}=\\{|1\rangle|1\rangle|\psi_{j}^{(3)}\rangle|\phi_{j}^{(3)}\rangle|2\rangle|1\rangle\cdots|1\rangle\\}_{j=1}^{n_{3}},\\\\[5.69054pt]
\ \ \ \ \ \ \ \ \ \ \ \vdots\\\\[5.69054pt]
\mathcal{S}_{L-1}=\\{|2\rangle|1\rangle|1\rangle|1\rangle\cdots|1\rangle|\psi_{j}^{(L-1)}\rangle|\phi_{j}^{(L-1)}\rangle\\}_{j=1}^{n_{L-1}}\end{array}$
(2)

is also a genuinely nonlocal set of product states in
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_{L}}$.

To construct multipartite genuinely nonlocal sets, instead of using bipartite
nonlocal product states, we can also start with some known genuinely nonlocal
sets of product states in tripartite systems.

###### Proposition 3

Let $L\geq 3$ be an integer. Let
$\\{|\psi_{j}\rangle|\phi_{j}\rangle|\chi_{j}\rangle\\}_{j=1}^{n}\subseteq\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$
be a set of product states that is genuinely nonlocal. Then the union of the
following sets (See Fig. 7)

$\begin{array}[]{lllll}\mathcal{S}_{1}=\\{|\psi_{j}\rangle|\phi_{j}\rangle|\chi_{j}\rangle|1\rangle|1\rangle|1\rangle\cdots|1\rangle|2\rangle|2\rangle\\}_{j=1}^{n},\\\\[5.69054pt]
\mathcal{S}_{2}=\\{|\psi_{j}\rangle|2\rangle|2\rangle|\phi_{j}\rangle|\chi_{j}\rangle|1\rangle\cdots|1\rangle|1\rangle|1\rangle\\}_{j=1}^{n},\\\\[5.69054pt]
\mathcal{S}_{3}=\\{|\psi_{j}\rangle|1\rangle|1\rangle|2\rangle|2\rangle|\phi_{j}\rangle|\chi_{j}\rangle|1\rangle\cdots|1\rangle\\}_{j=1}^{n},\\\\[5.69054pt]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\\\\[5.69054pt]
\mathcal{S}_{L}=\\{|\psi_{j}\rangle|1\rangle|1\rangle|1\rangle|1\rangle\cdots|2\rangle|2\rangle|\phi_{j}\rangle|\chi_{j}\rangle\\}_{j=1}^{n},\end{array}$

is also a genuinely nonlocal set of product states in
$\bigotimes_{i=1}^{2L+1}\mathcal{H}_{A_{i}}$ where
$\mathcal{H}_{A_{i}}=\mathbb{C}^{3}$.

Figure 7: States structure corresponding to the Proposition 3.

In fact, the above constructions can be extended to much more general
settings. Let $L\geq 3$ be an integer and $\mathcal{P}:=\\{1,2,3,\cdots,L\\}$.
Let $\mathcal{H}:=\otimes_{j\in\mathcal{P}}\mathcal{H}_{j}$ be an $L$-parties
quantum system. Suppose there are $s$ proper subsets of $\mathcal{P}$:
$\mathcal{P}_{1},\mathcal{P}_{2},\cdots,\mathcal{P}_{s}$ and we denote
$\overline{\mathcal{P}}_{i}:=\mathcal{P}\setminus\mathcal{P}_{i}$ for each
$i$. We make the following assumptions:

  1. (a)

${S}_{i}=\\{|\Psi_{j}\rangle_{\mathcal{P}_{i}}\\}_{j=1}^{n_{i}}$ is a
genuinely nonlocal product set in
$\mathcal{H}_{\mathcal{P}_{i}}:=\otimes_{j\in\mathcal{P}_{i}}\mathcal{H}_{j}$
for each $i\in\\{1,2,\cdots,s\\}$.

  2. (b)

There is a fully product state
$|\Phi_{i}\rangle_{\overline{\mathcal{P}}_{i}}\in\otimes_{j\in\overline{\mathcal{P}}_{i}}\mathcal{H}_{j}$
for each $i$ such that the states in the union of the sets
$\mathcal{S}_{i}=\\{|\Psi_{j}\rangle_{\mathcal{P}_{i}}|\Phi_{i}\rangle_{\overline{\mathcal{P}}_{i}}\\}_{j=1}^{n_{i}}$
($1\leq i\leq s$) are mutually orthogonal to each other.

We use the notation
$\mathfrak{P}:=(\mathcal{P},\\{\mathcal{P}_{1},\mathcal{P}_{2},\cdots,\mathcal{P}_{s}\\})$.
For each $\mathfrak{P}$, We attach it a graph
$G_{\mathfrak{P}}=(V_{\mathfrak{P}},E_{\mathfrak{P}})$ defined as follows: its
vertex set is $V_{\mathfrak{P}}=\mathcal{P}$ and its edge set is

$E_{\mathfrak{P}}=\bigcup_{i=1}^{s}\\{(u_{i},v_{i})|u_{i},v_{i}\in\mathcal{P}_{i}\text{
and }u_{i}\neq v_{i}\\}.$

###### Theorem 4

Under the notation and assumptions of the two paragraphs previous and suppose
that $G_{\mathfrak{P}}$ is connected, then the set
$\mathcal{S}:=\cup_{i=1}^{s}\mathcal{S}_{i}$ is a genuinely nonlocal set of
product states in $\otimes_{j\in\mathcal{P}}\mathcal{H}_{j}$.

_Proof._ Suppose not, there exist a nontrivial bipartition of $\mathcal{P}$,
say $U\ |\ V$ (both $U$ and $V$ are nonempty subset of $\mathcal{P}$), such
that the set $\mathcal{S}$ is locally distinguishable when considering as a
set of bipartite states in
$(\otimes_{j\in{U}}\mathcal{H}_{j})\bigotimes(\otimes_{j\in
V}\mathcal{H}_{j}).$ As the connectivity of $G_{\mathfrak{P}}$, there must
exist some edge $(u,v)\in E_{\mathfrak{P}}$ which connects the two sets $U$
and $V$, i.e. $u\in U$ and $v\in V$. By the definition of $E_{\mathfrak{P}}$,
there exist some $i$ such that $u,v\in\mathcal{P}_{i}$. However, the set
$S_{i}$ is genuinely nonlocal in $\mathcal{H}_{\mathcal{P}_{i}}$ by the
assumption (a) above. So it is locally indistinguishable for the partition

$(U\cap\mathcal{P}_{i})\ |\ (V\cap\mathcal{P}_{i})$

of $\mathcal{P}_{i}$ as both $U\cap\mathcal{P}_{i}$ and $V\cap\mathcal{P}_{i}$
are nonempty. By Observation 1, the set $\mathcal{S}_{i}$ is locally
indistinguishable in the bipartite system
$(\otimes_{j\in{U}}\mathcal{H}_{j})\bigotimes(\otimes_{j\in
V}\mathcal{H}_{j}).$ However, $\mathcal{S}_{i}\subseteq\mathcal{S}$ implies
that $\mathcal{S}$ must be locally indistinguishable as bipartite states
$(\otimes_{j\in{U}}\mathcal{H}_{j})\bigotimes(\otimes_{j\in
V}\mathcal{H}_{j}).$ Hence we obtain a contradiction. So the set $\mathcal{S}$
must be genuinely nonlocal.

##  IV Conclusion and Discussion

We study a strong form of locally indistinguishable set of fully product
states called genuinely nonlocal set. We generalize the results of locally
indistinguishable product states in bipartite system in Ref. Xu20a  but
provide a much more elegant proof. Based on a simple observation, we extend
the results of Zhang _et al._ in Ref. Zhang17  to the cases of genuinely
nonlocal sets. Moreover, we extend these results to a much more general
setting by relating the construction of genuinely nonlocal sets with the
connectivity of some graphs. As a consequence, we can show that there always
exists some genuinely nonlocal set of fully product states in
$\otimes_{i=1}^{L}\mathbb{C}^{d_{i}}$ provided $d_{i}\geq 3$ for all $i$. One
should note that the genuinely nonlocal set we constructed here maybe locally
reducible under the concept introduced in Ref. Halder19 . Therefore, it is
interesting to find some method to characterize the locally irreducible
settings.

Acknowledgments This work is supported by National Natural Science Foundation
of China (11771419, 11875160, 11901084, 12005092, and U1801661), the China
Postdoctoral Science Foundation (2020M681996), the Natural Science Foundation
of Guang-dong Province (2017B030308003), the Key R$\&$D Program of Guangdong
province (2018B030326001), the Guang-dong Innovative and Entrepreneurial
Research TeamProgram (2016ZT06D348), the Science, Technology and Innovation
Commission of Shenzhen Municipality (JCYJ20170412152620376 and
JCYJ20170817105046702 and KYTDPT20181011104202253), the Economy, Trade and
Information Commission of Shenzhen Municipality (201901161512), the Research
startup funds of DGUT (GC300501-103), the Fundamental Research Funds for the
Central Universities, and Anhui Initiative in Quantum Information Technologies
under Grant No. AHY150200.

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