# Local unitary equivalence of arbitrary-dimensional multipartite quantum
states

Qing Zhou  Yi-Zheng Zhen  Xin-Yu Xu  Hefei National Research Center for
Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei 230026, China  CAS Center
for Excellence in Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei 230026, China  Shuai Zhao  Hefei
National Research Center for Physical Sciences at the Microscale and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
School of Cyberspace, Hangzhou Dianzi University, Hangzhou 310018, China  Wen-
Li Yang<EMAIL_ADDRESS>Institute of Modern Physics, Northwest University,
Xi’an 710069, China  Shao-Ming Fei<EMAIL_ADDRESS>School of Mathematical
Sciences, Capital Normal University, Beijing 100048, China  Li Li
<EMAIL_ADDRESS>Nai-Le Liu<EMAIL_ADDRESS>Kai Chen<EMAIL_ADDRESS>Hefei National Research Center for Physical Sciences at the Microscale and
School of Physical Sciences, University of Science and Technology of China,
Hefei 230026, China  CAS Center for Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei 230026,
China  Hefei National Laboratory, University of Science and Technology of
China, Hefei 230088, China

(Feb 21, 2024)

###### Abstract

Local unitary equivalence is an important ingredient for quantifying and
classifying entanglement. Verifying whether or not two quantum states are
local unitary equivalent is a crucial problem, where only the case of
multipartite pure states is solved. For mixed states, however, the
verification of local unitary equivalence is still a challenging problem. In
this paper, based on the coefficient matrices of generalized Bloch
representations of quantum states, we find a variety of local unitary
invariants for arbitrary-dimensional bipartite quantum states. These
invariants are operational and can be used as necessary conditions for
verifying the local unitary equivalence of two quantum states. Furthermore, we
extend the construction to the arbitrary-dimensional multipartite case. We
finally apply these invariants to estimate concurrence, a vital entanglement
measure, showing the practicability of local unitary invariants in
characterizing entanglement.

##  I Introduction

Entanglement as an important quantum resource plays a crucial role in quantum
information technologies Nielsen and Chuang (2000), such as quantum
teleportation Bennett  _et al._ (1993); Bouwmeester  _et al._ (1997), quantum
secret sharing Horodecki  _et al._ (2009), quantum cryptography Ekert (1991)
and quantum repeaters Briegel  _et al._ (1998). However, the quantification
and classification of entanglement remain a challenging problem. The local
unitary (LU) equivalence of the quantum state is an effective ingredient to
characterize entanglement, the amount of which is invariant under LU
transformations Dür  _et al._ (2000). The LU invariants of a quantum state can
thus be adopted to detect entanglement or entanglement classes and to derive,
e.g., entanglement measures Chen  _et al._ (2005a, b) and separability
criteria Chen and Wu (2002, 2003); Rudolph (2003); Albeverio  _et al._
(2003a). If one can completely solve the problem of the LU equivalence class,
there will be a major breakthrough in understanding and making good use of
entanglement resource.

In recent years, much effort has been devoted to characterizing and
understanding the LU equivalence of quantum states Albeverio  _et al._ (2003b,
2005a); Sun  _et al._ (2005); Zhang  _et al._ (2013); Li  _et al._ (2014);
Albeverio  _et al._ (2005b, c, 2007); Zhang  _et al._ (2013); Martins (2015);
Linden  _et al._ (1999a); Albeverio  _et al._ (2005a); Sun  _et al._ (2017);
Cui  _et al._ (2017); Jing  _et al._ (2015); Li and Qiao (2013); Jing  _et
al._ (2014); Zhang  _et al._ (2015). If two quantum states are LU equivalent,
there must be some invariants under LU transformation. Thus, the basic method
to determine whether or not two quantum states are LU equivalent is by
checking the LU invariants. Up to now, many novel methods for computing the LU
invariants have been developed Grassl  _et al._ (1998); Rains (2000). For pure
states, an elegant result of the LU equivalence problem for multipartite qubit
states has been obtained by Kraus Kraus (2010a, b). The problem for arbitrary-
dimensional pure states case has also been solved in Ref. Liu  _et al._
(2012). However, for mixed cases, only two-qubit states and some special cases
have been solved. For instance, a set of 18 polynomial invariants is proposed
to completely determine whether a two-qubit mixed quantum state is LU
equivalent to another state Makhlin (2002). As for arbitrary dimensional
bipartite quantum states, only a set of LU invariants for nondegenerate
density matrices is known Zhou  _et al._ (2012). In general, it is extremely
difficult to present a sufficient and necessary condition for certifying the
LU equivalence of two arbitrary quantum states.

In this paper, we focus on building LU invariants, which are the necessary
condition for the LU equivalence for arbitrary-dimensional multipartite mixed
quantum states. The previous results for LU problems are to calculate LU
invariants from the density matrices directly, which are inconvenient to deal
with the degeneracy problem. Here, we derive LU invariants based on the
singular value decomposition (SVD) of coefficient matrices under the
generalized Bloch representations of quantum states. In this manner, the
difficulty due to degeneration of the density matrix can be avoided. Moreover,
our method for deriving LU invariants of bipartite states can be conveniently
extended to arbitrary dimensional multipartite cases. As an example, we show
its application in the case of three-qudit states. Furthermore, the
concurrence as an entanglement measure Rungta  _et al._ (2001); Mintert  _et
al._ (2005, 2004); Uhlmann (2000); Wootters (1998); Piotr Badziag and Piotr
Deuar and Michał Horodecki and Paweł Horodecki and Ryszard Horodecki (2002)
can be derived from our LU invariants. A quadratic relation is presented here
between the LU invariants and the concurrence, which reveals the significance
and practicability of LU invariants in exploring entanglement.

The paper is organized as follows. In Sec. II, we derive necessary conditions
for the LU equivalence of any two bipartite mixed quantum states, from
generalized Bloch representations of the density matrices. In Sec. III, we
extend our methods to the multipartite case and study the LU equivalence for
three-qudit states. In Sec. IV, by analytical calculations, the relation
between the LU invariants and the concurrence is explored here for pure
bipartite states. Finally, we conclude with our results in Sec. V.

##  II Two-qudit case

In this section, based on the generalized Bloch representations of density
matrices of arbitrary dimensional bipartite quantum states Eltschka and
Siewert (2020); Appel  _et al._ (2020), we present the necessary conditions
for LU equivalence in matrix form and then try to find the operational
invariants under LU transformations.

Let ${\cal H}_{1}$ and ${\cal H}_{2}$ be two complex Hilbert spaces with ${\rm
dim}({\cal H}_{1})={\rm dim}({\cal H}_{2})=N$, and $\rho$ be a density matrix
of a bipartite quantum mixed state acting on ${\cal H}_{1}\otimes{\cal
H}_{2}$. Without loss of generality, $\rho$ can be expressed as:

$\rho=\frac{1}{N^{2}}I\otimes I+\sum_{i}R_{i}\lambda_{i}^{1}\otimes
I+\sum_{j}S_{j}I\otimes\lambda_{j}^{2}+\sum_{ij}T_{ij}\lambda_{i}^{1}\otimes\lambda_{j}^{2},$
(1)

where $T_{ij}=\frac{1}{4}Tr(\rho\lambda_{i}^{1}\otimes\lambda_{j}^{2})$,
$S_{j}=\frac{1}{2N}Tr(\rho I\otimes\lambda_{j}^{2})$,
$R_{i}=\frac{1}{2N}Tr(\rho\lambda_{i}^{1}\otimes I)$, $i,j=1,2,...,N^{2}-1$.
Here, $\lambda_{i}$ are the generators of $SU(N)$ with
$Tr(\lambda_{i}\lambda_{j})=2\delta_{ij}$. The $T_{ij},S_{j},R_{i}$ are all
real coefficients, as the state $\rho$ and $SU(N)$’s generators are all
Hermitian.

The two-qudit states $\rho$ and another two-qudit state
$\rho^{\prime}=\frac{1}{N^{2}}I\otimes
I+\sum_{i}R_{i}^{\prime}\lambda_{i}^{1}\otimes
I+\sum_{j}S_{j}^{\prime}I\otimes\lambda_{j}^{2}+\sum_{ij}T_{ij}^{\prime}\lambda_{i}^{1}\otimes\lambda_{j}^{2}$
are called LU equivalent, if there exist unitary operators $U_{1},U_{2}\in
SU(N)$, such that

$\rho^{\prime}=(U_{1}\otimes U_{2})\rho(U_{1}\otimes U_{2})^{{\dagger}}.$ (2)

Since the coefficients $\\{R_{i},S_{j},T_{ij}\\}_{i,j}$ capture complete
information of $\rho$, one can investigate the LU equivalence based on the
“feature matrix” $M(\rho)$ defined as
$M(\rho)={\left[\begin{array}[]{cc}1&S^{t}\\\ R&T\end{array}\right]}$. Here,
the $S=(S_{j})$ and $R=(R_{i})$ are $N^{2}-1$ dimensional column vectors and
$T=(T_{ij})$ is a $(N^{2}-1)\times(N^{2}-1)$ matrix. Similarly, one can define
$M(\rho^{\prime})={\left[\begin{array}[]{cc}1&S^{\prime t}\\\
R^{\prime}&T^{\prime}\end{array}\right]}$.

It is shown in Ref. Cui  _et al._ (2017) that if two bipartite mixed states
$\rho$ and $\rho^{\prime}$ are LU equivalent, then there are $O_{1},O_{2}\in
SO(N^{2}-1)$ such that
$R^{\prime}=O_{1}^{t}R,S^{\prime}=O_{2}^{t}S,T^{\prime}=O_{1}^{t}TO_{2}$. Then
the following Lemma can be presented.

######  Lemma 1.

If two bipartite mixed states $\rho$ and $\rho^{\prime}$ are LU equivalent,
there are $O_{1},O_{2}\in SO(N^{2}-1)$, such that $M(\rho^{\prime})={\rm
diag}\\{1,O_{1}^{t}\\}M(\rho){\rm diag}\\{1,O_{2}\\}$, where $t$ stands for
transposition.

From Lemma 1 we obtain the transformation of the quantum state’s feature
matrix under LU transformations. As the above Lemma is based on the
generalized Bloch representations, it will be restricted by the properties of
Bloch-vector space Kimura (2003). Hence, the special orthogonal matrices $O\in
SO(N^{2}-1)$ acting on the coefficient vectors of $\rho$ may transform $\rho$
into an unphysical form that does not satisfy the definition of a density
matrix. Hence, the Lemma 1 is a necessary but not sufficient condition for the
LU equivalence of quantum states when $N>2$.

To make the conditions for verifying LU equivalence operational, we will find
further the LU invariants based on Lemma 1. First, let us focus on matrix $T$
and make a SVD of it. As $T$ is a real matrix, one can obtain a real
orthogonal matrix $P$ and $Q$ with respect to the corresponding singular
values, such that $T=P\Sigma Q^{t},$ where $\Sigma={\rm
diag}\\{\mu_{1},\mu_{2},...,\mu_{d},0,...,0\\}.$ Here, the singular values are
in decreasing order, and $d$ is the $rank$ of matrix $T$. To deal with the
degenerate singular values, one can divide $\Sigma$ into a block matrix by
putting the same singular values in a submatrix, such that $\Sigma={\rm
diag}\\{\Sigma_{1},\Sigma_{2},...,\Sigma_{n},\Sigma_{n+1}\\}$, where $n+1$
means that there are $n+1$ different singular values. The $\Sigma_{m}$ is an
$a_{m}\times a_{m}$ matrix with $a_{1}+a_{2}+\dots+a_{n+1}=N^{2}-1$. For
$m\in\\{1,2,\dots,n\\}$, there is $\Sigma_{m}\propto I$. When it comes to the
case of $m=n+1$, one should consider whether there exists a zero singular
value of matrix $T$. If $T$ is a full rank matrix, the $\Sigma_{n+1}$ is also
proportional to the identity matrix, otherwise $\Sigma_{n+1}$ is a zero
matrix. Thus, the matrix $M$ can be rewritten as
$M(\rho)={\left[\begin{array}[]{cc}1&\\\
&P\end{array}\right]}{\left[\begin{array}[]{cc}1&\widetilde{S}^{t}\\\
\widetilde{R}&\Sigma\end{array}\right]}{\left[\begin{array}[]{cc}1&\\\
&Q^{t}\end{array}\right]},$ where $\widetilde{R}\equiv
P^{t}R,(\widetilde{S})^{t}\equiv S^{t}Q$. Let $\pi_{m}(R)$ denote the
projection of the $m$-th part of vector $R$ and and $\|*\|$ be the Euclid norm
Horn  _et al._ (1990). Here, each part $\pi_{m}(R)$ is an $a_{m}$ dimensional
vector, $m=1,2,\cdots,n+1$. With these notations, we present our main result
as the following theorem.

######  Theorem 1.

If two two-qudit states $\rho$ and $\rho^{\prime}$ are LU equivalent, one has
the following invariants:

$\begin{split}&(1)\Sigma,{\rm det}(T),{\rm det}(M(\rho)),\\\
&(2)\|\pi_{m}(\widetilde{R})\|,\|\pi_{m}(\widetilde{S})\|,m=1,2,...,n+1,\\\
&(3)\pi_{m}(\widetilde{S})^{t}\pi_{m}(\widetilde{R}),m=1,2,...,n^{\prime},\end{split}$
(3)

where $n^{\prime}$ is determined by matrix $T$. If $T$ has no zero singular
values, one has $n^{\prime}=n+1$, otherwise, $n^{\prime}=n$.

###### Proof.

From the Lemma 1, it is easy to get ${\rm det}(T^{\prime})={\rm det}(T)$ and
${\rm det}(M(\rho^{\prime}))={\rm det}(M(\rho))$, as ${\rm
det}(O_{1}^{t})={\rm det}(O_{2})={\rm det}({\rm diag}\\{1,O_{1}^{t}\\})={\rm
det}({\rm diag}\\{1,O_{2}\\})=1$ with $O_{1},O_{2}\in SO(N^{2}-1)$. Similarly,
one can perform a SVD of matrices $T^{\prime}$ with
$T^{\prime}=P^{\prime}\Sigma^{\prime}Q^{\prime t}$. If $\rho$ and
$\rho^{\prime}$ are LU equivalent, from Lemma 1 one has
$T^{\prime}=O_{1}^{t}P\Sigma Q^{t}O_{2}$. As $O_{1}^{t}P$ and $Q^{t}O_{2}$ are
orthogonal matrices, one has $\Sigma^{\prime}=\Sigma$. To prove other
invariants, the degeneracy of matrix $T$ should be considered. If matrix $T$
is degenerate, $P$ and $Q$ are not unique. Firstly, from Lemma 1, one can fix
$P^{\prime},Q^{\prime}$ as $\ P^{\prime}=O_{1}^{t}P,\ Q^{\prime}=O_{2}^{t}Q.$
By these relations, one has

$\widetilde{R}^{\prime}=P^{\prime
t}R^{\prime}=P^{t}O_{1}O_{1}^{t}R=\widetilde{R},\ \widetilde{S}^{\prime
t}=S^{\prime t}Q^{\prime}=\widetilde{S}^{t}.$ (4)

Next, let us discuss the general case when $T$ has zero singular values. For
$m=1,2,...,n$, as $\Sigma_{m}\propto I$, the $\Sigma_{m}$-preserving
operations are orthogonal matrices $G_{m}$ with
$G_{m}\Sigma_{m}G_{m}^{t}=\Sigma_{m}$. For $m=n+1$, the orthogonal matrices
$G_{n+1}$, $\mathcal{G}_{n+1}$ are $\Sigma_{m}$-preserving operations due to
$G_{n+1}\Sigma_{n+1}\mathcal{G}^{t}_{n+1}=\Sigma_{n+1}$. Here, there is no
need for the orthogonal matrices $G_{n+1}$ and $\mathcal{G}_{n+1}$ to be the
same, as $\Sigma_{n+1}$ is the zero matrix. Thus, one can construct
$\widetilde{O}={\rm diag}\\{G_{1},G_{2},...,G_{n},G_{n+1}\\}$ and
$\widetilde{\mathcal{O}}={\rm
diag}\\{G_{1},G_{2},...,G_{n},\mathcal{G}_{n+1}\\}$, such that
$\widetilde{O}\Sigma\widetilde{\mathcal{O}}^{t}=\Sigma$, where
$\widetilde{O},\ \widetilde{\mathcal{O}}$ are orthogonal matrices. For a
general SVD of matrix $T$, the feature matrix can be written as

$M(\rho)={\left[\begin{array}[]{cc}1&\\\
&P\widetilde{O}\end{array}\right]}{\left[\begin{array}[]{cc}1&\widetilde{S}^{\S
t}\\\
\widetilde{R}^{\S}&\Sigma\end{array}\right]}{\left[\begin{array}[]{cc}1&\\\
&\widetilde{\mathcal{O}}^{t}Q^{t}\end{array}\right]},$ (5)

where $\widetilde{R}^{\S}\equiv\widetilde{O}^{t}\widetilde{R},\
\widetilde{S}^{\S t}\equiv\widetilde{S}^{t}\widetilde{\mathcal{O}}$.
Similarly, for state $\rho^{\prime}$, one has
$\widetilde{R}^{\prime\S}\equiv\widetilde{O}^{\prime
t}\widetilde{R}^{\prime},\ \widetilde{S}^{\prime\S
t}\equiv\widetilde{S}^{\prime t}\widetilde{\mathcal{O}^{\prime}}$, with
orthogonal matrices $\widetilde{O^{\prime}}={\rm
diag}\\{G^{\prime}_{1},G^{\prime}_{2},...,G^{\prime}_{n},G^{\prime}_{n+1}\\}$
and $\widetilde{\mathcal{O}^{\prime}}={\rm
diag}\\{G^{\prime}_{1},G^{\prime}_{2},...,G^{\prime}_{n},\mathcal{G}^{\prime}_{n+1}\\}$.
From Eq. (4), one has
$\pi_{m}(\widetilde{R}^{\prime})=\pi_{m}(\widetilde{R}),\
\pi_{m}(\widetilde{S}^{\prime})=\pi_{m}(\widetilde{S}).$ Then, from the
definition of $\widetilde{R}^{\S},\widetilde{S}^{\S}$, it is easy to check
that
$\|\pi_{m}(\widetilde{R}^{\S})\|=\|\pi_{m}(\widetilde{R}^{\prime\S})\|,\|\pi_{m}(\widetilde{S}^{\S})\|=\|\pi_{m}(\widetilde{S}^{\prime\S})\|$,
where $m=1,2,\dots,n+1$. Furthermore, for $m=1,2,\dots,n$, one can also obtain
$\pi_{m}(\widetilde{S}^{\S})^{t}\pi_{m}(\widetilde{R}^{\S})=\pi_{m}(\widetilde{S}^{\prime\S})^{t}\pi_{m}(\widetilde{R}^{\prime\S})$
by direct calculation. Moreover, if $T$ has no zero singular values, the
$\Sigma_{n+1}$ is not a zero matrix. In this case, one has
$\mathcal{G}_{n+1}=G_{n+1}$ and then
$\pi_{n+1}(\widetilde{S}^{\S})^{t}\pi_{n+1}(\widetilde{R}^{\S})=\pi_{n+1}(\widetilde{S}^{\prime\S})^{t}\pi_{n+1}(\widetilde{R}^{\prime\S})$.
Without loss of generality, we can erase the notation $\S$ and summarize the
results in Theorem 1. ∎

From the Theorem 1, one can verify that two two-qudit states are not LU
equivalent, if the invariants shown in the Eq. (3) are not equal for two
states. Compared with the existing results, the invariants in the Theorem 1
covers the one of Ref. Cui  _et al._ (2017) and have advantages in verifying
the LU equivalence of high-dimensional mixed states.

######  Example 1.

By introducing certain noise into a nonmaximally entangled two-qutrit state
$|\psi\rangle=\frac{\sqrt{2}}{4}|00\rangle+\frac{\sqrt{2}}{4}|11\rangle+\frac{\sqrt{3}}{2}|22\rangle$,
one has ${\rho}=q|\psi\rangle\langle\psi|+\frac{1-q}{6}(|01\rangle\langle
01|+|10\rangle\langle 10|+|02\rangle\langle 02|+|20\rangle\langle
20|+|12\rangle\langle 12|+|21\rangle\langle 21|)$. Assuming $q=\frac{4}{17}$,
the coefficients of Bloch representation can be obtained as
$R=S=(0,0,0,0,0,0,0,-5\sqrt{3}/306)$ and

$T=\begin{bmatrix}\frac{1}{68}&0&0&0&0&0&0&0\\\
0&\frac{\sqrt{6}}{68}&0&0&0&0&0&0\\\ 0&0&-\frac{1}{68}&0&0&0&0&0\\\
0&0&0&-\frac{5}{102}&0&0&0&0\\\ 0&0&0&0&\frac{\sqrt{6}}{68}&0&0&0\\\
0&0&0&0&0&-\frac{\sqrt{6}}{68}&0&0\\\ 0&0&0&0&0&0&-\frac{\sqrt{6}}{68}&0\\\
0&0&0&0&0&0&0&0\\\ \end{bmatrix}.$

Now, one can construct another states ${\rho}^{\prime}$ with coefficients
$R^{\prime}=R,\ S^{\prime}=S,\ T^{\prime}=-T$.

In this case, as the LU invariants in Ref. Cui  _et al._ (2017) for the above
two states are all the same, one cannot verify whether or not the two states
$\rho$ and $\rho^{\prime}$ are LU equivalent. However, by our method, one can
check that ${\rm det}(M(\rho))\neq{\rm det}(M(\rho^{\prime}))$, which means
$\rho$ and $\rho^{\prime}$ have different invariants. Hence, $\rho$ and
$\rho^{\prime}$ are not LU equivalent. Our results cover the invariants in
Ref. Cui  _et al._ (2017) and we have an extra invariant ${\rm det}(M(\rho))$.
The invariants derived in Ref. Cui  _et al._ (2017) are based on the orbit of
group, where it cannot naturally lead to the discovery of an invariant such as
${\rm det}(M(\rho))$. However, our construction of invariants is based on a
feature matrix, which binds together the coefficient vectors $R$, $S$ and
coefficient matrix $T$. Moreover, by performing a SVD of matrix $T$, we
explore the fine-grained relation between vectors $R$ and $S$ according to
singular values of $T$. In most of the existing works, they study the
coefficient vectors and coefficient matrix in isolation or bind them by the
orbit of the group. The construction of the feature matrix brings different
points of view to explore the LU invariants, which can give a more refined
characterization of LU equivalence.

By the above discussion, our criteria for verifying the LU equivalence of
arbitrary dimensional mixed states are superior, as they can be applied to
more scenarios.

##  III three-qudit case

To show that the method can be applied to a multipartite case, we firstly
consider the three-qudit system. Then, with a simple extension, the
generalization to the general multipartite case is immediate. The generalized
Bloch representation of a density matrix for a three-qudit state can be
written as

$\begin{split}\rho=&\frac{1}{N^{3}}I^{1}I^{2}I^{3}+\sum_{i}T_{i}^{1}\lambda_{i}^{1}I^{2}I^{3}+\sum_{j}T_{j}^{2}I^{1}\lambda_{j}^{2}I^{3}\\\
&+\sum_{k}T_{k}^{3}I^{1}I^{2}\lambda_{k}^{3}+\sum_{ij}T_{ij}^{12}\lambda_{i}^{1}\lambda_{j}^{2}I^{3}+\sum_{ik}T_{ik}^{13}\lambda_{i}^{1}I^{2}\lambda_{k}^{3}\\\
&+\sum_{jk}T_{jk}^{23}I^{1}\lambda_{j}^{2}\lambda_{k}^{3}+\sum_{ijk}T_{ijk}^{123}\lambda_{i}^{1}\lambda_{j}^{2}\lambda_{k}^{3},\end{split}$
(6)

where $\lambda_{i}^{1},\lambda_{j}^{2},\lambda_{k}^{3}$ are the generators of
$SU(N)$ with $i,j,k=1,2,\dots,N^{2}-1$. Here, the $T_{1}=\\{T_{i}^{1}\\},\
T_{2},\ T_{3}$ are three $N^{2}-1$ dimensional coefficient vectors, the
$T_{12}=\\{T_{ij}^{12}\\},\ T_{13},\ T_{23}$ are three
$(N^{2}-1)\times(N^{2}-1)$ dimensional coefficient matrices and
$T_{123}=\\{T_{ijk}^{123}\\}$ is a coefficient tensor. Here, one can write
$I\otimes I\otimes I$ as $I^{1}I^{2}I^{3}$ by omitting the notation $\otimes$.
Some results in the three-qudit system are similar to the two-qudit case. If
two mixed states $\rho$ and $\rho^{\prime}$ are LU equivalent, there are some
$O_{1},O_{2},O_{3}\in SO(N^{2}-1)$, such that

$\displaystyle{T}_{1}^{\prime}$ $\displaystyle={O}_{1}^{t}{T}_{1},\
{T}_{2}^{\prime}={O}_{2}^{t}{T}_{2},\ {T}_{3}^{\prime}={O}_{3}^{t}{T}_{3},$
(7) $\displaystyle{T}_{12}^{\prime}$
$\displaystyle={O}_{1}^{t}{T}_{12}{O}_{2},\
{T}_{13}^{\prime}={O}_{1}^{t}{T}_{13}{O}_{3},\
{T}_{23}^{\prime}={O}_{2}^{t}{T}_{23}{O}_{3},$
$\displaystyle{T}_{123}^{\prime}$
$\displaystyle=({O}_{1}^{t}\otimes{O}_{2}^{t}\otimes{O}_{3}^{t}){T}_{123}.$

Now, let the elements in $T_{123}$ be arranged as
$(N^{2}-1)\times(N^{2}-1)^{2}$ matrices $T_{1|23},T_{2|13}$, and $T_{3|12}$,
whose details are presented in the Appendix. For example,

$T_{1|23}=\begin{bmatrix}T_{111}&T_{112}&\cdots&T_{1(N^{2}-1)(N^{2}-1)}\\\
T_{211}&T_{212}&\cdots&T_{2(N^{2}-1)(N^{2}-1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
T_{(N^{2}-1)11}&T_{(N^{2}-1)12}&\cdots&T_{(N^{2}-1)(N^{2}-1)(N^{2}-1)}\end{bmatrix},$
(8)

where we let the first subscript be the row index. From Eq. (7), one has

${T}_{\alpha|\beta\gamma}^{\prime}={O}_{\alpha}^{t}{T}_{\alpha|\beta\gamma}{O}_{\beta}\otimes{O}_{\gamma},\
{T}_{\beta\gamma}^{\prime}={O}_{\beta}^{t}{T}_{\beta\gamma}{O}_{\gamma},$ (9)

where $(\alpha,\beta,\gamma)=(1,2,3),(2,1,3),(3,1,2)$. For tripartite case,
the coefficient vectors and matrices can not be put into a single matrix in a
meaningful way. Nevertheless, one can still define some feature matrices as
the ${M}_{\beta\gamma}={\left[\begin{array}[]{cc}1&{T}_{\gamma}^{t}\\\
{T}_{\beta}&{T}_{\beta\gamma}\end{array}\right]}$ and
${M}_{\alpha|\beta\gamma}={\left[\begin{array}[]{cc}1&{T}_{\beta}^{t}\otimes{T}_{\gamma}^{t}\\\
{T}_{\alpha}&{T}_{\alpha|\beta\gamma}\end{array}\right]}.$ Moreover, letting
$T_{\beta\gamma}$ be a column vector $vec(T_{\beta\gamma})$, which is arranged
by all columns of the matrix $T_{\beta\gamma}$, one can construct
${\hat{M}}_{\alpha|\beta\gamma}={\left[\begin{array}[]{cc}1&{\rm
vec}({T}_{\beta\gamma})^{t}\\\
{T}_{\alpha}&{T}_{\alpha|\beta\gamma}\end{array}\right]}$ which has the
relation ${\rm
vec}({T}_{\beta\gamma}^{\prime})={O}_{\beta}^{t}\otimes{O}_{\gamma}^{t}{\rm
vec}({T}_{\beta\gamma})$ from Eq. (9). By performing a SVD of matrices
$T_{\beta\gamma},T_{\alpha|\beta\gamma}$, there are orthogonal matrices
$P_{\beta\gamma},Q_{\beta\gamma},P_{\alpha|\beta\gamma}$, and
$Q_{\alpha|\beta\gamma}$ such that

${T}_{\beta\gamma}={P}_{\beta\gamma}{\Sigma}_{\beta\gamma}{Q}_{\beta\gamma}^{t},{T}_{\alpha|\beta\gamma}={P}_{\alpha|\beta\gamma}{\Sigma}_{\alpha|\beta\gamma}{Q}_{\alpha|\beta\gamma}^{t},$
(10)

where the singular values in $\Sigma_{\beta\gamma}$ and
$\Sigma_{\alpha|\beta\gamma}$ are all in decreasing order. Similar to the two-
qudit case, one can divide the singular-value matrix $\Sigma_{\beta\gamma}$
($\Sigma_{\alpha|\beta\gamma}$) into a block matrix with $n_{\beta\gamma}$
($n_{\alpha|\beta\gamma}$) submatrices, by putting the same singular value
into a single submatrix. Define vectors
$u_{1}=P_{\beta\gamma}^{t}T_{\beta},v_{1}=Q_{\beta\gamma}^{t}T_{\gamma},u_{2}=P_{\alpha|\beta\gamma}^{t}T_{\alpha},v_{2}=Q_{\alpha|\beta\gamma}^{t}(T_{\beta}\otimes
T_{\gamma}),u_{3}=u_{2},v_{3}=Q_{\alpha|\beta\gamma}^{t}vec(T_{\beta\gamma})$
and $n_{1}=n_{\beta\gamma},n_{2}=n_{3}=n_{\alpha|\beta\gamma}$. From Eqs.
(9,10), one has the following theorem.

######  Theorem 2.

If two three-qudit states $\rho$ and $\rho^{\prime}$ are LU equivalent, one
has the following invariants:

$\displaystyle\begin{split}&(1)\Sigma_{\beta\gamma},\Sigma_{\alpha|\beta\gamma},{\rm
det}(T_{\beta\gamma}),{\rm det}(M_{\beta\gamma}),\\\
&(2)\|\pi_{m}(u_{s})\|,\|\pi_{m}(v_{s})\|,m=1,2,\dots,n_{s},\\\
&(3)(\pi_{m}(u_{s}))^{t}(\pi_{m}(v_{s})),m=1,2,\dots,n_{s}^{\prime},\end{split}$

where $s\in\\{1,2,3\\}$, $(\alpha,\beta,\gamma)=(1,2,3),(2,1,3),(3,1,2)$. In
addition, $n_{1}^{\prime}$ is determined by $T_{\beta\gamma}$ and
$n_{2}^{\prime}=n_{3}^{\prime}$ are determined by $T_{\alpha|\beta\gamma}$. If
$T_{\beta\gamma}$ has no zero singular values, one has $n_{1}^{\prime}=n_{1}$,
otherwise, $n_{1}^{\prime}=n_{1}-1$. Similar definitions are applied to
$n_{2}^{\prime}$ and $n_{3}^{\prime}$.

###### Proof.

The proof of Theorem 2 is similar to the one of Theorem 1, as the LU
invariants are obtained from the feature matrices. ∎

Theorem 2 leads to more invariants than existing results, e.g. Ref. Cui  _et
al._ (2017). Therefore, it can be adopted to verify the LU equivalence of the
states that cannot have been verified before. Moreover, the result presents a
general approach to derive the necessary criteria of LU equivalence for the
multipartite qudit case, based on the coefficient vectors and feature matrices
constructed from the generalized Bloch representation. Although, for
simplicity, we discussed the qudit case with the same dimensions, cases with
different dimensions can be studied similarly. Therefore, this method can
always be used to derive the necessary conditions for the LU equivalence of
arbitrary-dimensional multipartite mixed quantum states.

##  IV Invariants and entanglement

Entanglement is one of the most important quantum resources, which is
generally applied into various quantum information processing tasks. However,
the complete characterization of entanglement is extremely challenging. It is
known that a complete set of invariants under LU transformation can be adopted
to depict the properties of entanglement. For pure states, the constructed
invariants can be used to derive the concurrence. Here, we show how our
invariants lead to the estimation of concurrence.

For simplicity, let us consider a two-qudit pure state $\rho$. It is known
that the concurrence of a pure bipartite state is given by
$C_{N}(\rho)=\sqrt{(N/(N-1))(1-Tr(\rho_{1}^{2}))}$ Piotr Badziag and Piotr
Deuar and Michał Horodecki and Paweł Horodecki and Ryszard Horodecki (2002);
Chen  _et al._ (2013), where $\rho_{1}$ is the partial trace of state $\rho$
by tracing out the subsystem ${\cal H}_{2}$. By combining the Eq. (1), one has
$1-C_{N}^{2}(\rho)=\frac{2N^{3}}{N-1}\|R\|^{2}$. From the proof of Theorem 1,
one can obtain that
$\|R\|=\|\widetilde{R}\|=\sqrt{\sum_{m}\|\pi_{m}(\widetilde{R})\|^{2}}$. As
$\rho$ is a pure state, we have
$\|R\|=\|S\|=\|\widetilde{S}\|=\sqrt{\sum_{m}\|\pi_{m}(\widetilde{S})\|^{2}}$.
Therefore, one has

$\sum_{m}\|\pi_{m}(\widetilde{R})\|^{2}=\sum_{m}\|\pi_{m}(\widetilde{S})\|^{2}=\frac{N-1}{2N^{3}}(1-C_{N}^{2}(\rho)).$

By adopting the invariants
$\|\pi_{m}(\widetilde{R})\|,\|\pi_{m}(\widetilde{S})\|$, one can measure the
entanglement of any pure two-qudit states. Therefore, the result simply
reveals that the concurrence can be expressed by our LU invariants in the
scenario of bipartite pure states. The close relation between LU invariants
and entanglement is naturally constructed. Such an intimately deep connection
will promote the research of entanglement measure for mixed states based on LU
invariants.

##  V Conclusion

In this paper, we have presented the necessary criteria for the local unitary
equivalence of arbitrary dimensional multipartite mixed quantum states. By
using the criteria, one can verify the local unitary equivalence of states
with density matrices even having degenerate eigenvalues. It can be easily
extended to the multipartite case, which gives rise to the necessary criteria
for the local unitary equivalence of multipartite states based on feature
matrices. In addition, the relation between the local unitary invariants and
entanglement, is shown with one example of pure bipartite states. In a similar
way, one can find other relations between local unitary invariants and
entanglement measure for any pure multipartite states. We hope that a full
characterization of the entanglement properties for mixed states can be
motivated from the constructed invariants.

In the future, constructing more local unitary invariants to obtain sufficient
and necessary conditions for verifying the local unitary equivalence of
arbitrary-dimensional multipartite mixed quantum states is interesting and
challenging. Meanwhile, more local unitary invariants can promote the
exploration for characterizing entanglement properties.

## Acknowledgments

This work has been supported by the National Natural Science Foundation of
China (Grants No. 62375252, No. 62031024, No. 11874346, No. 12174375, No.
12005091, No. 12075159, No. 12171044), the National Key R$\&$D Program of
China (2019YFA0308700), the Anhui Initiative in Quantum Information
Technologies (AHY060200), the Innovation Program for Quantum Science and
Technology (No. 2021ZD0301100), the Academician Innovation Platform of Hainan
Province, the Research Startup Foundation of Hangzhou Dianzi University (No.
KYS275623071) and Zhejiang Provincial Natural Science Foundation of China
under Grant No. LQ24A050005.

## Appendix A: details for matrices $T_{2|13},T_{3|12}$

Here, the detailed forms of matrices $T_{2|13},T_{3|12}$ are presented as
following

$T_{2|13}=\begin{bmatrix}T_{111}&T_{112}&\cdots&T_{(N^{2}-1)1(N^{2}-1)}\\\
T_{121}&T_{122}&\cdots&T_{(N^{2}-1)2(N^{2}-1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
T_{1(N^{2}-1)1}&T_{1(N^{2}-1)2}&\cdots&T_{(N^{2}-1)(N^{2}-1)(N^{2}-1)}\end{bmatrix},$
(A1)
$T_{3|12}=\begin{bmatrix}T_{111}&T_{121}&\cdots&T_{(N^{2}-1)(N^{2}-1)1}\\\
T_{112}&T_{122}&\cdots&T_{(N^{2}-1)(N^{2}-1)2}\\\
\vdots&\vdots&\ddots&\vdots\\\
T_{11(N^{2}-1)}&T_{12(N^{2}-1)}&\cdots&T_{(N^{2}-1)(N^{2}-1)(N^{2}-1)}\end{bmatrix}.$
(A2)

For any multipartite case, the coefficient matrix
$T_{\alpha|\beta\gamma\dots}$ can be arranged in a similar way, where the
$\alpha$-th subscript is used as the row index.

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