#  Local discrimination of generalized Bell states via commutativity

Mao-Sheng Li  School of Mathematics, South China University of Technology,
Guangzhou 510641, China  Fei Shi  School of Cyber Security, University of
Science and Technology of China, Hefei, 230026, People’s Republic of China
Yan-Ling Wang<EMAIL_ADDRESS>School of Computer Science and Technology,
Dongguan University of Technology, Dongguan, 523808, China

###### Abstract

We study the distinguishability of generalized Bell states under local
operations and classical communication. We introduce the concept of maximally
commutative set (MCS), subset of generalized Pauli matrices whose elements are
mutually commutative, and there is no other generalized Pauli matrix that is
commute with all the elements of this set. We find that MCS can be considered
as a detector for local distinguishability of set $\mathcal{S}$ of generalized
Bell states. In fact, we get an efficient criterion. That is, if the
difference set of $\mathcal{S}$ is disjoint with or completely contain in some
MCS, then the set $\mathcal{S}$ is locally distinguishable. Furthermore, we
give a useful characterization of MCS for arbitrary dimension, which provides
great convenience for detecting the local discrimination of generalized Bell
states. Our method can be generalized to more general settings which contains
lattice qudit basis. Results in [Phys. Rev. Lett. 92, 177905 (2004)], [Phys.
Rev. A 92, 042320 (2015)] and a recent work [arXiv: 2109.07390] can be deduced
as special cases of our result.

PACS numbers

03.67.Hk,03.65.Ud

##  I Introduction

Quantum states discrimination is a fundamental task in quantum information
processing. It is well known that a set of quantum states can be perfectly
distinguished by global measurement if and only if the states of given set are
mutually orthogonal nils04 . However, our quantum states are usually
distributed in composite systems with long distances, so only local operations
and classical communication (LOCC) are allowed. In such setting, a state is
chosen from a known orthogonal set of quantum states in a composite systems
and the task is to identify the state under LOCC. If the task can be
accomplished perfectly, we say that the set is _locally distinguishable_ ,
otherwise, _locally indistinguishable_. If an orthogonal set is locally
indistinguishable, we also called that the set presents some kind of
nonlocality Bennett99  in the sense that more quantum information could be
inferred from global measurement than that from local operations. Any two
orthogonal multipartite states are showed to be locally distinguishable
Walgate-2000 . Bennett _et al._ Bennett99  presented the first example of
orthogonal product states that are locally indistinguishable which reveals the
phenomenon of “quantum nonlocality without entanglement”. Results on the local
distinguishability of quantum states have been practically applied in quantum
cryptography primitives such as data hiding Terhal01 ; DiVincenzo02  and
secret sharing Markham08 ; Rahaman15 ; WangJ17 .

For general orthogonal sets of quantum states, it is difficult to give a
complete characterization of whether they are locally distinguishable or not.
Therefore, most studies (See Bennett99 ; ben99u ; Walgate-2000 ; Ghosh-2001 ;
walgate-2002 ; divin03 ; HSSH ; Ghosh-2004 ; rin04 ; Watrous-2005 ; fan-2005 ;
Nathanson-2005 ; Hayashi-etal-2006 ; nis06 ; Duan2007 ; Duan-2009 ; feng09 ;
Bandyo-2011 ; BGK-2011 ; Yu-Duan-2012 ; Nathanson13 ; BN-2013 ; Cosentino-2013
; Yus15 ; B-IQC-2015 ; childs13 ; Cosentino-Russo-2014 ; Ha21 ; B-IQC-2015 ;
Li15 ; Yang13 ; zhang14 ; wang15 ; zhang15 ; zhang16-1 ; Xu-16-2 ; zhang16 ;
Wang-2017-Qinfoprocess ; Zhang-Oh-2017 ; halder ; Xu20b ; Li20 ; Li18 ;
Halder1909 ; Halder20c ; Halder19 ; Zhang1906 ; Shi20S ; Tian20 ; Wang21 ;
Banik21 ; Ha21  for an incomplete list) focus on two extreme cases: sets of
product states or sets of maximally entangled states. In this paper, we
restrict ourselves to the settings of maximally entangled cases.

Bell states are the most famous maximally entangled states and their local
distinguishability has been well understood. In fact, any two Bell states are
locally distinguishable but any three or four are not Ghosh-2001 . Nathanson
Nathanson-2005  showed that any three maximally entangled states in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ can be locally distinguished. Moreover,
any $l>d$ maximally entangled states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$
are known to be locally indistinguishable Nathanson-2005 . Therefore, it is
interesting to consider whether set of maximally entangled states with
cardinality $l\leq d$ can be locally distinguishable or not? Interestingly,
using the fact that applying local unitary operation does not change the local
distinguishability, Fan fan-2005  showed that any $l$ generalized Bell states
(GBSs) in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ are locally distinguishable if
$(l-1)l\leq 2d$ provided that $d$ is a prime number. Fan’s result was extended
by Tian et al. to the prime power dimensional quantum system in Tian15  where
they restricted themselves to the mutually commuting qudit lattice states.
Since Fan’s result, there has been lots of works Tian15_2 ; Tian15 ; Tian16 ;
Singal15 ; Wang17 ; Yuan20 ; Yang21 ; Hashimoto21  paid attention to the
locally distinguishability of GBSs. However, the complete classification of
local distinguishablity of GBSs is still difficult to achieve. On the other
hand, set of GBSs is an important and special subset of maximally entangled
states, which makes the problem of local distinguishability of GBSs important
and interesting. Motivated by a recent work Yuan21 , we find that the local
distinguishability of GBSs can be detected by maximally commutative set (MCS)
of GBSs.

The rest of this article is organized as follows. In Sec. II, we introduce the
matrices representation of generalized Bell states. Then we give a brief
review of some known results on the sufficient conditions of locally
distinguishable set of GBSs. In Sec. III, we present the definition of
maximally commutative set and show that it is useful for judging the locally
distinguishability. After that, we present some examples of MCS and study the
properties of general MCS. Finally, we draw a conclusion and presented some
questions in Sec. IV.

##  II A Review of local distinguishability of $\mathrm{GBSs}$

Throughout this paper, we will use the following notations. Let $d\geq 2$ be
an integer. Denote $\mathbb{Z}_{d}$ to be the ring defined over
$\\{0,1,\cdots,d-1\\}$ with the sum operation “$+$” (here $i+j$ should be
equal to the element $(i+j)\ \mathrm{mod}\ d$) and multiplication operation
(computing the usual multiplication first then taking module $d$). Consider a
bipartite quantum system $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ with both
local dimensions equal to $d$. Suppose that
$\\{|0\rangle,|1\rangle,\cdots,|d-1\rangle\\}$ is the computational basis of a
single qudit. Under this computational basis, the standard maximally entangled
state in this system can be expressed as
${|\Psi_{00}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|ii\rangle}.$ Generally,
any maximally entangled state can be written in the form
$|\Psi_{U}\rangle=(I\otimes U)|\Psi_{00}\rangle$ for some unitary matrix $U$
of dimensional $d$. We often call $U$ the defining unitary matrix of the
maximally entangled state $|\Psi_{U}\rangle$. To define the generalized Bell
states, we define the following two operations

$X_{d}=\displaystyle\sum_{i=0}^{d-1}|i+1\text{ mod }d\rangle\langle i|,\text{
and }Z_{d}=\displaystyle\sum_{i=0}^{d-1}\omega^{i}|i\rangle\langle i|,$

where $\omega=e^{\frac{2\pi\sqrt{-1}}{d}}$. Then the following $d^{2}$
orthogonal MESs are called as generalized Bell states:

$\\{|\Psi_{m,n}\rangle=(I\otimes
X_{d}^{m}Z_{d}^{n})|\Psi_{00}\rangle\big{|}m,n\in\mathbb{Z}_{d}\\}.$ (1)

And matrices in $\\{X_{d}^{m}Z_{d}^{n}\big{|}\ m,n\in\mathbb{Z}_{d}\\}$ are
called the generalized Pauli matrices (GPMs). For simplicity, we also use $X$
and $Z$ to represent $X_{d}$ and $Z_{d}$ when the dimension is known. Due to
the one-one correspondence of MES and its defining unitary matrix, for
convenience, we will treat the following three sets equally without
distinction

$\mathcal{S}:=\\{|\Psi_{m_{i},n_{i}}\rangle\\}_{i=1}^{l}=\\{X^{m_{i}}Z^{n_{i}}\\}_{i=1}^{l}=\\{(m_{i},n_{i})\\}_{i=1}^{l}.$

Our aim in this paper is to provide some sufficient condition such that the
set $\mathcal{S}$ is locally distinguishable. Now we gave a brief review of
the relative results.

Fan fan-2005  noted that if all $m_{i}$ ($i=1,\cdots,l$) are distinct, the set
$\mathcal{S}$ can be locally distinguished and set with this property is
called $F$-type Hashimoto21 . For each $\alpha\in\mathbb{Z}_{d}$, defining
$H_{\alpha}$ be the matrix whose $jk$ entry is $w^{-jk-\alpha s_{k}}/\sqrt{d}$
for $j,k=1,\cdots d-1$ and $s_{k}:=\sum_{i=k}^{d-1}i$. Then $H_{\alpha}$ is a
unitary matrix and $H_{\alpha}\otimes H_{\alpha}^{t}$ transfers
$|\Psi_{m_{i},n_{i}}\rangle$ to $|\Psi_{\alpha m_{i}+n_{i},-m_{i}}\rangle.$ He
found that if $d$ is a prime and $(l-1)l\leq 2d$, there exists an $\alpha$
such that $H_{\alpha}\otimes H_{\alpha}^{t}$ can transfer $\mathcal{S}$ to a
set of $F$-type.

There is a useful sufficient condition for local distinguishability of general
maximally entangled states. Denote $\mathcal{S}$ as the defining unitary
matrices set, if there exists some nontrivial vector
$|v\rangle\in\mathbb{C}^{d}$ such that

$\langle v|U^{\dagger}V|v\rangle=0$ (2)

for any different $U,V\in\mathcal{S}$, then the set of maximally entangled
states corresponding to $\mathcal{S}$ is one-way distinguishable (hence
locally distinguishable) Ghosh-2004 ; Nathanson13 . If the set $\mathcal{S}$
is $F$-type, the vector $|v\rangle$ can be chosen as any vector of the
computational basis, i.e., $|i\rangle,i\in\mathbb{Z}_{d}$. We define
difference set $\Delta\mathcal{S}$ of
$\mathcal{S}:=\\{U_{i}|i=1,2,\cdots,l\\}$ as

$\Delta\mathcal{S}=\\{U_{i}^{\dagger}U_{j}\ |\ 1\leq i<j\leq l\\}.$ (3)

Noticing that

$(X^{m_{i}}Z^{n_{i}})^{\dagger}X^{m_{j}}Z^{n_{j}}=\omega^{-(m_{j}-m_{i})n_{i}}X^{m_{j}-m_{i}}Z^{n_{j}-n_{i}}.$

Up to a phase, we can identify $\Delta\mathcal{S}$ as the set
$\\{(m_{j}-m_{i},n_{j}-n_{i})|1\leq i<j\leq l\\}.$ In order to find some
nonzero vector $|v\rangle$ such that Eqs. (2) are satitsfied, the following
lemma is important (See also in Ref. Yuan21 ).

###### Lemma 1

For two unitary matrices $U$ and $V$, if they satisfy $UV=zVU$ where $z$ is a
complex number and are not commutative, i.e., $z\neq 1$, then each eigenvector
$|v\rangle$ of $V$ satisfies $\langle v|U|v\rangle=0$.

In fact, suppose that $V|v\rangle=\lambda|v\rangle$ where
$\overline{\lambda}\lambda=1$. We also have $\langle
v|V^{\dagger}=\overline{\lambda}\langle v|.$ Therefore, $\langle
v|U|v\rangle=\langle v|\overline{\lambda}U\lambda|v\rangle=\langle
u|V^{\dagger}UV|u\rangle=z\langle v|U|v\rangle.$ Hence, $\langle
v|U|v\rangle=0$ as $z\neq 1.$ A pair of unitaries that satisfy the first
condition are called Weyl commutative.

Fortunately, any pair of generalized Pauli matrices are Weyl commutative. In
fact, for two pairs of $(m_{i},n_{i})$ and $(m_{j},n_{j})$ in
$\mathbb{Z}_{d}\times\mathbb{Z}_{d}$, we always have

$X^{m_{i}}Z^{n_{i}}X^{m_{j}}Z^{n_{j}}=\omega^{m_{j}n_{i}-m_{i}n_{j}}X^{m_{j}}Z^{n_{j}}X^{m_{i}}Z^{n_{i}}.$

Moreover, $X^{m_{i}}Z^{n_{i}}$ and $X^{m_{j}}Z^{n_{j}}$ are commutative if and
only if ${m_{j}n_{i}-m_{i}n_{j}}\equiv 0\ \mathrm{mod}\ d$. This condition can
be formulated as the detminant equation

$\left|\begin{array}[]{cc}m_{i}&n_{i}\\\ m_{j}&n_{j}\end{array}\right|\equiv
0\mod d.$ (4)

We also call that $(m_{i},n_{i})$ and $(m_{j},n_{j})$ are commutative if this
condition is satisfied.

For any GBS set $\mathcal{S}$, if there is a generalized Pauli matrix $V$
which is not commutative to every GPM $U\in\Delta\mathcal{S}$, by Eqs. (2),
(3), and Lemma 1, each eigenvector $|v\rangle$ of $V$ satisfies $\langle
v|U|v\rangle=0$ and therefore the set $\mathcal{S}$ is locally
distinguishable.

Let $m,n\in\mathbb{Z}_{d}$, denote the solution set of the following
congruence equation by $S(m,n)$,

$\displaystyle nx-my=0\mod d.$ (5)

Therefore, $S(m,n)$ denote the set of elements in
$\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ that are commute with $(m,n).$ In order
to check whether is there any GPM $V$ that do not commute with all the
elements in $\Delta\mathcal{S}.$ The authors of Ref. Yuan21  defined the set

$\displaystyle\mathcal{D}(\mathcal{S})\triangleq(\mathbb{Z}_{d}\times\mathbb{Z}_{d})\setminus\bigcup_{(m,n)\in\Delta\mathcal{S}}S(m,n).$

By definition, $\mathcal{D}(\mathcal{S})$ denotes the set of all elements in
$\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ that do not commute with all the elements
in $\Delta\mathcal{S}.$ Under this definition, they proved the following
results.

######  Theorem 1 (See Ref. Yuan21 )

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 $\mathbb{Z}_{d}$.

The nonemptyness of $\mathcal{D}(\mathcal{S})$ implies the local
distinguishability of $\mathcal{S}$. Therefore, the set
$\mathcal{D}(\mathcal{S})$ can be called a $\it{discriminant\ set}$ of
$\mathcal{S}$.

##  III Detector for local distinguishability of $\mathrm{GBSs}$

In the first case of Theorem 1, the nonempty of the discriminant set
$\mathcal{D}(\mathcal{S})$ is equivalent to that there exists some
$(s,t)\in\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ such that

$ms-nt\neq 0,\ \ \forall(m,n)\in\Delta\mathcal{S}.$

That is, $X^{s}Z^{t}$ is not commute with $X^{m}Z^{n}$. Therefore, any nonzero
eigenvector $|v\rangle$ of $X^{s}Z^{t}$ satisfies

$\langle v|X^{m}Z^{n}|v\rangle=0$

from which one can conlude that the set $\mathcal{S}$ is locally
distinguishable. From this point, we can call $X^{s}Z^{t}$ as a detector of
local discrimination of GBS. Simply, the ability of the detector $X^{s}Z^{t}$
can be defined as the set

$\mathcal{D}e(X^{s}Z^{t})\triangleq(\mathbb{Z}_{d}\times\mathbb{Z}_{d})\setminus
S(s,t).$ (6)

This denotes the set of all elements in $\mathbb{Z}_{d}\times\mathbb{Z}_{d}$
that do not commute with $(s,t)$. Then the one-way local distinguishability of
$\mathcal{S}$ can be detected by $X^{s}Z^{t}$ if and only if
$\Delta\mathcal{S}\subseteq\mathcal{D}e(X^{s}Z^{t}).$

In fact, we can introduce a stronger detector by the following observation. If
a set of detectors $\\{X^{s_{i}}Z^{t_{i}}\\}_{i=1}^{n}$ are commutative, they
can share a common eigenbasis $\\{|v_{j}\rangle\\}_{j=1}^{d}$. Therefore, if

$\Delta\mathcal{S}\subseteq\bigcup_{i=1}^{n}\mathcal{D}e(X^{s_{i}}Z^{t_{i}})$

we can also conclude that the set $\mathcal{S}$ is one-way distinguishable.
Therefore, the more elements of the detected set, the stronger its
distinguishingability which motivates the following definition.

A subset $\\{X^{s_{i}}Z^{t_{i}}\\}_{i=1}^{n}$ of GBSs is call _maximally
commutative_ if the elements of the given subset are mutually commute and
there is no other GBS which can commute with all the elements of the set. This
can be written as the coordinates
$\\{(s_{i},t_{i})\\}_{i=1}^{n}\subseteq\mathbb{Z}_{d}\times\mathbb{Z}_{d}$
such that $s_{i}t_{j}=t_{i}s_{j}$ for every $i,j$ but there is no
$(s,t)\in\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ such that $s_{i}t=t_{i}s$ for
every $i$.

For any maximally commutative set of GBSs
$\mathcal{C}:=\\{X^{s_{i}}Z^{t_{i}}\\}_{i=1}^{n}$, we defined a detector as

$\mathcal{D}e(\mathcal{C}):=\bigcup_{(s,t)\in\mathcal{C}}\mathcal{D}e(X^{s}Z^{t}).$

Therefore, one conclude that if
$\Delta\mathcal{S}\subseteq\mathcal{D}e(\mathcal{C})$, the set $\mathcal{S}$
is one-way distinguishable. On the other hand, one finds that
$\mathcal{D}e(\mathcal{C})$ is equal to $\mathcal{P}_{d}\setminus\mathcal{C}$
where $\mathcal{P}_{d}:=\\{X^{m}Z^{n}|m,n\in\mathbb{Z}_{d}\\}.$ In fact, every
element in $\mathcal{P}_{d}$ but outside $\mathcal{C}$ must be not commute
with one of element $X^{s}Z^{t}$ in $\mathcal{C}$. That is, it belongs to
$\mathcal{D}e(X^{s}Z^{t})$. It means that
$\mathcal{P}_{d}\setminus\mathcal{C}\subseteq\mathcal{D}e(\mathcal{C})$.
Obviously,
$\mathcal{D}e(\mathcal{C})\subseteq\mathcal{P}_{d}\setminus\mathcal{C}$. Thus,
$\mathcal{D}e(\mathcal{C})=\mathcal{P}_{d}\setminus\mathcal{C}.$ Therefore,
$\Delta\mathcal{S}\subseteq\mathcal{D}e(\mathcal{C})$ if and only if
$\Delta\mathcal{S}\cap\mathcal{C}=\emptyset.$ Moreover, if
$\Delta\mathcal{S}\subseteq\mathcal{C},$ that is, the elements in
$\Delta\mathcal{S}$ are mutually commutative. By Theorem 1, the set
$\mathcal{S}$ is also locally distinguishable.

###### Theorem 2

Let $\mathcal{S}$ be a GBS set in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and
$\mathcal{C}$ be a set of maximally commutative GBS of dimensional $d$. If
$\Delta\mathcal{S}\cap\mathcal{C}=\emptyset$ or
$\Delta\mathcal{S}\subseteq\mathcal{C},$ then the set $\mathcal{S}$ is locally
distinguishable (see Fig. 1 for an intuitive view of the conditions).

Figure 1: Here $\mathcal{C}$ represents a maximally commutative set of GBS and
$\Delta\mathcal{S}$ is the difference set of $\mathcal{S}$. If
$\Delta\mathcal{S}$ and the detector $\mathcal{C}$ are in one of the above
relations , then the set $\mathcal{S}$ is locally distinguishable.

The result of Theorem 2 implies that of Theorem 1. If
$\mathcal{D}(\mathcal{S})$ is nonempty (that is the distinguishability of
$\mathcal{S}$ can be detected by the first condition of Theorem 1), then there
must exist some maximally commutative set $\mathcal{C}$ of GBS satisfies the
condition of Theorem 2. In fact, the nonempty of the discriminant set
$\mathcal{D}(\mathcal{S})$ is equivalent to the existence of $X^{s}Z^{t}$ that
do not commute with every element of $\Delta\mathcal{S}$ but such an
$X^{s}Z^{t}$ can be extended to be a maximally commutative set $\mathcal{C}$
of GBS. As the elements in $\mathcal{C}$ are all commute with $X^{s}Z^{t}$,
therefore, $\Delta\mathcal{S}\cap\mathcal{C}=\emptyset.$ Moreover, if $d$ is a
composite number, and suppose that $d=pq$ where $p,q\geq 2$ are two integers.
Clearly, $X^{p}$ is commute with $Z^{q}$, therefore, they can extend to a
maximally commutative set of GBS, said, $\mathcal{C}$. if $s$ or $t$ is
invertible in $\mathbb{Z}_{d}$, we claim that $X^{s}Z^{t}\notin\mathcal{C}.$
In fact, as $ZX=\omega XZ$, if $s$ is invertable, then
$Z^{q}(X^{s}Z^{t})=\omega^{sq}(X^{s}Z^{t})Z^{q}\neq(X^{s}Z^{t})Z^{q}$. If $t$
is invertible, then $(X^{s}Z^{t})X^{p}=\omega^{tp}X^{p}(X^{s}Z^{t})\neq
X^{p}(X^{s}Z^{t}).$ It also means that
$\Delta\mathcal{S}\cap\mathcal{C}=\emptyset.$ Therefore, if
$\Delta\mathcal{S}$ contains those elements one of whose coordinates is
invertible in $\mathbb{Z}_{d}$, then the set $\mathcal{C}$ can detect the one-
way distinguishability of $\mathcal{S}$.

Therefore, it is important to find out all the maximally commutative sets of
GBSs. Now we present some examples in the low dimensional cases.

###### Example 1

There are exactly four classes of maximally commute sets of GBSs in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}.$

$\begin{array}[]{rlccc}\mathcal{C}_{1}&=&\\{(0,0),&(0,1),&(0,2)\\},\\\
\mathcal{C}_{2}&=&\\{(0,0),&(1,0),&(2,0)\\},\\\
\mathcal{C}_{3}&=&\\{(0,0),&(1,1),&(2,2)\\},\\\
\mathcal{C}_{4}&=&\\{(0,0),&(1,2),&(2,1)\\}.\end{array}$

###### Example 2

There are exactly seven classes of maximally commute sets of GBSs in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}.$

$\begin{array}[]{rlcccc}\mathcal{C}_{1}&=&\\{(0,0),&(0,1),&(0,2),&(0,3)\\},\\\
\mathcal{C}_{2}&=&\\{(0,0),&(0,2),&(2,0),&(2,2)\\},\\\
\mathcal{C}_{3}&=&\\{(0,0),&(0,2),&(2,1),&(2,3)\\},\\\
\mathcal{C}_{4}&=&\\{(0,0),&(1,0),&(2,0),&(3,0)\\},\\\
\mathcal{C}_{5}&=&\\{(0,0),&(1,1),&(2,2),&(3,3)\\},\\\
\mathcal{C}_{6}&=&\\{(0,0),&(1,2),&(2,0),&(3,2)\\},\\\
\mathcal{C}_{7}&=&\\{(0,0),&(1,3),&(2,2),&(3,1)\\}.\\\ \end{array}$

###### Example 3

There are exactly 15 classes of maximally commute sets of GBSs in
$\mathbb{C}^{8}\otimes\mathbb{C}^{8}.$ Here we do not write out the coordinate
$(0,0)$ which belongs to all the 15 sets.

$\begin{array}[]{rlcccccccc}\mathcal{C}_{1}&=&\\{(0,1),&(0,2),&(0,3),&(0,4),&(0,5),&(0,6),&(0,7)\\},\\\
\mathcal{C}_{2}&=&\\{(0,2),&(0,4),&(0,6),&(4,0),&(4,2),&(4,4),&(4,6)\\},\\\
\mathcal{C}_{3}&=&\\{(0,2),&(0,4),&(0,6),&(4,1),&(4,3),&(4,5),&(4,7)\\},\\\
\mathcal{C}_{4}&=&\\{(0,4),&(2,0),&(2,4),&(4,0),&(4,4),&(6,0),&(6,4)\\},\\\
\mathcal{C}_{5}&=&\\{(0,4),&(2,1),&(2,5),&(4,2),&(4,6),&(6,3),&(6,7)\\},\\\
\mathcal{C}_{6}&=&\\{(0,4),&(2,2),&(2,6),&(4,0),&(4,4),&(6,2),&(6,6)\\},\\\
\mathcal{C}_{7}&=&\\{(0,4),&(2,3),&(2,7),&(4,2),&(4,6),&(6,1),&(6,5)\\},\\\
\mathcal{C}_{8}&=&\\{(1,0),&(2,0),&(3,0),&(4,0),&(5,0),&(6,0),&(7,0)\\},\\\
\mathcal{C}_{9}&=&\\{(1,1),&(2,2),&(3,3),&(4,4),&(5,5),&(6,6),&(7,7)\\},\\\
\mathcal{C}_{10}&=&\\{(1,2),&(2,4),&(3,6),&(4,0),&(5,2),&(6,4),&(7,6)\\},\\\
\mathcal{C}_{11}&=&\\{(1,3),&(2,6),&(3,1),&(4,4),&(5,7),&(6,2),&(7,5)\\},\\\
\mathcal{C}_{12}&=&\\{(1,4),&(2,0),&(3,4),&(4,0),&(5,4),&(6,0),&(7,4)\\},\\\
\mathcal{C}_{13}&=&\\{(1,5),&(2,2),&(3,7),&(4,4),&(5,1),&(6,6),&(7,3)\\},\\\
\mathcal{C}_{14}&=&\\{(1,6),&(2,4),&(3,2),&(4,0),&(5,6),&(6,4),&(7,2)\\},\\\
\mathcal{C}_{15}&=&\\{(1,7),&(2,6),&(3,5),&(4,4),&(5,3),&(6,2),&(7,1)\\}.\\\
\end{array}$

Using these MCSs, we can show that Theorem 2 is strictly powerful than Theorem
1 when the dimension $d=8$. Set
$\mathcal{S}:=\\{\mathbb{I},X^{5}Z^{6},X^{6}Z^{3},X^{6}Z^{5},X^{7}Z^{6}\\}$
whose difference set $\Delta\mathcal{S}$ is

$\begin{array}[]{l}(5,6),(6,3),(6,5),(7,6),(1,5),\\\
(1,7),(2,0),(0,2),(1,3),(1,1).\end{array}$

One can check that $\mathcal{D}(\mathcal{S})=\emptyset$, $\Delta\mathcal{S}$
is non-commutative and neither coordinates of $(2,0)$ are invertible in
$\mathbb{Z}_{8}$. Therefore, Theorem 1 fails to detect the distinguishability
of this set. However, one finds that
$\Delta\mathcal{S}\cap\mathcal{C}_{6}=\emptyset.$ That is, the local
distinguishablity of $\mathcal{S}$ can be detected by $\mathcal{C}_{6}$.
Moreover, one can check that neither
$\Delta\mathcal{S}\cap\mathcal{C}_{i}=\emptyset$ nor
$\Delta\mathcal{S}\subseteq\mathcal{C}_{i}$ when $i\neq 6$. That is, among the
15 classes, $\mathcal{C}_{6}$ is the only detector that can detect the local
distinguishability of $\mathcal{S}.$ More numerical results comparing the
power of Theorem 1 and Theorem 2 can be seen in the figure 2 (we randomly
generated $N$ sets of $d$ dimensional GBSs with cardinality $n$ and find out
the numbers $N_{1}$ and $N_{2}$ of sets whose local distinguishability can be
detected by Theorem 1 and Theorem 2 respectively. The corresponding successful
rates are defined by $N_{1}/N$ and $N_{2}/N$).

Figure 2: For $n=5,6,7,8,9,10,11$, we randomly generate $N=100000$ sets of
GBSs whose cardinality are all $n$ for $d=7-20$ respectively. The solid lines
represent the successful rates by Theorem 1 and the dot lines represent the
successful rates by Theorem 2. The lines from above to below represent sets
with cardinality being from $5$ to $11$ respectively. Here the starting point
of each curve with respect to $n$ is with dimension $d\geq n$.

###### Proposition 1

Let $p\geq 2$ be a prime. Then there are exactly $p+1$ classes of maximally
commutative sets of GBSs in $\mathbb{C}^{p}\otimes\mathbb{C}^{p}.$

In fact, these sets are characterized by $(0,1)\mathbb{Z}_{p}$,
$(1,0)\mathbb{Z}_{p}$, and $(1,i)\mathbb{Z}_{p}$, $1\leq i\leq p-1$ where
$(a,b)\mathbb{Z}_{p}:=\\{(ai,bi)|i\in\mathbb{Z}_{p}\\}$. By this proposition
and Theorem 2, one would deduce Fan’s result again. That is, if $\mathcal{S}$
is a set of $p$ dimensional GBSs with $l$ elements and $l(l-1)/2\leq p$, then
$\mathcal{S}$ is locally distinguishable. In fact, in this setting, the number
of elements in $\Delta\mathcal{S}$ (which does not contain $(0,0)$) is less or
equal than $l(l-1)/2$. However, $(0,1)\mathbb{Z}_{p}\setminus\\{(0,0)\\}$,
$(1,0)\mathbb{Z}_{p}\setminus\\{(0,0)\\}$, and
$(1,i)\mathbb{Z}_{p}\setminus\\{(0,0)\\}$ ($1\leq i\leq p-1$) are $p+1$
classes of mutually disjoint sets. Therefore, there must exists some MCS
$\mathcal{C}$ such that $\mathcal{C}\cap\Delta\mathcal{S}=\emptyset$.

###### Lemma 2

Each maximally commutative set of GBSs in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ must be with cardinality less or equal
than $d$.

This can be obtained by observing that commutative set of unitary matrices can
be simultaneously diagonalized and the elements of GBSs are mutually
orthogonal. Moreover, one could easily verify the following lemma.

###### Lemma 3

Let $\mathcal{C}$ be a maximally commutative set of GBSs in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. If $(i,j)$ belongs to $\mathcal{C},$ so
does $(ik,jk)$ where $k\in\mathbb{Z}_{d},$ i.e.,
$(i,j)\mathbb{Z}_{d}\subseteq\mathcal{C}.$ Moreover, if both $(i_{1},j_{1})$
and $(i_{2},j_{2})$ belongs to $\mathcal{C},$ so does
$(i_{1}+i_{2},j_{1}+j_{2}).$

From this lemma, one can conclude that each maximally commutative set
$\mathcal{C}$ can be written as the forms

$\mathcal{C}=\bigcup_{k=1}^{n}(i_{k},j_{k})\mathbb{Z}_{d},\ \text{ or }\
\mathcal{C}=\sum_{k=1}^{n}(i_{k},j_{k})\mathbb{Z}_{d}.$ (7)

Here $A+B:=\\{a+b\big{|}\ a\in A,b\in B\\}$ where $A,B$ are subsets of a
group.

We find that the number of MCSs of GBSs is related to is an interesting
function in number theory which is known as sigma function. The sigma function
is usually denoted by the Greek letter sigma ($\sigma$). This function
actually denotes the sum of all divisors of a positive integer. For examples,
$\sigma(6)=1+2+3+6=12,$ and $\sigma(16)=1+2+4+8+16=31.$ Generally, let
$d=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{l}^{n_{l}},$ then

$\sigma(d)=\prod_{k=1}^{l}(1+p_{k}+\cdots+p_{k}^{n_{k}}).$

######  Theorem 3 (Structure Characterization of MCS)

Let $d\geq 3$ be an integer. For each pair $(i,j)$ in
$\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ where $i\neq 0,$ we define the following
set

$\mathcal{C}_{i,j}:=\\{(x,y)\in\mathbb{Z}_{d}\times\mathbb{Z}_{d}\big{|}\ \
\left|\begin{array}[]{cc}i&j\\\ x&y\end{array}\right|\equiv 0\mod d,x\in
i\mathbb{Z}_{d}\\}.$

Then $\mathcal{C}_{i,j}$ is a MCS of GBSs in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ with exactly $d$ elements. Moreover, if
we define $\mathcal{C}_{0,0}:=\\{(0,y)|y\in\mathbb{Z}_{d}\\}$, then every MCS
of GBSs in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ must be one of
$\mathcal{C}_{i,j}$ with $i\neq 0$ or $\mathcal{C}_{0,0}.$ There are exactly
$\sigma({d})$ classes of MCSs which can be listed as follows

$\mathcal{MCS}_{d}:=\\{\mathcal{C}_{i,j}|d=ik,0\leq j\leq
k-1\\}\cup\\{\mathcal{C}_{0,0}\\}.$

###### Proof.

First, we show that the cardinality of each $\mathcal{C}_{i,j}$ is equal to
$d$. Denote $d_{i}$ as the greatest common divisor of $i$ and $d$. Then the
set $i\mathbb{Z}_{d}:=\\{ij\in\mathbb{Z}_{d}\ |\ j\in\mathbb{Z}_{d}\\}$ has
exactly $d/d_{i}$ elements. More exactly,

$i\mathbb{Z}_{d}=\\{ik\big{|}\ \ k=0,1,\cdots,\frac{d}{d_{i}}-1\\}.$

For each $x=ik$ ($k=0,1,\cdots,\frac{d}{d_{i}}-1$), there are exactly $d_{i}$
solutions of $y\in\mathbb{Z}_{d}$ that satisfies

$\left|\begin{array}[]{cc}i&j\\\ x&y\end{array}\right|\equiv 0\mod d.$ (8)

In fact, the Eq. (8) is equivalent to $i(y-kj)\equiv 0\mod d$ whose solutions
can be expressed analytically as $y=kj+\frac{d}{d_{i}}l$ where $l=0,1,\cdots
d_{i}-1.$ Therefore, the set $\mathcal{C}_{i,j}$ can be expressed as

$\\{(ik,kj+\frac{d}{d_{i}}l)\big{|}\ k=0,1,\cdots,\frac{d}{d_{i}};l=0,1,\cdots
d_{i}-1\\}.$

One can check that for two different pairs of $(k_{1},l_{1})$ and
$(k_{2},l_{2})$ with the above conditions, the coordinates
$(ik_{1},k_{1}j+\frac{d}{d_{i}}l_{1})\neq(ik_{2},k_{2}j+\frac{d}{d_{i}}l_{2}).$
Therefore, the cardinality of $\mathcal{C}_{i,j}$ is equal to $d.$

Now we show that the elements in $\mathcal{C}_{i,j}$ are mutually commutative.
In fact, for any two solutions $(ik_{1},k_{1}j+\frac{d}{d_{i}}l_{1})$ and
$(ik_{2},k_{2}j+\frac{d}{d_{i}}l_{2})$, we have

$\left|\begin{array}[]{cc}ik_{1}&k_{1}j+\frac{d}{d_{i}}l_{1}\\\\[2.84526pt]
ik_{2}&k_{2}j+\frac{d}{d_{i}}l_{2}\end{array}\right|=\frac{i}{d_{i}}(k_{1}l_{2}-k_{2}l_{1})d$
(9)

which is always equal to $0\mod d$ as $d_{i}$ divides $i$.

Therefore, each $\mathcal{C}_{i,j}$ is a commutative set of GBSs with
cardinality $d$. By Lemma 2, each $\mathcal{C}_{i,j}$ must also be maximally.

Next, we show that for every MCS $\mathcal{C}$, it must be one of
$\mathcal{C}_{i,j}$ with $i\neq 0$ or $C_{0,0}.$ For any maximally commutative
set $\mathcal{C}$ of GBSs in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, by Eq.
(7), there exists $(x_{k},y_{k})\in\mathcal{C}$ ($k=1,\cdots,n$), such that

$\mathcal{C}=\sum_{k=1}^{n}(x_{k},y_{k})\mathbb{Z}_{d}.$

If all $x_{k}$ are equal to 0, one must conclude that
$\mathcal{C}=\mathcal{C}_{0,0}.$ If not, let $i$ denote the greatest common
divisor of $x_{1},x_{2},\cdots,x_{n}$ and $d$, which is not equal to zero in
this case. There exist $r_{k}\in\mathbb{Z}_{d}$, such that
$i=\sum_{k=1}^{n}r_{k}x_{k}$ (by Ref. Nathanson2000 , p12, Theorem 1.4, we
have $i=R_{0}d+\sum_{k=1}^{n}R_{k}x_{k},R_{i}\in\mathbb{Z}$, then taking
module $d$). And we define $j=\sum_{k=1}^{n}r_{k}y_{k}.$ By Lemma 3, we have
$(i,j)\in\mathcal{C}.$ As both $(x_{k},y_{k})$ and $(i,j)$ are in
$\mathcal{C}.$ By the definition of $i$, for each $k$, the element $x_{k}\in
i\mathbb{Z}_{d}$. As both $(x_{k},y_{k})$ and $(i,j)$ are in $\mathcal{C}$, we
have $iy_{k}-jx_{k}\equiv 0\mod d.$ Therefore, by definition of
$\mathcal{C}_{i,j}$, for each $k$, the element
$(x_{k},y_{k})\in\mathcal{C}_{i,j}.$ By Lemma 3 again, one have
$\mathcal{C}\subseteq\mathcal{C}_{i,j}.$ However, both sets are maximally
commutative sets of GBSs. Therefore, $\mathcal{C}$ must equal to
$\mathcal{C}_{i,j}$.

In the following, we show that each $\mathcal{C}_{x,y}$ ($x\neq 0$) is in fact
lie in one of $\mathcal{MCS}_{d}$. Set $d_{x}$ denote the greatest common
divisor of $x$ and $d$ (we might assume $x=c_{x}d_{x}$ where
$c_{x}\in\mathbb{Z}$). So $d_{x}=qx+rd$ for some integers $q,r$. There exist a
unique $j\in\\{0,1,\cdots,k_{x}-1\\}$ (where $k_{x}d_{x}=d$) such that

$qy-j\in k_{x}\mathbb{Z}_{d}.$

That is, $qy-j=k_{x}l_{x}$ for some $l_{x}\in\mathbb{Z}_{d}.$ For this $j$, we
have the following equation

$\left|\begin{array}[]{cc}d_{x}&j\\\
x&y\end{array}\right|=\left|\begin{array}[]{cc}d_{x}&qy-k_{x}l_{x}\\\
x&y\end{array}\right|=(d_{x}-qx)y+l_{x}k_{x}x$

which is equal $(ry+l_{x}c_{x})d\equiv 0$ under $\mod d.$ By definition, $x\in
d_{x}\mathbb{Z}_{d}$. Therefore, we have $(x,y)\in\mathcal{C}_{d_{x},j}.$ As
the elements in $\mathcal{C}_{d_{x},j}$ are commute with each other, for any
$(x_{1},y_{1})\in\mathcal{C}_{d_{x},j}$, we have

$\left|\begin{array}[]{cc}x&y\\\ x_{1}&y_{1}\end{array}\right|\equiv 0\mod d.$

Note that $d_{x}\mathbb{Z}_{d}=x\mathbb{Z}_{d}$, we have $x_{1}\in
x\mathbb{Z}_{d}.$ Therefore,
$\mathcal{C}_{d_{x},j}\subseteq\mathcal{C}_{x,y}.$ By the maximality, we must
have $\mathcal{C}_{x,y}=\mathcal{C}_{d_{x},j}.$ Therefore, one conclude that
every $\mathcal{C}_{x,y}$ must be one of the elements in $\mathcal{MCS}_{d}.$

On the other hand, we need to show that the sets in $\mathcal{MCS}_{d}$ are
mutually different. Clearly, $\mathcal{C}_{0,0}$ is different from all the
other sets. Let $\mathcal{C}_{i_{1},j_{1}}$ and $\mathcal{C}_{i_{2},j_{2}}$ be
any two members of $\mathcal{MCS}_{d}$ where $(i_{1},j_{1})\neq(i_{2},j_{2})$
and $i_{1},i_{2}$ are nonzero. As $d=i_{1}k_{1},$ if $i_{1}=i_{2},$ we have
$i_{1}j_{2}-i_{2}j_{1}=i_{1}(j_{2}-j_{1})$ which lies between $-(d-1)$ and
$d-1$ but not equal to zero. Hence, $(i_{1},j_{1})$ and $(i_{2},j_{2})$ are
not commute. Therefore,
$\mathcal{C}_{i_{1},j_{1}}\neq\mathcal{C}_{i_{2},j_{2}}.$ If $i_{1}\neq
i_{2}$, we can assume that $i_{1}<i_{2}$ without loss of generality. As both
$i_{1}$ and $i_{2}$ are divisors of $D$, one can check that $i_{1}\notin
i_{2}\mathbb{Z}_{d}$. Therefore, by definition,
$(i_{1},j_{1})\notin\mathcal{C}_{i_{2},j_{2}}.$ Hence, we also have
$\mathcal{C}_{i_{1},j_{1}}\neq\mathcal{C}_{i_{2},j_{2}}.$

For each divisor $i(1\leq i<d)$ of $d$, it contributes to $d/i$ classes of
MCSs to $\mathcal{MCS}_{d}.$ Therefore,

$|\mathcal{MCS}_{d}|=1+\sum_{i|d,1\leq i<d}d/i=\sum_{i|d}d/i=\sigma(d).$

This completes the proof.  $\sqcap$$\sqcup$

Figure 3: We consider the cases $d=20$ and $n=9,10$ or $11$ (which correspond
to curves with color green, purple and pink respectively). We randomly
generate 100000 sets (which are separated into 10 equal classes) of GBSs whose
cardinality are all $n=9,10,11$ respectively. Each solid curve shows the
successful rates for the $\sigma(20)=42$ classes of MCSs. The dot lines
represent the successful rates for the $42$ classes of MCSs such that the
local distinguishability of the samples can only detected by one class of MCS
itself.

From the above theorem, we know that there are $\sigma(d)$ classes of MCSs of
$d$ dimensional GBSs. Are there any differences in the ability of these MCSs
to detect generalized Bell state sets? Is there any redundant MCS in detecting
the local discrimination of generalized Bell sets? We present some numerical
results for the two questions.

The three solid lines in Fig. 3 imply that the successful rates of all MCSs
are almost equal to each other (one should compare this with the detectors
defined in Eq. (6), see Fig. 4). The three dot lines imply that each class of
MCSs is irredudant in the sense that for each MCS $\mathcal{C}$, there exist
some set $\mathcal{S}$ whose local distinguishability can only be detected by
$\mathcal{C}$ but not by other MCSs.

Figure 4: We consider the case $d=20$ and $n=10$. The figure shows the
successful rates for each detectors (see Eq. (6)) indicated by the coordinates
$(i,j)\in\mathbb{Z}_{20}\times\mathbb{Z}_{20}$. We randomly generate 100000
sets of GBSs whose cardinality are all $n=10$.

##  IV Conclusion and discussion

In this paper, we studied the problem of local distinguishability of
generalized Bell states. Firstly, we gave a review of some important methods
for detecting the local distinguishability of GBSs. Motivated by a recent
method derived by Yuan et. al. Yuan21 , we introduced the concept maximally
commutative set of GBSs. Surprisingly, we found that each MCS is useful for
detecting the local distinguishability of GBSs. More exactly, given a set
$\mathcal{S}$ of GBSs, if there exists some MCS $\mathcal{C}$ such that the
difference set of $\mathcal{S}$ is disjoint with or contains in $\mathcal{C},$
then the set $\mathcal{S}$ can be one-way distinguishable. This method is
stronger than that in Ref. Yuan21 . This motivates us to find out all the MCSs
of given dimension. Indeed, we presented a complete structure characterization
of MCS in Theorem 3.

However, MCS only gives a sufficient condition for locally distinguishable, it
is not necessary. It is interesting to derive an easy checking condition for
local distinguishability of GBSs which are both sufficient and necessary. In
addition, it is interesting to check whether Fan’s results can be extended to
systems without the assumption on the dimension of local systems. A weaker
form is that: given any integer $l$, do there exist some $D$ (which depends on
$l$) such that if $d\geq D$, then any $l$ GBSs in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ are locally distinguishable? As far as
we known, this problem is only solved for the case $l=3$. We conjecture that
this holds for all other cases.

Note that our method here can be generalized to any maximally entangled basis
whose defining unitary matices $\mathcal{B}$ satisfies: for any
$U,V\in\mathcal{B}$ there exists some $W\in\mathcal{B}$ such that
$U^{\dagger}V\propto W$. The lattice qudits basis Tian15  is such an example.
From their proof, any locally distinguishable set of lattices qudits basis
that can be detected by Ref. Tian15  can be always detected by a MCS of
lattice qudits basis. Therefore, our method can be also seen as a
generalization of theirs. Therefore, it is also interesting to give a complete
characterization of the MCS of lattice qudits basis and study its application
to local discrimination.

###### Acknowledgements.

M.S.L. and Y.L.W. were supported by National Natural Science Foundation of
China under Grant No. (12005092, 11901084, 61773119), the China Postdoctoral
Science Foundation (2020M681996), the Research startup funds of DGUT
(GC300501-103). F.S. was supported by the NSFC under Grants No. 11771419 and
No.12171452, the Anhui Initiative in Quantum Information Technologies under
Grant No. AHY150200, and the National Key Research and Development Program of
China 2020YFA0713100.

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# Balanced Masked and Standard Face Recognition

Delong Qi, Kangli Hu, Weijun Tan*, Qi Yao, Jingfeng Liu
Shenzhen Deepcam Information Technologies, Shenzhen, China
<EMAIL_ADDRESS>
*LinkSprite Technologies, USA
<EMAIL_ADDRESS>
###### Abstract

We present the improved network architecture, data augmentation, and training
strategies for the Webface track and Insightface/Glint360K track of the masked
face recognition challenge of ICCV2021. One of the key goals is to have a
balanced performance of masked and standard face recognition. In order to
prevent the overfitting for the masked face recognition, we control the total
number of masked faces by not more than 10% of the total face recognition in
the training dataset. We propose a few key changes to the face recognition
network including a new stem unit, drop block, face detection and alignment
using YOLO5Face, feature concatenation, a cycle cosine learning rate etc. With
this strategy, we achieve good and balanced performance for both masked and
standard face recognition.

##  1 Introduction

Face recognition, as a method to identify or verify a person’s identity using
a face image, has made tremendous progresses in recent years, thanks to deep
learning, particularly deep CNN. However, there are still some challenges, one
of which is low quality of images. The factors that cause low quality images
include pose, blur, occlusion, illumination etc. As the outbreak of the
disastrous COVID-19 and inspired by the motivation to prevent virus spreading,
face recognition has become one way to trace COVID-19 patients and close
contacts [25]. After a patient is confirmed and his identity is recognized,
safety measures can be taken to control the virus spreading. Face mask, as an
effective way to prevent the virus spreading, poses a new challenge to
traditional face recognition. Since the mask covers a large part of face where
abundant features are present, traditional face recognition algorithm may not
work effectively. This drives a need to understand how face recognition
algorithm deals with masked faces, as a special case of occlusion.

To cope with the challenge arising from wearing masks, it is crucial to
improve the existing face recognition algorithms. Even though some commercial
providers have claimed the availability of face recognition algorithms capable
of handling face masks, and an increasing number of research publications have
surfaced, there is still no publicly available masked face recognition
benchmark. The ICCV2021-MFR Workshop organises Masked Face Recognition (MFR)
challenge [7, 17, 32, 33] and the goal is to bench-marking face recognition
algorithms for masked faces.

In this paper we present our solution for this MFR challenge [17]. We notice
that one challenge is how to balance the performance of the masked and
standard face recognition. At the beginning of the challenge, the MFR error
rate was used as the ranking metric. To achieve best MFR performance, people
tend to put a lot of masked face images in their training dataset. This causes
a overfitting problem such that the performance of the MFR is very good, but
that of the standard face recognition is getting worse. Later the organizer
realized this problem and changed there ranking metric to a mixture of the
error rates of the MFR and standard face recognition. In our solution, we
realize this problem from the beginning and keep the balance of the MFR and
standard face recognition. Our strategy is to control the percentage of the
masked face image, by no more than 10% of the total number of face images in
the training dataset.

Other than this key strategy, we propose a few changes to the backbone network
architecture. We use the ArcFace [8] as our baseline with Resnet [14] as the
backbone. We test the ArcFace loss [8], cosFace loss [26], and other loss
functions. We design a new stem unit, and add the drop block [12] in the last
two layers of the Resnet network. We also propose to use YOLO5Face [22] to do
face alignment, and to concatenate features from multiple models to improve
the recognition accuracy.

As of August 3 2021, we achieve the All-Masked MFR metric 0.1056 in the
Webface track and TAR@Mask 0.84327, TAR@Mask-All 0.92702 in the
Insightface/Glint360k track.

##  2 Related Work

Face recognition has been widely studied and deployed in practical
applications in recent years. For a latest review, please refer to [27]. For
face recognition in the wild, more and more attentions are on poor quality
face images by pose, blur, occlusion, illumination etc. Some examples are [28,
24, 19].

MFR is a challenging task since a large part of the face is occluded, so
abundant features on mouth, nose, and lower cheek are all lost. As a result,
the face recognition algorithm has to focus on the eyes, ears, upper cheek,
forehead to identity a person’s identity.

The work [2, 11] provide a tool to generate masked face images as a data
augmentation for MFR.

In [3], normal face detector is used to detect masked face, and a pretrained
VGGFace2 [6] is used to extract features for face recognition. They treat the
MFR as a normal face recognition problem.

In [13], the author first removes the masked region, then use pre-trained
networks to extract the features of the eyes and forehead, and finally use a
bag of words and a MLP to do the classification.

In [21] a Resnet50 [14] and ArcFace [8] are used. They introduce an
probability of mask usage, and add a mask usage classification loss to the
ArcFace loss.

In [10], the authors collect some MFR dataset, and propose a latent part
detection to locate the latent facial part which is robust to mask wearing,
and the latent part is further used to extract discriminative features.

In [18] the authors explore the Convolutional Block Attention Module in a
Resnet50 network. They also suggest removing the masked region helps the MFR.
They test their algorithm on a variety of MFR datasets.

##  3 Methods

All major changes are described in this section. These include the data
augmentation for masked face images, a new stem unit, a DropBlock, YOLO5Face
[22] detection and alignment, and feature concatenation.

###  3.1 Data Augmentation for Masked Face Images

Generation of masked face images is one type of data augmentation methods.
However, for the importance of a balanced masked and standard face
recognition, we put it here as a separate subsection. In our early study, we
find that in order to get the best performance of MFR, people is temped to use
as many as possible masked face images in the training dataset. As a result,
the face recognition model is over-fitted for MFR, but does not work well on
normal face recognition.

Therefore, we control the balance of the MFR and standard face recognition by
controlling the percentage of masked face images in the training dataset.
After some experiment we find that 10% is a good trade-off.

In stead of using public available MFR dataset, we use a synthetic tool to
generate MFR dataset. The tool we use is the FMA-3D [11]. This tool can
generate masked face images online or offline. We find that generating masked
face images online affects the throughout of the training substantially, we so
decide to generate masked face images offline. Shown in Figure 1 are some
examples of the generated masked face images.

Figure 1: Example of synthetically generated masked face images.

###  3.2 Network Architecture

We use the ArcFace framework [8] with Resnet [14] as backbone. The network
architecture is shown in Figure 2. All changed blocks are highlighted in
green. Figure 2 (a) is the overall architecture, where YOLO5Face [22] is used
only on the test data. The training data provided by Webface [33] or Glint360k
[1] are used with data augmentation. The final feature map is flatten, then a
512-neuron full-connection (FC) layer is used to generate the feature. Lastly
another FC layer is used for classification. Figure 2 (b) is the feature
concatenation of two models, which will be described later.

Figure 2: Network architecture. All key changes are highlighted in green.

###  3.3 Stem Unit

In face recognition backbone likes ResNet [14] there is a stem unit - a
component whose goal is to quickly reduce the input resolution. Typically, the
input is transformed from [b, c, h, w] to [b, c*2, h//2, w//2], where b,c,h,w
are batch,channel,height,and width of the input images. ResNet50 [14] stem is
comprised of a stride-2 conv7x7 followed by a max pooling layer which reduces
the input resolution by a factor of 4. The ResNet50-D [15] stem design is more
elaborate - the conv7x7 is replaced by three conv3x3 layers. The new
ResNet50-D stem design improves accuracy, but at a cost of lowering the
training throughput. In the TResNet [23], the stem unit is called a
DepthToSpace transformation layer, which aims to replace the traditional
convolution-based downscaling unit by a fast and seamless layer, with little
information loss as possible. In YOLOv5 [29] the author wants to reduce the
cost of Conv2d computation and use a tensor reshaping to reduce space
resolution and increase the depth. It has been shown that this focus layer has
a good impact on the YOLOv5 performance.

Inspired by the stem unit in Resnet50 [14] and the focus layer in the YOLOv5
[29], we design a new stem unit of down-sampling rate 2 similar to the focus
layer in YOLOv5. There are two parallel stem units C1 and C2, whose output are
added up. In the first unit C1, the input image is first passed to an average-
pooling layer with kernel size=2 and stride=2 to reduce the space resolution.
After that, a Conv layer with kernel size=2, stride=2 is applied. In the
second unit C2, the input image is 2x down-sampled on the dimensions h and w.
The even and odd index on the dimension h and w are mutually combined to form
four images. These four images are concatenated to form an image whose
resolution is 2x reduced but the depth is 4x increased. This image is sent to
a Conv layer whose kernel=3x3, stride=2, padding=1. In both stem units, a
batchNorm layer and a PReLU layer are used after the Conv layer. The
architecture of this stem unit is shown in Figure 3.

Figure 3: Architecture of the stem unit. The parameters are kernel (k), stride
(s), and padding (p).

###  3.4 DropBlock

In deep learning, dropout is a widely used regularization method. However, it
is often less effective for convolutional layers. This is perhaps due to the
fact that activation units in convolutional layers are spatially correlated so
information can still flow through convolutional networks despite dropout.
Thus a structured form of dropout is needed to regularize convolutional
networks. In [12], A DropBlock, where units in a contiguous region of a
feature map are dropped, is proposed. It is found that applying the DropBlock
increases the accuracy.

We add this DropBlock into our face recognition network. This block is added
after the Conv layer and the skip connection in the last two layers of the
Resnet [14], as shown in Figure 1 (a).

###  3.5 Face Alignment using YOLO5Face

Face images are aligned using landmarks before they are sent to face
recognition. Before RetinaFace becomes available, MTCNN [30] is widely used
for face image alignment. In Webface dataset, the RetinaFace [9] is used as
the standard for face alignment. YOLO5Face [22] is a delicate redesigned face
detector from the YOLOv5 [29] object detector. In addition to the bounding box
and confidence score, it also outputs five-point landmark, similar to MTCNN
and RetinaFace. We use the YOLO5Face for face image alignment in some of our
studies.

Some qualitative examples are shown in Figure 4. The landmarks from the
YOLO5Face [22] and RetinaFace [9] on a set of face images with large pose are
shown. It is not hard to see that the landmarks of YOLO5Face are better.
Please note that the some of the detection scores are small because these
faces have large poses.

Figure 4: Examples of landmark, first row is from RetinaFace [9], and second
row from the YOLO5Face [22].

###  3.6 Feature Concatenation

The left-right face flipping has been shown to improve the face recognition
performance. It is not allowed to use in this MFR competition. We borrow this
idea to use extracted feature vectors from multiple models. We choose two best
models, extract their feature vectors, and concatenate them as the feature
vector for the face recognition, as shown in Figure 2 (b). This is like an
model ensemble concept. Our experiment shows that the performance can be
improved.

##  4 Experiments

###  4.1 Dataset

In early time of the first phase, the Glint360K dataset [1] is used as the
preliminary training dataset. After we meet the required baseline performance,
30% of the WebFace260M [33] are released however are not used in our training.
Other than the masked face images already included in the dataset, more masked
face images are generated, as described in Section 3.5. The total number of
masked face images is not exceeding 10% of the total number of face images.

###  4.2 Implementation details

The ArcFace framework [8] in Pytorch is used in our study. Input image size is
set to 112x112. We use the Resnet34 [14] as our backbone. We expect larger
models like Resnet50, Resnet101 [14] will give better performance. But for
faster training speed, we use Resnet34 in most of our studies.

Other data augmentation methods we use include random cropping, random
flipping, and the more complex Albumentations [5], which include Affine
transforms (scaling, translation, rotation, distortion), noise, blurring,
brightness and contrast adjustment.

Our model is trained on four Nvidia RTX GPUs. The training runs 24 epochs with
batch size 512. We use the cyclic cosine decay [20, 4] for the learning rate
strategy as shown in Figure 5. The initial learning rate is 0.1, the momentum
is 0.9, and the weight decay is 5E-4. We use 0.1 epoch as warm-up, the
learning rate is reduced to the minimum decay learning rate in 16 epochs.

Figure 5: Cycle cosine decay learning rate, where one epoch is around 16100
steps.

###  4.3 Ablation Study

In the Insightface/Glint360k track, we use Resnet34 [14] backbone for its fast
training speed. We use this track as our ablation study tool in addition to
submission to the competition leader board.

Technique | MFR-Mask | MFR-All | Time
---|---|---|---
Baseline | 65.10 | 83.02 | 3.285
\+ Cycle Cosine LR | 67.38 | 83.08 | 3.285
\+ Data Aug | 77.20 | 82.65 | 3.285
\+ Stem Unit | 77.13 | 82.45 | 3.165
\+ SE | 78.37 | 83.12 | 4.294
\+ DropBlock | 79.21 | 84.22 | 4.294
\+ EMA | 79.35 | 84.64 | 4.294
Table 1: Ablation study on the Insightface/Glint360k track with backbone
Resnet34 [14]. Time refers to the average inference time of the model.

We test a variety of techniques in this study in an incremental manner, and
the results are listed in Table 1. In the table, the cyclic cosine learning
rate is described in Section 4.2; data augmentation refers to random cropping
plus the Albumentations augmentation with random horizontal flipping,
blurring, Gaussian blurring, motion blurring, RGB shift, and image
compression, where the probability is 0.5 for the horizontal flipping, and
0.05 for all others. We also test the Squeeze-and-Excitation (SE) network
[16], and the Exponential Moving Average (EMA) gradient update.

We see that the cycle cosine learning rate brings a 2.2% performance
improvement; the data augmentation brings the biggest improvement of nearly
10%; all other techniques brings 0.1-0.2% improvement except for the stem
unit. The stem unit does not improve the accuracy, but it improves the
inference time.

Face alignment with YOLO5Face [22]. We do this ablation study on the Webface
track. We use the best configuration from the previous ablation study as
baseline. Instead of Resnet34, Resnet124 is used as backbone. The results are
listed in Table 2. First we train two models $R124\\_1$ and $R124\\_2$, then
we concatenate the features from them and form the third model. In these three
models, we use the RetinaFace [9] face detector and alignment. Then we keep
using the concatenation model, and replace the RetinaFace with the YOLO5Face
[22]. We test two models, a small-sized model YOLO5-S, and a medium sized
model YOLO5-M. Both models give better performance than the third model. This
demonstrates that the YOLO5Face gives better face detection and landmark
prediction as we qualitatively show in Figure 3.

Model | FaceDetect | All-Masked | Dim
---|---|---|---
R124_1 | Retina | 0.1132 | 512
R124_2 | Retina | 0.1142 | 512
$R124\\_1\|R124\\_2$ | Retina | 0.1065 | 1024
$R124\\_1\|R124\\_2$ | YOLO5-S | 0.1061 | 1024
$R124\\_1\|R124\\_2$ | YOLO5-M | 0.1056 | 1024
Table 2: Ablation study on the Webface track with backbone Resnet124 [14].
Retina refers to the RetinaFace [31]. Two YOLO5Face models [22], a small model
YOLO5-S, and a medium model YOLO5-M are used. The symbol $\|$ denotes feature
concatenation.

###  4.4 Benchmark Results

To avoid overfitting on masked/standard face recognition, the Webface track
organizer decide to revise the formula for calculating all three MFR metrics
in the leaderboard. New MFR metrics are designed to show a weighted sum to
consider both masked and standard faces at the same time. The new formulas are
shown below,

  * •

New All-Masked (MFR) = 0.25 * Old All-Masked (MFR) + 0.75 * All (SFR).

  * •

New Wild-Masked (MFR) = 0.25 * Old Wild-Masked (MFR) + 0.75 * Wild (SFR).

  * •

New Controlled-Masked (MFR) = 0.25 * Old Controlled-Masked (MFR) + 0.75 *
Controlled (SFR).

As of August 3, 2021, our best results on the two tracks are listed in Table
3.

Track | Metric | Time
---|---|---
Insightface/Glint260k | (84.327, 92.702) | 17.835
Webface | (0.1056, 0.0187) | 996.00
Table 3: Benchmark results. For the Insight/Glint360 track, the metrics are
(Mask, MR-All), and the time is the inference time in millisecond. For the
Webface track, the metrics are (All-Masked MFR, All SFR), and the time is
total time in millisecond.

##  5 Conclusions

In this paper we present a solution for the ICCV2021 MFR challenge [17]. Our
solution has a good balance of the masked MFR and the standard face
recognition.

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