# Necessary and sufficient conditions for local discrimination of generalized
Bell states: finding out all locally indistinguishable sets of generalized
Bell states

Jiang-Tao Yuan12<EMAIL_ADDRESS>Ying-Hui Yang2<EMAIL_ADDRESS>Cai-
Hong Wang12<EMAIL_ADDRESS>1School of Sciences, Zhejiang University of
Science and Technology, Hangzhou, 310023, China
2School of Mathematics and Information Science, Henan Polytechnic University,
Jiaozuo, 454000, China

###### Abstract

In general, for a bipartite quantum system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and an integer $k$ such that $4\leq
k\leq d$, there are few necessary and sufficient conditions for local
discrimination of sets of $k$ generalized Bell states (GBSs) and it is
difficult to locally distinguish $k$-GBS sets. In this paper, we consider the
local discrimination of GBS sets and the purpose is to completely solve the
problem of local discrimination of GBS sets in some bipartite quantum systems,
specifically, we show some necessary and sufficient conditions for local
discrimination of GBS sets by which the local discrimination of GBS sets can
be quickly determined. Firstly some sufficient conditions are given, these
sufficient conditions are practical and effective. Fan,s and Wang et al.,s
results (Phys Rev Lett 92:177905, 2004: Phys Rev A 99:022307, 2019) can be
deduced as special cases of these conditions. Secondly in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, a necessary and sufficient condition
for local discrimination of GBS sets is provided, all locally
indistinguishable 4-GBS sets are found, and then we can quickly determine the
local discriminability of an arbitrary GBS set. In
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, a concise necessary and sufficient
condition for one-way local discrimination of GBS sets is obtained, which
gives an affirmative answer to the case $d=5$ of the problem proposed by Wang
et al. (Phys Rev A 99:022307, 2019).

###### pacs:

03.65.Ud, 03.67.Hk

##  I Introduction

In quantum mechanics, any set of orthogonal quantum states can be
discriminated globally. In general, for a bipartite composite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, local operations and classical
communication (LOCC) is not sufficient to distinguish sets of orthogonal
quantum states benn1999pra ; walg2000prl ; walg2002prl ; gho2001prl , that is,
it is a difficult task to locally distinguish sets of orthogonal states. Any
two orthogonal states can be perfectly (or deterministically) distinguished by
LOCC walg2000prl , a complete set of orthogonal maximally entangled states
(MESs) is locally indistinguishable (deterministically or probabilistically),
and $d+1$ or more MESs in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ are locally
indistinguishable horo2003prl ; fan2004prl ; fan2007pra ; gho2004pra ;
nath2005jmp , and any three generalized Bell States (GBSs) in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\ (d\geq 3)$ are always locally
distinguishable wang2017qip . The nonlocal nature of quantum information is
revealed when a set of orthogonal states of a composite quantum system cannot
be perfectly distinguished by LOCC. The study of local quantum state
discrimination is widely applied to many fields such as data hiding, quantum
secret sharing, and quantum private query divi2002trans-inform -wei2018trans-
comp . It is also helpful to explore the power and limitations of LOCC which
is of inherent interest as a tool to understand entanglement and nonlocality.

It is natural to ask whether or not a set of $k(4\leq k\leq d)$ orthogonal
MESs in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ can be (perfectly) distinguished
by LOCC band2011njp . Many advances have been made in this field. The
researchers have constructed many examples of locally indistinguishable GBS
sets band2011njp ; yu2012prl ; zhang2015pra ; wang2016qip ; yang2018pra ;
yuan2019qip , for example, Bandyopadhyay et al. band2011njp  showed one-way
LOCC indistinguishable 4-GBS sets in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ and
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$. On the other hand, some sufficient
conditions for local discrimination of GBSs are obtained by using admissible
solution set and nonadmissible solution set wang2019pra ; yang2021qip .

Although many advances have been made, they are mainly concrete examples of
local indistinguishability and sufficient conditions for local
distinguishability. There are few necessary and sufficient conditions for
local discrimination of orthogonal state sets, which makes it difficult to
locally distinguish a general GBS set. Even for a general GBS set in a low
dimensional bipartite system, it is difficult to judge its local
distinguishability.

Based on these advances, in this paper, the local discrimination of GBS sets
is considered. Firstly, some practical and effective sufficient conditions for
GBS sets in an arbitrary dimensional system are given. These sufficient
conditions are easy to use and widely applicable, they cover the previous
conclusions including Fan,s and Wang et al.,s results fan2004prl ; wang2019pra
. And then we show that these sufficient conditions are also necessary for
local discrimination of all 4-GBS sets in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, and we give a table containing all the
locally indistinguishable (standard) 4-GBS sets, which can be used to judge
the local distinguishability of a 4-GBS set quickly. In
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, one of the sufficient conditions is
proved to be necessary, thus a necessary and sufficient condition for one-way
local discrimination of GBS sets is provided. The necessary and sufficient
condition is concise and computable, and it gives an affirmative answer to the
case $d=5$ of the problem proposed by Wang et al. wang2019pra ; yang2018qip .

The rest of this paper is organized as follows. In Section II, we recall some
relevant notions and results. In Section III, we present three suffient
conditions for local discrimination of GBS sets in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. In Section IV, for 4-GBS sets in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, a necessary and sufficient condition
for local state discrimination is provided, and all locally indistinguishable
(standard) 4-GBS sets are found. Further, in Section V, for 4-GBS and 5-GBS
sets in $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, we show a simple and direct
necessary and sufficient condition for one-way local discrimination, which
gives an affirmative answer to case $d=5$ of the problem proposed by Wang et
al. Lastly, in Section VI, we draw the conclusions.

##  II Preliminaries

Consider a $d$-dimensional Hilbert space, $\\{|j\rangle\\}_{j=0}^{d-1}$ is the
computational basis and $\mathbb{Z}_{d}=\\{0,1,\ldots,d-1\\}$. Let
$U_{m,n}=X^{m}Z^{n},m,n\in\mathbb{Z}_{d}$ be generalized Pauli matrices (GPMs)
constituting a basis of unitary operators where $X|j\rangle=|j+1$ mod
$d\rangle$, $Z|j\rangle=\omega^{j}|j\rangle$ and $\omega=e^{2\pi i/d}$, which
are generalizations of Pauli matrices. Denote the set of all GPMs on a
$d$-dimensional Hilbert space by $P(d)$.

In a quantum system $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, the canonical
maximally entangled state $|\Phi\rangle$ is
$|\Phi_{00}\rangle=(1/\sqrt{d})\sum_{j=0}^{d-1}|jj\rangle$. It is known that
$(I\otimes U)|\Phi\rangle=(U^{T}\otimes I)|\Phi\rangle$, where $T$ means
matrix transposition. Any MES can be written as $|\Psi\rangle=(I\otimes
U)|\Phi\rangle$ where $U$ is unitary. If $U=X^{m}Z^{n}$, the states

$\displaystyle|\Phi_{m,n}\rangle=(I\otimes U_{m,n})|\Phi\rangle$

are called generalized Bell states (GBSs). Note that there is a one-to-one
correspondence between the MESs and unitaries, and the corresponding unitaries
of GBSs are GPMs. It is convenient to denote a GBS set $\\{(I\otimes
X^{m_{i}}Z^{n_{i}})|\Phi\rangle\\}$ by $\\{X^{m_{i}}Z^{n_{i}}\\}$ or
$\\{(m_{i},n_{i})\\}$. For a given GBS set
$S=\\{X^{m_{j}}Z^{n_{j}}|m_{j},n_{j}\in\mathbb{Z}_{d}\\}$, the difference set
$\Delta S$ of the GBS set $S$ means

$\displaystyle\Delta
S=\\{(m_{j}-m_{k},n_{j}-n_{k})|(m_{j},n_{j}),(m_{k},n_{k})\in S,j\neq k\\}.$

It plays an important role in local state discrimination tian2015pra ;
yang2018pra . This section consists of three parts: Wely commutation relation
and congruence equation, Clifford operator, and LU-equivalence.

###  II.1 Weyl commutation relation and congruence equation

Two unitaries $A$ and $B$ are called Weyl commutative if $AB=zBA$ where $z$ is
a complex number petz2008book .

######  Lemma 1.

yuan2020jpa ; wang-yuan2021jmp  For two unitary matrices $A$ and $B$, if they
are not commutative and satisfy Weyl commutation relation, then each
eigenvector $u$ of $A$ satisfies $\langle u|B|u\rangle=0$. Especially, for an
arbitrary GBS set $S$, if there is a GPM $T$ which is not commutative to every
GPM $U$ in $\Delta S$, then each eigenvector $|v\rangle$ of $T$ satisfies
$\langle v|U|v\rangle=0$ and the set $S$ is one-way LOCC distinguishable.

In order to locally distinguish a GBS set, according to Lemma 1, we need to
find the GPM $T$ in Lemma 1. For two GPMs $(m,n)$ and $(s,t)$, they are
commutative if and only if $ns-mt=0\mod d$. So we prepare the following result
about congruence equation to accomplish this task.

######  Lemma 2.

Let $m,n\in\mathbb{Z}_{d}$, $\gcd(m,n)$ be the greatest common divisor of $m$
and $n$, and

$\displaystyle nx-my=0\mod d$ (1)

be a congruence equation. Then there is always a solution
$(\frac{m}{\gcd(m,n)},\frac{n}{\gcd(m,n)})$ to the congruence equation and the
following assertions holds.

  1. (1)

If $m=0,n\neq 0$, then solution set of the equation is
$\\{(\frac{d}{\gcd(n,d)}t,k)|k\in\mathbb{Z}_{d},t\in\mathbb{Z}_{\gcd(n,d)}\\}$.

  2. (2)

If $m\neq 0,n=0$, then solution set of the equation is
$\\{(k,\frac{d}{\gcd(m,d)}t)|k\in\mathbb{Z}_{d},t\in\mathbb{Z}_{\gcd(m,d)}\\}$.

  3. (3)

If $m\neq 0,n\neq 0$, then solution set of the equation is
$\\{(\frac{m}{\gcd(m,n)}k+\frac{d}{\gcd(m,n,d)}t,\frac{n}{\gcd(m,n)}k+\frac{d}{\gcd(m,n,d)}t^{\prime})\\}$
where
$k=0,\cdots,\frac{d}{\gcd(m,n,d)}-1;t,t^{\prime}\in\mathbb{Z}_{\gcd(m,n,d)}$.

For a given GPM $(m,n)$ in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, there are
$d\cdot\gcd(m,n,d)$ solutions to the congruence equation (1).

###  II.2 Clifford operator

A Clifford operator is defined as a unitary operator which maps the Pauli
group to itself under conjugation hos2005pra ; far2014jpa . The classical (or
symplectic) representation $W$ of a single-qudit Clifford operator is a unique
two by two symplectic matrix (up to a global phase) over $\mathbb{Z}_{d}$,
that is,

$\displaystyle W=\left[\begin{array}[]{llll}a_{1}&b_{1}\\\
a_{2}&b_{2}\end{array}\right]$

where the entries are over $\mathbb{Z}_{d}$. It is known that a $2\times 2$
matrix $W$ is symplectic if and only if $\det(W)=a_{1}b_{2}-a_{2}b_{1}\equiv
1$ (mod $d$). For a GBS $(m,n)$, the Clifford operator $W$ maps $(m,n)$ to
$(a_{1}m+b_{1}n,a_{2}m+b_{2}n)$, or
$X^{m}Z^{n}\mathop{\sim}\limits^{W}X^{a_{1}m+b_{1}n}Z^{a_{2}m+b_{2}n}$. Two
basic Clifford operators are quantum Fourier transform (QFT) gate $R$ and
phase-shift gate $P$:

$\displaystyle R=\left[\begin{array}[]{cccc}0&-1\\\
1&0\end{array}\right],P=\left[\begin{array}[]{llll}1&0\\\
1&1\end{array}\right]$

The phase-shift and QFT gates are a necessary and sufficient set of gates to
generate (up to global phase) the entire single-qudit Clifford group in any
finite dimension far2014jpa .

###  II.3 LU-equivalence

When discussing a large number of GBS sets, we need the concept of local
unitary (LU) equivalence. Let’s recall the LU-equivalence of two MES sets.

######  Definition 1 (wu-tian2018pra ).

Let $\\{|\phi_{1}\rangle,|\phi_{2}\rangle,\cdots,|\phi_{n}\rangle\\}$ and
$\\{|\psi_{1}\rangle,|\psi_{2}\rangle,\cdots,|\psi_{n}\rangle\\}$ be two sets
of MESs. Their corresponding unitary matrix sets are
$M=\\{M_{1},M_{2},\cdots,M_{n}\\}$ and $N=\\{N_{1},N_{2},\cdots,N_{n}\\}$. If
there exist two unitary operators $U_{A}$, $U_{B}$ and a permutation $\sigma$
over $\\{1,2,\cdots,n\\}$ such that $|\psi_{i}\rangle\approx(U_{A}\otimes
U_{B})|\phi_{\sigma(i)}\rangle$, where $\approx$ denotes “equal up to some
global phase”, then these two MES sets are called LU-equivalent. Meanwhile the
corresponding unitary matrix sets are called $U$-equivalent, that is

$\displaystyle LMR\approx N,$

where $L=U_{B}$ and $R=U_{A}^{T}$, denoted by $M\sim N$. Especially, when
$R=L^{\dagger}$, the two sets $M$ and $N$ are called unitary conjugate
equivalent (UC equivalent), denoted by $M\mathop{\sim}\limits^{L}N$.

In general we can study LU-equivalence of two MES sets by studying the
$U$-equivalence of their unitary matrix sets. For convenience, we call a GBS
set containing the standard MES a standard GBS set, and a GPM set containing
the identity matrix a standard GPM set. Let $M=\\{M_{1},M_{2},\cdots,M_{n}\\}$
and $N=\\{N_{1},N_{2},\cdots,N_{n}\\}$ be two GPM sets. If $M$ is
$U$-equivalent to $N$, then there exist two unitaries $L$, $R$ and a
permutation $\sigma$ over $\\{1,2,\cdots,n\\}$ such that $LM_{i}R\approx
N_{\sigma(i)}$. Thus for a fixed $i$,

$\displaystyle N_{\sigma(j)}$ $\displaystyle\thickapprox
LM_{j}R=(LM_{i})(M_{i}^{\dagger}M_{j})(LM_{i})^{\dagger}(LM_{i}R)$
$\displaystyle=(LM_{i})(M_{i}^{\dagger}M_{j})(LM_{i})^{\dagger}N_{\sigma(i)}(j=1,2,\cdots,n).$

Furthermore, if some $M_{i}=I\in M$, then we have

$\displaystyle N_{\sigma(j)}\approx(LM_{j}L^{\dagger})N_{\sigma(i)},\hbox{or}\
L^{\dagger}(N_{\sigma(j)}N_{\sigma(i)}^{{\dagger}})L\approx M_{j}.$ (4)

The formula (4) tells us that, for any standard GPM set $M$, if $M$ is
U-equivalent to $N$ then $M$ consists of elements of the shape of
$L^{\dagger}(N_{j}N_{i}^{{\dagger}})L$. In particular, if $L$ is a Clifford
transformation, then the GPM set
$M=\\{L^{\dagger}(N_{j}N_{i}^{{\dagger}})L\\}$ is U-equivalent to $N$. For a
UC transformation that maps one GPM set $A$ to another GPM set $B$, if the set
$A$ or $B$ containing nondegenerate $X^{s}$ and $Z^{t}$, then the
transformation must be a Clifford operator. These facts lead to the following
assertion.

######  Lemma 3.

Let $N=\\{N_{j}\\}$ be a GPM set, if $L$ is a Clifford transformation and
$\sigma$ is a permutation, then the standard GPM set
$M=\\{L^{\dagger}(N_{\sigma(j)}N_{\sigma(i)}^{{\dagger}})L\\}$ is U-equivalent
to $N$. Especially, if $N$ contains two nondegenerate GPMs $X^{s}$ and
$Z^{t}$, then any standard GPM set $M$ that is U-equivalent to $N$ has the
form $M=\\{L^{\dagger}(N_{\sigma(j)}N_{\sigma(i)}^{{\dagger}})L\\}$, where $L$
is a Clifford operator and $\sigma$ is a permutation.

For a given GPM set $N$, Lemma 3 can be used to construct lots of GPM sets
which are $U$-equivalent to $N$. Each $U$-equivalent set of $N$ in Lemma 3
only depends on Clifford operators and $N$ itself, and then it gives an
effective way to find lots of standard GPM sets that is U-equivalent to $N$.

##  III Sufficient conditions for arbitrary dimensional systems

In this section, we will show three sufficient conditions for local
discrimination of GBS sets in arbitrary dimensional systems and some examples
which can not be determined by previous known sufficient conditions.

In order to find the GPMs in Lemma 1, for a GBS set $\mathcal{S}$ and a GPM
$(m,n)$ in the difference set $\Delta\mathcal{S}$, denote the solution set of
the corresponding congruence equation (1) by $S(m,n)$, then every element in
the set

$\displaystyle\mathcal{D}(\mathcal{S})\triangleq
P(d)\setminus\bigcup_{(m,n)\in\Delta\mathcal{S}}S(m,n)$

satisfies the condition in Lemma 1 and can be used to locally distinguish the
set $\mathcal{S}$. Therefore, the set $\mathcal{D}(\mathcal{S})$ can be called
a $\it{discriminant\ set}$ of $\mathcal{S}$.

######  Theorem 1.

Let $\mathcal{S}=\\{(m_{i},n_{i})|4\leq i\leq l\leq d\\}$ be a GBS set in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, then the set $\mathcal{S}$ is local
distinguishable when any of the following conditions is true.

  1. (1)

The discriminant set $\mathcal{D}(\mathcal{S})$ is not empty.

  2. (2)

The set $\Delta\mathcal{S}$ is commutative.

  3. (3)

The dimension $d$ is a composite number, and for each
$(m,n)\in\Delta\mathcal{S}$, $m$ or $n$ is invertible in $\mathbb{Z}_{d}$.

Since the commutativity of the set $\mathcal{S}$ ensures the commutativity of
its difference set $\Delta\mathcal{S}$, according to Theorem 1 (2),
$\mathcal{S}$ is local disdinguishable when $\mathcal{S}$ is commutative. The
condition (3) in Theorem 1 only applies when $d$ is composite.

Proof. From the previous discussion, the conclusion (1) is obvious. Let’s
consider (2), since the set $\Delta\mathcal{S}$ is commutative, there exists a
unitary matrix $U$ which maps each GPM $A$ in $\Delta\mathcal{S}$ to a
diagonal matrix $U^{\dagger}AU$. Let
$|\beta\rangle\triangleq\frac{1}{\sqrt{d}}|(1,\cdots,1)\rangle$ be a
$d$-dimensional unit vector, then the vector $|\alpha\rangle=U|\beta\rangle$
satisfies
$\langle\alpha|A|\alpha\rangle=\langle\beta|U^{\dagger}AU|\beta\rangle=\hbox{tr}{A}=0$,
thus the set $S$ is one-way LOCC distinguishable. Now consider the last
condition, let $d$ be a composite number and $d=st$ be a decomposition. For
$(m,n)\in\Delta\mathcal{S}$ and $m$ is invertible in $\mathbb{Z}_{d}$, the GPM
$Z^{t}$ is not commutative to $(m,n)$, then each eigenstate $|\alpha\rangle$
of $Z^{t}$ satisfies $\langle\alpha|X^{m}Z^{n}|\alpha\rangle=0$. Similarly, if
$n$ is invertible in $\mathbb{Z}_{d}$, then each eigenstate $|\beta\rangle$ of
$X^{s}$ satisfies $\langle\beta|X^{m}Z^{n}|\beta\rangle=0$. Since $X^{s}$ and
$Z^{t}$ are commutative, they have common eigenstates. Obviously, each common
eigenstate $|\gamma\rangle$ satisfies
$\langle\gamma|X^{m}Z^{n}|\gamma\rangle=0$. Therefore the set $S$ is one-way
LOCC distinguishable.

We give some examples to show that Theorem 1 has a wide range of applications.

For a given GBS set $\mathcal{L}=\\{(m_{i},n_{i})|1\leq i\leq l\leq d\\}$ in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and the congruence equation

$\displaystyle m_{i}y+n_{i}=m_{j}y+n_{j}\mod d$ (5)

where $i<j$ and $(m_{i},n_{i}),(m_{j},n_{j})\in\mathcal{L}$, let $S_{ij}$ be
the solution set of the equation (5), $S_{\mathcal{L}_{y}}$ be the the union
of all $S_{ij}$, and $S_{A}$ be $\\{0,\cdots,d-1,\infty\\}$. The two sets
$S_{A}$ and $S_{\mathcal{L}_{y}}$ are called admissible solution set and
nonadmissible solution set for $\mathcal{L}$ respectively wang2019pra ;
yang2021qip . If the set $S_{A}\backslash S_{\mathcal{L}_{y}}$ is not empty,
then it is easy to check that, for each element $y$ in $S_{A}\backslash
S_{\mathcal{L}_{y}}$, the GPM $(d-1,y)$ is also an element of the set
$\mathcal{D}(\mathcal{L})$, and then the set $\mathcal{L}$ is local
distinguishable. This shows that Theorem 1 (1) leads to Wang et al.,s result
which is an extension of Fan,s result (fan2004prl, , Theorem). But Wang et
al.,s result (wang2019pra, , Theorem 1) is not applicable to the following
three examples.

######  Example 1.

Consider the 5-GBS set $\mathcal{L}_{1}=\\{(0,0),(0,1),(1,0),(1,4),(5,5)\\}$
in $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$. By using the congruence equation
(5), it is easy to check that the set $S_{A}\backslash
S_{\mathcal{L}_{1_{y}}}$ is empty and Wang et al. s result is not applicable.
Now we apply Theorem 1 (1) to the set $\mathcal{L}_{1}$. By using congruence
equation (1) and simple calculation, the difference set
$\Delta\mathcal{L}_{1}=\\{(0,1),(1,0),(1,4),(5,5),(1,5),(1,3),(5,4),(0,4),(4,5),\\\
(4,1),(0,5),(5,0),(5,2),(1,1),(5,1),(5,3),(1,2),(0,2),\\\ (2,1),(2,5)\\}$ and
the GPM $(2,3)$ belongs to the discriminant set
$\mathcal{D}(\mathcal{L}_{1})$, and then the set $\mathcal{L}_{1}$ is local
distinguishable.

######  Example 2.

Consider the GBS set $\mathcal{L}_{2}=\\{(1,2),(1,0),(3,2),(3,0)\\}$ in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ and the set
$\mathcal{L}_{3}=\\{(2,3),(2,0),(5,3),(5,0)\\}$ in
$\mathbb{C}^{6}\otimes\mathbb{C}^{6}$. The difference sets are
$\Delta\mathcal{L}_{2}=\\{(0,2),(2,0),(2,2)\\}$ and
$\Delta\mathcal{L}_{3}=\\{(0,3),(3,0),(3,3)\\}$ respectively. By Lemma 2 and
the congruence equation (5), it is easy to check that all four sets
$S_{A}\backslash S_{\mathcal{L}_{2_{y}}}$, $S_{A}\backslash
S_{\mathcal{L}_{3_{y}}}$, $\mathcal{D}(\mathcal{L}_{2})$ and
$\mathcal{D}(\mathcal{L}_{3})$ are empty, and then Wang et al. s result and
Theorem 1 (1) are not applicable. Fortunately, both sets
$\Delta\mathcal{L}_{2}$ and $\Delta\mathcal{L}_{3}$ are commutative. By
Theorem 1 (2), $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ are local
distinguishable.

######  Example 3.

Consider the GBS set $\mathcal{L}_{4}=\\{(1,2),(1,3),(2,2),(0,1)\\}$ in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$. The difference set are
$\Delta\mathcal{L}_{4}=\\{(0,1),(1,0),(3,3),(1,3),(3,2),(2,3),(0,3),\\\
(3,0),(1,1),(3,1),(1,2),(2,1)\\}$. It is easy to check that both sets
$S_{A}\backslash S_{\mathcal{L}_{4_{y}}}$ and $\mathcal{D}(\mathcal{L}_{1})$
are empty, and then Wang et al. s result and Theorem 1 (1)-(2) are not
applicable. Obviously, Theorem 1 (3) is applicable and $\mathcal{L}_{4}$ is
local distinguishable.

It is interesting that all the three examples can not be discriminated by Wang
et al.,s result. Example 1 can only be determined by Theorem 1 (1), Example 2
can only be determined by Theorem 1 (2), and only Theorem 1 (3) applies to
Example 3.

##  IV Necessary and sufficient conditions for 4-GBS sets in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$

It is obvious that every GBS set is LU-equivalent to a standard GBS set. In
this section, we will show that if a 4-GBS set in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ is local distinguishable then one of the
three conditions in Theorem 1 is true, that is, the three sufficient
conditions in Theorem 1 are also necessary. Moreover, we will provide a list
of all local indistinguishable standard 4-GBS sets. The list and the necessary
and sufficient conditions allow us to quickly judge the local
distinguishability of any 4-GBS set in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$.

######  Theorem 2.

Let $\mathcal{S}=\\{(m_{i},n_{i})|1\leq i\leq 4\\}$ be a GBS set in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, then $\mathcal{S}$ is local
distinguishable if and only if one of the following conditions is true.

  1. (1)

The discriminant set $\mathcal{D}(\mathcal{S})$ is not empty.

  2. (2)

The set $\Delta\mathcal{S}$ is commutative.

  3. (3)

For each $(m,n)\in\Delta\mathcal{S}$, $m$ or $n$ is invertible in
$\mathbb{Z}_{4}$.

Proof. Let $\mathcal{S}=\\{(m_{i},n_{i})|1\leq i\leq 4\\}$ be a local
distinguishable 4-GBS set, we need to show that the set $\mathcal{S}$ meets
one of the three conditions. It is known that all 4-GBS sets can be classified
into ten locally inequivalent classes and the representative elements of these
equivalence classes are

$\displaystyle\mathcal{K}=\\{I,X^{2},Z^{2},X^{2}Z^{2}\\},$
$\displaystyle\mathcal{L}=\\{I,X,X^{2},X^{3}\\},$
$\displaystyle\Gamma^{1}_{20}=\\{I,X,Z,X^{2}\\},$
$\displaystyle\Gamma^{1}_{31}=\\{I,X,Z,X^{3}Z\\},$
$\displaystyle\Gamma^{1}_{33}=\\{I,X,Z,X^{3}Z^{3}\\},$
$\displaystyle\Gamma^{2}_{12}=\\{I,X,Z^{2},XZ^{2}\\},$
$\displaystyle\Gamma^{2}_{30}=\\{I,X,Z^{2},X^{3}\\},$
$\displaystyle\Gamma^{1}_{12}=\\{I,X,Z,XZ^{2}\\},$
$\displaystyle\Gamma^{2}_{20}=\\{I,X,Z^{2},X^{2}\\},$
$\displaystyle\Gamma^{2}_{32}=\\{I,X,Z^{2},X^{3}Z^{2}\\}.$

The first 7 sets are one-way LOCC distinguishable, while the last 3 sets are
one-way LOCC indistinguishable (also two-way LOCC indistinguishable). Hence
the local distinguishable set $\mathcal{S}$ is LU-equivalent to one of the
first 7 sets tian2016pra :
$\mathcal{K},\mathcal{L},\Gamma^{1}_{20},\Gamma^{1}_{31},\Gamma^{1}_{33},\Gamma^{2}_{12},\Gamma^{2}_{30}$.
By the formula (4) and Lemma 3, the three conditions in Theorem 2 are
invariant under LU-equivalence. Now we only need to consider the above seven
sets. The two sets $\mathcal{K}$ and $\mathcal{L}$ are commutative, then their
difference sets are also commutative and they meet the condition (2). Since
the difference set
$\Delta\Gamma^{1}_{20}=\\{(1,0),(0,1),(2,0),(3,1),(2,3),(3,0),(0,3),(1,3),(2,1)\\}$,
it is easy to check that $(1,1)\in\mathcal{D}(\Gamma^{1}_{20})$, and then
$\Gamma^{1}_{20}$ satisfies the condition (1). Similarly, it is easy to check
that $\\{(1,1),(1,2)\\}\subseteq\mathcal{D}(\Gamma^{1}_{31})$,
$\\{(1,1),(1,3)\\}\subseteq\mathcal{D}(\Gamma^{2}_{12})$, and
$\\{(1,1),(1,3)\\}\subseteq\mathcal{D}(\Gamma^{2}_{30})$, then the three sets
also meet the condition (1). About the set $\Gamma^{1}_{33}$, since the
difference set
$\Delta\Gamma^{1}_{33}=\\{(1,0),(0,1),(3,3),(1,3),(2,3),(3,2),(3,0),(0,3),(1,1),\\\
(3,1),(2,1),(1,2)\\}$, it is clear that it meets the condition (3).

Next we will use the LU-equivalence to give a list of all local
indistinguishable standard 4-GBS sets. We first give a protocol according to
Lemma 3 to determine the U-equivalence of two GPM sets $M$ and $N$.

Protocol. Let
$M=\\{M_{1},\cdots,M_{l}\\}=\\{(s_{1},t_{1}),\cdots,(s_{l},t_{l})\\}$ and
$N=\\{(m_{1},n_{1}),\cdots,(m_{l},n_{l})\\}$ be two sets of GPMs and
$I,X^{s},Z^{t}$ ($s$, $t$ are invertible in $\mathbb{Z}_{d}$) are contained in
$N$.

1\. Do $i=1$.

2\. Let
$L_{i}=\\{(s_{1}-s_{i},t_{1}-t_{i}),\cdots,(s_{l}-s_{i},t_{l}-t_{i})\\}$.

3\. Use a program to apply a Clifford operator
$F=\left[\begin{array}[]{cccc}a_{1}&b_{1}\\\ a_{2}&b_{2}\end{array}\right]$ to
the set $L_{i}$, then we obtain the set

$\displaystyle\\{(a_{1}(s_{1}-s_{i})+b_{1}(t_{1}-t_{i}),a_{2}(s_{1}-s_{i})+b_{2}(t_{1}-t_{i})),\cdots,$
$\displaystyle(a_{1}(s_{l}-s_{i})+b_{1}(t_{l}-t_{i}),a_{2}(s_{l}-s_{i})+b_{2}(t_{l}-t_{i}))\\}(\textrm{mod}\
d)$

for all $0\leq a_{1},a_{2},b_{1},b_{2}\leq d-1$ and
$a_{1}b_{2}-a_{2}b_{1}\equiv 1$ (mod $d$).

4\. For any $1\leq i\leq l$ and Clifford operator $F$, print all the sets
obtained in step 3, then we obtain all standard GPM sets that are U-equivalent
to $N$.

According to this protocol and the ten 4-GBS sets in (tian2016pra, , Theorem
2), we get ten equivalence classes
$\widetilde{\mathcal{K}},\widetilde{\mathcal{L}},\widetilde{\Gamma^{1}_{20}},\widetilde{\Gamma^{1}_{31}},\widetilde{\Gamma^{1}_{33}},\widetilde{\Gamma^{2}_{12}},\widetilde{\Gamma^{2}_{30}},\widetilde{\Gamma^{1}_{12}},\widetilde{\Gamma^{2}_{20}},\widetilde{\Gamma^{2}_{32}}$.
See Table I for the number of standard GBS sets in each equivalence class.

Table 1:  Numbers of standard GBS sets in equivalence classes $\widetilde{\mathcal{K}}$ | $\widetilde{\mathcal{L}}$ | $\widetilde{\Gamma^{1}_{20}}$ | $\widetilde{\Gamma^{1}_{31}}$ | $\widetilde{\Gamma^{1}_{33}}$ | $\widetilde{\Gamma^{2}_{12}}$ | $\widetilde{\Gamma^{2}_{30}}$ | $\widetilde{\Gamma^{1}_{12}}$ | $\widetilde{\Gamma^{2}_{20}}$ | $\widetilde{\Gamma^{2}_{32}}$ | Total
---|---|---|---|---|---|---|---|---|---|---
1 | 6 | 192 | 48 | 16 | 12 | 24 | 96 | 48 | 12 | 455

The ten 4-GBS sets are not LU equivalent to each other, and the ten
equivalence classes contain a total of 455 standard GBS sets. Since there are
$C_{15}^{3}(=455$) standard 4-GBS sets in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, the ten equivalence classes make up a
complete classification, and all 4-GBS sets in the three equivalence classes
$\widetilde{\Gamma^{1}_{12}},\widetilde{\Gamma^{2}_{20}},\widetilde{\Gamma^{2}_{32}}$
constitute all local indistinguishable standard 4-GBS sets in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$. That is, there are 156 local
indistinguishable standard 4-GBS sets, see Table II for details. Using Table
II, we can immediately determine the distinguishability of an arbitrary 4-GBS
set.

Table 2: All local indistinguishable standard 4-GBS sets in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ (156 items) $d$ |
---|---
|
{(0,0),(0,1),(1,0),(1,2)},{(0,0),(0,1),(1,0),(2,1)},{(0,0),(0,1),(1,0),(2,2)},{(0,0),(0,1),(1,1),(1,3)},{(0,0),(0,1),(1,1),(2,0)},
$\widetilde{\Gamma^{1}_{12}}$ | {(0,0),(0,1),(1,1),(2,3)},{(0,0),(0,1),(1,2),(2,1)},{(0,0),(0,1),(1,2),(2,2)},{(0,0),(0,1),(1,3),(2,0)},{(0,0),(0,1),(1,3),(2,3)},
(96 items) | {(0,0),(0,1),(2,0),(3,1)},{(0,0),(0,1),(2,0),(3,3)},{(0,0),(0,1),(2,1),(3,0)},{(0,0),(0,1),(2,1),(3,2)},{(0,0),(0,1),(2,2),(3,0)},
|
{(0,0),(0,1),(2,2),(3,2)},{(0,0),(0,1),(2,3),(3,1)},{(0,0),(0,1),(2,3),(3,3)},{(0,0),(0,1),(3,0),(3,2)},{(0,0),(0,1),(3,1),(3,3)},
|
{(0,0),(0,2),(1,0),(1,1)},{(0,0),(0,2),(1,0),(1,3)},{(0,0),(0,2),(1,0),(3,1)},{(0,0),(0,2),(1,0),(3,3)},{(0,0),(0,2),(1,1),(1,2)},
| {(0,0),(0,2),(1,1),(3
0)},{(0,0),(0,2),(1,1),(3,2)},{(0,0),(0,2),(1,2),(1,3)},{(0,0),(0,2),(1,2),(3,1)},{(0,0),(0,2),(1,2),(3,3)},
|
{(0,0),(0,2),(1,3),(3,0)},{(0,0),(0,2),(1,3),(3,2)},{(0,0),(0,2),(3,0),(3,1)},{(0,0),(0,2),(3,0),(3,3)},{(0,0),(0,2),(3,1),(3,2)},
|
{(0,0),(0,2),(3,2),(3,3)},{(0,0),(0,3),(1,0),(1,2)},{(0,0),(0,3),(1,0),(2,2)},{(0,0),(0,3),(1,0),(2,3)},{(0,0),(0,3),(1,1),(1,3)},
| {(0,0),(0,3),(1,1),(2
0)},{(0,0),(0,3),(1,1),(2,1)},{(0,0),(0,3),(1,2),(2,2)},{(0,0),(0,3),(1,2),(2,3)},{(0,0),(0,3),(1,3),(2,0)},
|
{(0,0),(0,3),(1,3),(2,1)},{(0,0),(0,3),(2,0),(3,1)},{(0,0),(0,3),(2,0),(3,3)},{(0,0),(0,3),(2,1),(3,1)},{(0,0),(0,3),(2,1),(3,3)},
|
{(0,0),(0,3),(2,2),(3,0)},{(0,0),(0,3),(2,2),(3,2)},{(0,0),(0,3),(2,3),(3,0)},{(0,0),(0,3),(2,3),(3,2)},{(0,0),(0,3),(3,0),(3,2)},
|
{(0,0),(0,3),(3,1),(3,3)},{(0,0),(1,0),(1,1),(3,1)},{(0,0),(1,0),(1,1),(3,2)},{(0,0),(1,0),(1,2),(2,1)},{(0,0),(1,0),(1,2),(2,3)},
|
{(0,0),(1,0),(1,3),(3,2)},{(0,0),(1,0),(1,3),(3,3)},{(0,0),(1,0),(2,1),(2,2)},{(0,0),(1,0),(2,2),(2,3)},{(0,0),(1,0),(3,1),(3,2)},
|
{(0,0),(1,0),(3,2),(3,3)},{(0,0),(1,1),(1,2),(3,0)},{(0,0),(1,1),(1,2),(3,1)},{(0,0),(1,1),(1,3),(2,1)},{(0,0),(1,1),(1,3),(2,3)},
|
{(0,0),(1,1),(2,0),(2,1)},{(0,0),(1,1),(2,0),(2,3)},{(0,0),(1,1),(3,0),(3,1)},{(0,0),(1,1),(3,1),(3,2)},{(0,0),(1,2),(1,3),(3,0)},
|
{(0,0),(1,2),(1,3),(3,3)},{(0,0),(1,2),(2,1),(2,2)},{(0,0),(1,2),(2,2),(2,3)},{(0,0),(1,2),(3,0),(3,1)},{(0,0),(1,2),(3,0),(3,3)},
|
{(0,0),(1,3),(2,0),(2,1)},{(0,0),(1,3),(2,0),(2,3)},{(0,0),(1,3),(3,0),(3,3)},{(0,0),(1,3),(3,2),(3,3)},{(0,0),(2,0),(2,1),(3,1)},
|
{(0,0),(2,0),(2,1),(3,3)},{(0,0),(2,0),(2,3),(3,1)},{(0,0),(2,0),(2,3),(3,3)},{(0,0),(2,1),(2,2),(3,0)},{(0,0),(2,1),(2,2),(3,2)},
|
{(0,0),(2,1),(3,0),(3,2)},{(0,0),(2,1),(3,1),(3,3)},{(0,0),(2,2),(2,3),(3,0)},{(0,0),(2,2),(2,3),(3,2)},{(0,0),(2,3),(3,0),(3,2)}.
| {(0,0),(2,3),(3,1),(3,3)}.
|
{(0,0),(0,1),(0,2),(2,0)},{(0,0),(0,1),(0,2),(2,2)},{(0,0),(0,1),(0,3),(2,1)},{(0,0),(0,1),(0,3),(2,3)},{(0,0),(0,1),(2,0),(2,2)},
$\widetilde{\Gamma^{2}_{20}}$ | {(0,0),(0,1),(2,1),(2,3)},{(0,0),(0,2),(0,3),(2,0)},{(0,0),(0,2),(0,3),(2,2)},{(0,0),(0,2),(1,0),(2,0)},{(0,0),(0,2),(1,0),(2,2)},
(48 items) | {(0,0),(0,2),(1,1),(2,0)},{(0,0),(0,2),(1,1),(2,2)},{(0,0),(0,2),(1,2),(2,0)},{(0,0),(0,2),(1,2),(2,2)},{(0,0),(0,2),(1,3),(2,0)},
|
{(0,0),(0,2),(1,3),(2,2)},{(0,0),(0,2),(2,0),(2,1)},{(0,0),(0,2),(2,0),(2,3)},{(0,0),(0,2),(2,0),(3,0)},{(0,0),(0,2),(2,0),(3,1)},
| {(0,0),(0,2),(2,0),(3
2)},{(0,0),(0,2),(2,0),(3,3)},{(0,0),(0,2),(2,1),(2,2)},{(0,0),(0,2),(2,2),(2,3)},{(0,0),(0,2),(2,2),(3,0)},
|
{(0,0),(0,2),(2,2),(3,1)},{(0,0),(0,2),(2,2),(3,2)},{(0,0),(0,2),(2,2),(3,3)},{(0,0),(0,3),(2,0),(2,2)},{(0,0),(0,3),(2,1),(2,3)},
|
{(0,0),(1,0),(1,2),(3,0)},{(0,0),(1,0),(1,2),(3,2)},{(0,0),(1,0),(2,0),(2,2)},{(0,0),(1,0),(3,0),(3,2)},{(0,0),(1,1),(1,3),(3,1)},
|
{(0,0),(1,1),(1,3),(3,3)},{(0,0),(1,1),(2,0),(2,2)},{(0,0),(1,1),(3,1),(3,3)},{(0,0),(1,2),(2,0),(2,2)},{(0,0),(1,2),(3,0),(3,2)},
|
{(0,0),(1,3),(2,0),(2,2)},{(0,0),(1,3),(3,1),(3,3)},{(0,0),(2,0),(2,1),(2,2)},{(0,0),(2,0),(2,2),(2,3)},{(0,0),(2,0),(2,2),(3,0)}.
|
{(0,0),(2,0),(2,2),(3,1)},{(0,0),(2,0),(2,2),(3,2)},{(0,0),(2,0),(2,2),(3,3)}.
|
{(0,0),(0,1),(2,0),(2,3)},{(0,0),(0,1),(2,1),(2,2)},{(0,0),(0,2),(1,0),(3,2)},{(0,0),(0,2),(1,1),(3,1)},{(0,0),(0,2),(1,2),(3,0)},
$\widetilde{\Gamma^{2}_{32}}$ | {(0,0),(0,2),(1,3),(3,3)},{(0,0),(0,3),(2,0),(2,1)},{(0,0),(0,3),(2,2),(2,3)},{(0,0),(1,0),(1,2),(2,2)},{(0,0),(1,1),(1,3),(2,0)},
(12 items) | {(0,0),(2,0),(3,1),(3,3)},{(0,0),(2,2),(3,0),(3,2)}.

##  V Necessary and sufficient conditions for GBS sets in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$

In this section, we will show that if a GBS set $\mathcal{L}$ in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ is one-way LOCC distinguishable then the
condition (1) in Theorem 1 is true. If the dimension $d$ is a prime, it is
easy to check that the condition (1) in Theorem 1 is equivalent to a simple
conclusion: the cardinality of the set
$\\{m^{-1}n|(m,n)\in\Delta\mathcal{L},m^{-1}n\triangleq\infty\hbox{ if
}m=0\\}$ is less than $d+1$. The cardinality can be calculated directly from
the difference set without solving the congruence equation (1), and then the
necessary and sufficient conditions allow us to quickly judge the one-way LOCC
distinguishability of any GBS set in $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$.
For convenience, the set
$\\{m^{-1}n|(m,n)\in\Delta\mathcal{L},m^{-1}n\triangleq\infty\hbox{ if
}m=0\\}$ can be called a $\it{discriminant\ index\ set}$ of $\mathcal{L}$.

######  Theorem 3.

Let $\mathcal{L}=\\{(m_{i},n_{i})\\}$ be a 4-GBS or 5-GBS set in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, then $\mathcal{L}$ is one-way LOCC
indistinguishable if and only if the discriminant set
$\mathcal{D}(\mathcal{L})$ is empty, or equivalently the cardinality of the
discriminant index set
$\\{m^{-1}n|(m,n)\in\Delta\mathcal{L},m^{-1}n\triangleq\infty\hbox{ if
}m=0\\}$ is 6.

Since the dimension $5$ is a prime, it is easy to check that each element
$m^{-1}n$ in Theorem 3 corresponds to a number $y\in S_{\mathcal{L}_{y}}$, and
each number $y\in S_{\mathcal{L}_{y}}$ also corresponds to an equivalent class
of commutative pairs defined in yang2018qip . So, we get the conclusion as
follows.

######  Corollary 1.

Let $\mathcal{L}=\\{(m_{i},n_{i})|1\leq i\leq 5\\}$ be a 4-GBS or 5-GBS set in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, then the following assertions are
equivalent to each other.

  1. (1)

$\mathcal{L}$ is one-way LOCC indistinguishable.

  2. (2)

The cardinality
$|\\{m^{-1}n|(m,n)\in\Delta\mathcal{L},m^{-1}n\triangleq\infty\hbox{ if
}m=0\\}|$ is 6.

  3. (3)

The cardinality $|S_{\mathcal{L}_{y}}|$ of the nonadmissible solution set
$S_{\mathcal{L}_{y}}$ is 6.

  4. (4)

Let $S$ be a maximal set containing the pairwise noncommuting pairs in
deference set, then the cardinality $|S|$ is equal to 6.

Corollary 1 implies that the following case $d=5$ of Wang et al’ problem
(wang2019pra, , Problem) is true.

######  Problem 1.

Let $\mathcal{L}=\\{X^{m_{i}}Z^{n_{i}}|1\leq i\leq l\\}$ be a $l$-GBS set of
dimension $d>2$. If $S_{A}\backslash S_{\mathcal{L}_{y}}=\emptyset$, is the
set $\mathcal{L}$ indistinguishable by one-way LOCC?

If $d$ is a prime, Problem 1 is equivalent to the conjecture posed in
yang2018qip .

######  Conjecture 1.

Let $d$ be a prime. If $|S|=d+1$, then the set $\mathcal{L}$ cannot be
distinguished by one-way LOCC.

When $d$ is a composite number, it obvious that Problem 1 is not true (see
Examples 1-3). When $d$ is a prime, Wang et al. wang2019pra  disproved the
case $d=101$ of Problem 1 or Conjecture 1 by finding a counterexample in
$\mathbb{C}^{101}\otimes\mathbb{C}^{101}$. The counterexample suggests that
Problem 1 (or Conjecture 1) may not be true. But the necessary and sufficient
conditions (Theorem 3 or Corollary 1) show that the case $d=5$ of Problem 1 is
true, so this is a surprising result, and it suggests that the problem is more
complex than expected. Now we give a proof of Theorem 3.

Proof. Let $\mathcal{L}=\\{(m_{i},n_{i})\\}$ be a one-way LOCC distinguishable
4-GBS or 5-GBS set, it is sufficient to show that the cardinality of the
discriminant index set
$\\{m^{-1}n|(m,n)\in\Delta\mathcal{L},m^{-1}n\triangleq\infty\hbox{ if
}m=0\\}$ is less than 6.

It is known that all quadruples of GBSs can be classified into eight locally
inequivalent classes and the representative elements of these equivalence
classes are
$H=\\{I,Z,Z^{2},Z^{3}\\},K=\\{I,Z,Z^{2},X\\},L=\\{I,Z,Z^{2},X^{2}\\},\Gamma^{1}_{1,1},\Gamma^{1}_{1,2},\Gamma^{2}_{2,1},\Gamma^{1}_{4,4},\Gamma^{1}_{2,2},$
where $\Gamma^{a}_{st}=\\{I,Z,X^{a},X^{s}Z^{t}\\}$. The first 6 sets are one-
way LOCC distinguishable, while the last 2 sets are one-way LOCC
indistinguishable wang-yuan2021jmp . Since the set $\mathcal{L}$ is a one-way
LOCC distinguishable, it is LU-equivalent to one of the six 4-GBS sets
$H,K,L,\Gamma^{1}_{1,1},\Gamma^{1}_{1,2},\Gamma^{2}_{2,1}$. Because of the
difference sets $\Delta\mathcal{H}=\\{(0,1),(0,2),(0,3),(0,4)\\}$ and
$\Delta\mathcal{K}=\\{(0,1),(0,2),(1,0),(1,4),(1,3),(0,4),(0,3),(4,0),(4,1),\\\
(4,2)\\}$, the cardinality of the corresponding discriminant index sets is 1
and 4 respectively. Similarly, the cardinality of the discriminant index sets
of $L,\Gamma^{1}_{1,1},\Gamma^{1}_{1,2}$ and $\Gamma^{2}_{2,1}$ is 4, 4, 5 and
4 respectively. Therefore, the cardinality of the six sets is less than 6.

It is known that all 5-GBS sets can be divided into 21 locally inequivalent
classes and the representative elements of these equivalence classes are

$\displaystyle H_{1}=\\{I,Z,Z^{2},Z^{3},Z^{4}\\},\ \ \
H_{2}=\\{I,Z,Z^{2},Z^{3},X\\};$ $\displaystyle K_{1}=\\{I,Z,Z^{2},X,X^{2}\\},\
\ \ \ K_{2}=\\{I,Z,Z^{2},X,X^{3}\\},$ $\displaystyle
K_{3}=\\{I,Z,Z^{2},X,X^{4}\\},\ \ \ \ K_{4}=\\{I,Z,Z^{2},X,XZ\\},$
$\displaystyle K_{5}=\\{I,Z,Z^{2},X,XZ^{2}\\},\ \
K_{6}=\\{I,Z,Z^{2},X,X^{2}Z\\},$ $\displaystyle
K_{7}=\\{I,Z,Z^{2},X,X^{2}Z^{2}\\},\ K_{8}=\\{I,Z,Z^{2},X,X^{3}Z^{2}\\},$
$\displaystyle K_{9}=\\{I,Z,Z^{2},X,X^{3}Z^{3}\\},\
K_{10}=\\{I,Z,Z^{2},X,X^{3}Z^{4}\\},$ $\displaystyle
K_{11}=\\{I,Z,Z^{2},X,X^{4}Z\\},\ K_{12}=\\{I,Z,Z^{2},X,X^{4}Z^{2}\\};$
$\displaystyle L_{1}=\\{I,Z,Z^{2},X^{2},X^{2}Z\\},$ $\displaystyle
L_{2}=\\{I,Z,Z^{2},X^{2},X^{2}Z^{2}\\},$ $\displaystyle
L_{3}=\\{I,Z,Z^{2},X^{2},X^{3}Z\\},$ $\displaystyle
L_{4}=\\{I,Z,Z^{2},X^{2},X^{3}Z^{2}\\};$
$\displaystyle\Gamma_{1}=\\{I,Z,X,XZ^{2},X^{2}Z\\},\
\Gamma_{2}=\\{I,Z,X,XZ^{2},X^{3}Z^{2}\\},$
$\displaystyle\Gamma_{3}=\\{I,Z,X,XZ^{2},X^{4}Z\\}.$

Among them nine sets $H_{1},H_{2},K_{1},K_{2},K_{4},K_{12},L_{1},L_{4},\\\
\Gamma_{2}$ are one-way LOCC distinguishable (also two-way LOCC
indistinguishable, see wang-yuan2021jmp ). If the set $\mathcal{L}$ is a one-
way LOCC distinguishable 5-GBS set, then it is LU-equivalence to one of the
nine 5-GBS sets. Using a similar method as the 4-GBS sets, we can deduce that
the cardinality of the discriminant index sets of the nine 5-GBS sets is 1, 5,
5, 5, 5, 4, 5, 4 and 5 respectively. So all of the nine 5-GBS sets have
cardinality less than 6.

##  VI Conclusions

It is well known that, for a bipartite quantum system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, there are few necessary and sufficient
conditions for local state discrimination and it is hard to locally
distinguish $k(4\leq k\leq d)$ GBSs. The purpose of this paper is to
completely solve the problem of local discrimination of GBS sets in some
bipartite quantum systems. To do this, we need to find some effective
sufficient and necessary conditions for local state discrimination. Firstly,
by using Weyl commutation relation and difference set, three practical
sufficient conditions are given, several known results can be deduced as
special cases of these conditions. Based on the three sufficient conditions
and by using LU equivalence, it is shown that the three conditions are also
necessary for local discrimination of GBS sets in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, and all 156 local indistinguishable
standard 4-GBS sets are found (see Table II for details). That is, the problem
of local discrimination of GBS sets in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$
is completely solved. In $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, one of the
three sufficient conditions is proved to be necessary for one-way LOCC
discrimination of GBS sets, that is, a GBS set $\mathcal{L}$ is one-way LOCC
indistinguishable if and only if the cardinality of the discriminant index set
$\\{m^{-1}n|(m,n)\in\Delta\mathcal{L},m^{-1}n\triangleq\infty\hbox{ if
}m=0\\}$ is 6. Since the cardinality of the discriminant index set is easy to
calculate, and two-way LOCC has no advantage over one-way LOCC in
distinguishing the sets of 5-GBS sets, the problem of one-way LOCC
discrimination of 4-GBS sets and two-way LOCC discrimination of 5-GBS sets in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ are completely solved. The necessary and
sufficient condition also shows that the case $d=5$ of the problem proposed by
Wang et al. (Phys Rev A 99:022307, 2019) has a positive answer. Despite these
advances, the problem of two-way LOCC discrimination of 4-GBS sets in
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ still remains unsolved. It is hoped that
the problem will be resolved in the near future.

###### Acknowledgements.

This work is supported by NSFC (Grant No. 11971151, 11901163).

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