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\title{\bf Separability of density matrices of graphs \\ for multipartite systems}

\author{Chen Xie  \quad Hui Zhao\thanks{Supported by NSFC11101017.} \\
\small College of Applied Sciences\\[-0.8ex]
\small Beijing University of Technology\\[-0.8ex]
\small  Beijing, China\\
\small\tt xiechen11happy@126.com   \qquad  \small\tt zhaohui@bjut.edu.cn\\
Zhi-Xi Wang\thanks{Supported by KZ201210028032.}\\
\small School of Mathematical Sciences\\[-0.8ex]
\small Capital Normal University\\[-0.8ex]
\small Beijing, China\\
\small\tt wangzhx@cnu.edu.cn}

\date{\dateline{}{}\\
\small Mathematics Subject Classifications: 05C90}

\begin{document}

\maketitle

\begin{abstract}

We investigate separability of Laplacian matrices of graphs when seen as density matrices.
This is a family of quantum states with many combinatorial properties.
We firstly show that the well-known matrix realignment criterion can be used to test separability of this type of quantum states.
The criterion can be interpreted as novel graph-theoretic idea.
Then, we prove that the density matrix of the tensor product of $N$ graphs is $N$-separable.
However, the converse is not necessarily true.
Additionally, we derive a sufficient condition for $N$-partite entanglement in star graphs and propose a
necessary and sufficient condition for separability of nearest point graphs.

\bigskip\noindent \textbf{Keywords:} density matrices of graphs, Laplacian matrices; separability
\end{abstract}

\section{Introduction}

Quantum entanglement is one of the most fascinating features of quantum
theory and it has numerous applications in quantum information processing,
secure communication and channels \cite{1, 2, 3}.
The theory of graphs, as a well-developed mathematical area,
has uncountably many applications in network systems, algorithms, optimization, and other fields \cite{4}.

For exploring combinatorial features of entanglement in mixed quantum states,
Braunstein et al. \cite{5} introduced a restricted class of states called
density matrices of graphs. These states are just normalized graph
Laplacians. It follows that study entanglement properties of Laplacians is a
combinatorial problem.

One of the reasons to study such a topic is that theorems about general sets
of quantum states may be more straightforward to prove when restricted to
Laplacian matrices. On the other side, graph-theoretic results may be
reinterpreted in terms of the language of quantum mechanics.

Braunstein et al. \cite{6} described a degree condition to test separability of density matrices of graphs.
In \cite{7}, the PPT criterion \cite{8} was proved to be
necessary and sufficient for separability in $H^2\otimes H^q$.
Further results on the multipartite separability of Laplacians were discussed in \cite{9}.
Ref. \cite{10} generalized the study of entanglement properties of mixed density matrices to tripartite states.

Ref. \cite{11} analyze in detail the separability and quantitative
characteristics of entanglement for Laplacians.
However, it is generally known that for the case of mixed bipartite states,
no single practical procedure can be guaranteed to detect entanglement,
even when we restrict the analysis to Laplacians.
It would be important to see if in this specific case there is indeed a method.
It is expected that the method would be purely combinatorial.

In this paper, we study multipartite entanglement properties.
By applying graph-theoretic ideas,
we are able to significantly expand our ability to
distinguish entangled and separable Laplacians.
It is well known that the realignment criterion \cite{12, 13, 14, 15}
(also called the cross-norm criterion) is a very strong one.
The criterion involves only straightforward matrix
manipulations and it is easy to apply. We consider realignment for graphs
and obtain a simple, computable necessary criterion for separability. This
is done in Section 2.

In Section 3, we prove that the density matrix of the tensor product of $N$
graphs is $N$-separable. While this seems to be an obvious statement, it is
not immediately clear why the converse should hold.
In fact, we can show that if a density matrix is $4$-separable,
it does not necessarily mean that
it can be written as the tensor product of four graphs.
In Section 4, we derive a sufficient condition for the entanglement of star graphs and give a
necessary and sufficient condition for the separability of the nearest point graph.
Comments and conclusions are given in Section 5.
It would be valuable to observe whether the difficulty of detecting entanglement for Laplacians
is determined by some graph-theoretic parameter,
as in the context of fixed-parameter tractability.

We hope that the results of the paper will contribute to diffuse the study
of separability criteria for graphs among mathematicians interested in
algebraic graph theory and graph products.


\section{The Laplacian Matrices of Graphs}

Let $G=(V(G),\ E(G))$ be a {\it graph} with a vertex set
$V(G)$ and an edge set $E(G)$, where $V(G)=\{v_1,\ v_2,\ \cdots,\
v_n\}$ is a non-empty and finite set of {\it vertices} and
$E(G)=\{\{v_i,\ v_j\}:\ v_i,\ v_j\in V\}$ is a non-empty set of
unordered {\it edges}. A {\it loop} is an edge of the form $\{v_i,\
v_i\}.$ We assume that $E(G)$ does not contain any loops in the
sequel. A graph $G$ is said to be on $n$ vertices if $|V(G)|=n$. The
{\it degree} of a vertex $v_i\in V(G)$ is the number of edges
adjacent to $v_i$, we denote it as $d_G(v_i)$.

\begin{definition}
The density matrix of graph $G$ is defined as the matrix
\begin{eqnarray}
\rho(G)=\frac{1}{d_{G}}L(G),
\end{eqnarray}
where $d_G=\displaystyle\sum_{i=1}^nd_G(v_i)$ is called the {\it
degree sum}. $L(G)$ is the {\it combinatorial Laplacian matrix} of
graph $G$ defined as
\begin{eqnarray}
L(G)=\Delta(G)-M(G),
\end{eqnarray}
where $\Delta(G)$ is the {\it degree matrix} with the diagonal entry $d_G(v_i)$.
$M(G)$ is the {\it adjacency matrix} of graph $G$ with the $ij$-th entry defined as
\begin{eqnarray}
[M(G)]_{i,j}=\left\{
\begin{array}{ll}
1, & \hbox{if $(v_{i},\ v_{j})\in E(G)$;}\\
0, & \hbox{if $(v_{i},\ v_{j})\notin E(G)$.}
\end{array}
\right.
\end{eqnarray}

Two distinct vertices $v_i$ and $v_j$
are said to be {\it adjacent}, if $\{v_i,\ v_j\}\in E(G)$. If every pair of vertices are
adjacent, the graph is {\it complete}. $K_n$ denotes the {\it
complete graph} on $n$ vertices.

The density matrix of a graph is a uniform mixture of pure density
matrices, that is, for a graph $G$ on $n$ vertices $v_1,\ v_2,\
\cdots,\ v_n,$ having $s$ edges $\{v_{i_1},\ v_{j_1}\},\ \{v_{i_2},\
v_{j_2}\},\ \cdots,\ \{v_{i_s},\ v_{j_s}\},$ where $1\leq i_1,\
j_1,\ i_2,\ j_2,\ \cdots,\ i_s,\ j_s\leq n,$
$$\rho(G)=\displaystyle\frac{1}{s}\sum_{k=1}^{s}\rho(H_{i_kj_k}),$$
here $H_{i_kj_k}$ is the factor of $G$ such that
$$[M(H_{i_kj_k})]_{u,\ w}=\left\{
\begin{array}{ll}
1, & \hbox{if}\ u=i_k\ \hbox{and}\ w=j_k\ \hbox{or}\
w=i_k\ \hbox{and}\ u=j_k;\\
0, & \hbox{otherwise.}
\end{array}
\right.$$ The density matrix $\rho(H_{i_kj_k})$ is {\it pure}.
\end{definition}

\begin{definition}
A {\it star graph} \cite{10} on $n$ vertices
$v_1,\ v_2,\ \cdots,\ v_n$ is the graph whose set of edges is
$\{\{v_1,\ v_i\}:\ i=2,\ 3,\ \cdots,\ n\}$.
\end{definition}

\begin{definition}
A {\it nearest point graph} \cite{10} is a
graph whose vertices are identified with the points of a cuboid and
the edges have length 1, $\sqrt{2}$ or $\sqrt{3}. $
\end{definition}

\begin{definition}
The state $\rho$ acting on $H=H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{N}$ is {\it separable} if it can be
written as
\begin{eqnarray}
\rho=\sum_{i}p_{i}\rho_{1}^{i}\otimes
\rho_{2}^{i}\otimes\cdots\otimes \rho_{N}^{i},
\end{eqnarray}
where $\rho_{j}^{i}$ is the density matrix on $H_{j}$ respectively. Otherwise, the state is {\it entangled}.
\end{definition}

\begin{definition}
The {\it tensor product of graphs} $G_1,\
\ldots,\ G_N$, denoted by $G_1\otimes \cdots\otimes G_N$, is the
graph whose adjacency matrix is $M(G_1\otimes \cdots\otimes
G_N)=M(G_1)\otimes \cdots\otimes M(G_N).$
\end{definition}

\begin{definition}
Let $G_j$ be a graph on $m_j$ vertices,
$v^j_1,\ v^j_2,\ \cdots,\ v^j_{m_j}$, and $n_j$ edges,
$\{v^j_{c^j_1}$,\ $v^j_{d^j_1}\}$,\ $\cdots,
\{v^j_{c^j_{n_j}}, v^j_{d^j_{n_j}}\} $ with $1\leq c^j_1,\
d^j_1,\ \cdots,\ c^j_{n_j},\ d^j_{n_j}\leq m_j$, $j=1,2,\cdots, N$.
Suppose Hilbert space ${H}_1$ is
the space spanned by the orthonormal basis $\{|v^{1}_1\ra,\
|v^{1}_{2}\ra,\ \cdots,\ |v^{1}_{m_1}\ra\}$ associated to $V(G_1)$;
$\cdots$; ${H}_N$ is the space spanned by the orthonormal basis
$\{|v^{N}_1\ra,$\ $|v^{N}_2\ra,$\ $\cdots,$\ $|v^{N}_{m_N}\ra\}$
associated to $V(G_N)$. The density matrix $\rho(G_1\otimes
\cdots\otimes G_N)$ is {\it $N$-separable} with respect to
Hilbert space ${H}_1\otimes \cdots\otimes {H}_N$ and any graph $G$
on $n$ vertices is {\it $N$-separable} with respect to
space ${C}^{m_1}\otimes \cdots\otimes {C}^{m_N}$, where
$n=m_1m_2\cdots m_N$.
\end{definition}

\begin{definition}
For an $m\times n$ matrix $A=[a_{ij}],$
where $a_{ij}$ is the matrix entry of $A$, the vector
$vec(A)$ is defined as
\begin{eqnarray}
vec(A)=[a_{11}, \cdots, a_{m1}, a_{12}, \cdots, a_{m2}, \cdots, a_{1n}, \cdots, a_{mn}]^{T}.
\end{eqnarray}

Let $Z$ be an $m\times m$ block matrix with block size $n\times n$. The realigned matrix $\widetilde{Z}$ of size $m^2 \times n^2$ is defined as
\begin{eqnarray}
\widetilde{Z}\equiv\begin{pmatrix}
vec(Z_{1,1}^{T})\\
\vdots\\
vec(Z_{m,1}^{T})\\
\vdots\\
vec(Z_{1,m}^{T})\\
\vdots\\
vec(Z_{m,m}^{T})
\end{pmatrix}.
\end{eqnarray}
\end{definition}

If an $m\times n$ bipartite density matrix $\rho_{AB}$ is separable,
then for the $m^2\times n^2$ matrix $\widetilde{\rho_{AB}}$ the Ky
Fan norm which is the sum of all the singular values of
$\widetilde{\rho_{AB}}$ is less than or equal to 1 \cite{14}. Next
applying this result, we present a separability criterion.

For a $pq\times pq$ matrix $A$, $A_{ij}$ denotes the $(i,j)$-th element
of  $A$. Let $f$ be the canonical bijection between
${1,\cdots, p}\times {1, \cdots, q}$ and ${1,\cdots, pq}:
f(i,j)=(i-1)q+j$. If $f(i_{1},j_{1})=k$ and $f(i_{2},j_{2})=l$, then
$A_{kl}$ can be written as  $A_{(i_{1},j_{1})(i_{2},j_{2})}$.

\begin{theorem}
For the graph $G$ with the Laplacian matrix
$A=(a_{ij})_{n^{2}\times n^{2}}$, the density matrix of the graph
$G$ is entangled if
\begin{eqnarray}
|\sum_{k=1}^{n}d_{G}(v_{k})-N(k_{1}, k_{2})|>1,
\end{eqnarray}
 for all vertices $v_{k}=f(k,k)=(k-1)n+k$,
$k=1 ,\cdots, n$, where $N(k_{1},k_{2})$ is the number of edges between $v_{k_{1}}$ and $v_{k_{2}}$, $v_{k_{1}} \neq v_{k_{2}}$.
\end{theorem}

\begin{proof} Suppose
\begin{eqnarray}
A=\begin{pmatrix}
a_{1 1}&\cdots&a_{1 n}&\cdots&a_{1 n^{2}-n}&\cdots&a_{1 n^{2}}\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{1 n}&\cdots&a_{n n}&\cdots&a_{n n^{2}-n}&\cdots&a_{n n^{2}}\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{1 n^{2}-n}&\cdots&a_{n n^{2}-n}&\cdots&a_{n^{2}-n n^{2}-n}&\cdots&a_{n^{2}-n n^{2}}\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{1 n^{2}}&\cdots&a_{n n^{2}}&\cdots&a_{n^{2}-n n^{2}}&\cdots&a_{n^{2} n^{2}}\\
\end{pmatrix}
\end{eqnarray}
is the Laplacian matrix of a graph, and the realigned matrix
$\widetilde{A}$ is
\begin{eqnarray}
\widetilde{A}=\begin{pmatrix}
a_{1 1}&\cdots&a_{1 n}&\cdots&a_{1 n}&\cdots&a_{n n}\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{1 n^{2}-n}&\cdots&a_{1 n^{2}}&\cdots&a_{n n^{2}-n}&\cdots&a_{n n^{2}}\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{1 n^{2}-n}&\cdots&a_{n n^{2}-n}&\cdots&a_{1 n^{2}}&\cdots&a_{n n^{2}}\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{n^{2}-n n^{2}-n}&\cdots&a_{n^{2}-n n^{2}}&\cdots&a_{n^{2}-n n^{2}}&\cdots&a_{n^{2} n^{2}}\\
\end{pmatrix}.
\end{eqnarray}
The diagonal elements of $\widetilde{A}$ are only  $N(k_{1}, k_{1})$
or $N(k_{1}, k_{2})$,  $k_{1}, k_{2}=1,\cdots, n$. So if
$|\sum_{k=1}^{n}d_{G}(v_{k})-N(k_{1}, k_{2})|>1,$ we have
$tr(\widetilde{A})>1$, the sum of the eigenvalue of $\widetilde{A}$
is greater than $1$. It is obvious that the sum of the singular
values of $\widetilde{A}$ is greater than $1$, therefore the density
matrix of the graph $G$ is entangled.
\end{proof}

\section{Separability of Graphs}

In Ref. \cite{10} the entanglement of mixed density matrices
for tripartite states was discussed. Now, we generalized their
results to multipartite quantum systems.

\begin{theorem} The density matrix of the tensor product of $N$
graphs is $N$-separable.
\end{theorem}

\begin{proof}
Using the Definitions 1 and 6, we have
$$\rho(G_1)=\frac{1}{n_1} \sum_{p_1=1}^{n_1} \rho(H_{c^{1}_{p_1} d^{1}_{p_1}}^{(1)}),
\rho(G_2)=\frac{1}{n_2} \sum_{p_2=1}^{n_2} \rho(H_{c^{2}_{p_2}
d^{2}_{p_2}}^{(2)}), \cdots, \rho(G_N)=\frac{1}{n_N}
\sum_{p_N=1}^{n_N} \rho(H_{c^{N}_{p_N} d^{N}_{p_N}}^{(N)}).$$
$H^{(i)}$ denote $H_{c^{i}_{p_i} d^{i}_{p_i}}^{(i)}$ with $i=1,
\cdots, N$, we have
\begin{eqnarray}
&&\rho(G_1\otimes G_2\otimes \cdots\otimes G_N) \nonumber \\
&=&\displaystyle\frac{1}{d_{G_1\otimes G_2\otimes \cdots \otimes G_N}}[\Delta(G_1\otimes G_2\otimes\cdots\otimes G_N)-M(G_1\otimes
G_2\otimes \cdots \otimes G_N)] \nonumber \\
&=&\displaystyle\frac{1}{2^{N}\cdot n_{1} n_{2} \cdots n_{N}} \sum_{p_1, p_2, \cdots, p_N=1}^{n_{1}, n_{2}, \cdots , n_{N}}
[\Delta(H^{(1)}) \otimes \Delta(H^{(2)}) \otimes \cdots \otimes \Delta(H^{(N)}) \nonumber \\
&&-M(H^{(1)}) \otimes M(H^{(2)}) \otimes \cdots \otimes M(H^{(N)})] \nonumber \\
&=&\displaystyle\frac{1}{2^{N-1}\cdot n_{1} n_{2} \cdots n_{N}} \sum_{p_1, p_2, \cdots, p_N=1}^{n_{1}, n_{2}, \cdots ,n_{N}}
[\rho(H^{(1)})\otimes \rho_+(H^{(2)})\otimes \cdots\otimes \rho_+(H^{(N)}) \nonumber \\
&&+\rho_+(H^{(1)})\otimes \rho(H^{(2)})\otimes \cdots\otimes \rho_+(H^{(N)})+\cdots
+\rho_+(H^{(1)})\otimes \rho_+(H^{(2)})\otimes \cdots\otimes \rho(H^{(N)}) \nonumber \\
&&+\cdots +\rho(H^{(1)})\otimes \rho(H^{2})\otimes \cdots\otimes
\rho_+(H^{(N)})],
\end{eqnarray}

where $\rho_+(H^{(i)})\stackrel{\rm
def}{=}\Delta(H^{(i)})-\rho(H^{(i)})
=\frac{1}{2}\big(\Delta(H^{(i)})+M(H^{(i)})\big)$.

Let $\rho_+(G_i)=\frac{1}{n_i}\sum_{p_i=1}^{n_i}\rho_+(H_{c_{p_i}^{i}
d_{p_i}^{i}}^{(i)})$, $i=1, \cdots, N$, we have
\begin{eqnarray}
&&\rho(G_1\otimes G_2\otimes \cdots \otimes G_N) \nonumber \\
&=&\displaystyle\frac{1}{2^{N-1} n_{1} n_{2} \cdots n_{N}}[\rho(G_1)\otimes \rho_+(G_2)\otimes \cdots\otimes \rho_+(G_N)
+\rho_+(G_1)\otimes \rho(G_2)\otimes \cdots\otimes \rho_+(G_N) \nonumber \\
&&+\cdots+\rho_+(G_1)\otimes \rho_+(G_2)\otimes \cdots\otimes
\rho(G_N)+\cdots+\rho(G_1)\otimes \rho(G_2)\otimes \cdots\otimes
\rho_+(G_N)].
\end{eqnarray}

Therefore $\rho(G)$ is $N$-separable.
\end{proof}

In order to describe the theorem in detail, we give the following example.

\begin{example}
The density matrix acting on $H_{1}\otimes H_{2}\otimes H_{3}\otimes
H_{4}$ of the tensor of four graphs is $4$-separable.

Let $G_i$ be a graph on $m_i$ vertices and $n_i$ edges, $i=1,2,3,4$, and denote $H^{(i)}=H_{c^{i}_{p_i} d^{i}_{p_i}}^{(i)}$, we have
\begin{eqnarray}
\rho_+(G_1)=\frac{1}{n_1}\sum_{p_1=1}^{n_1}\rho_+(H^{(1)}),
\rho_+(G_2)=\frac{1}{n_2}\sum_{p_2=1}^{n_2}\rho_+(H^{(2)}), \nonumber \\
\rho_+(G_3)=\frac{1}{n_3}\sum_{p_3=1}^{n_3}\rho_+(H^{(3)}),
\rho_+(G_4)=\frac{1}{n_4}\sum_{p_4=1}^{n_4}\rho_+(H^{(4)}),
\end{eqnarray}
where $\rho_+(H^{(i)})\stackrel{\rm
def}{=}\Delta(H^{(i)})-\rho(H^{(i)})
=\frac{1}{2}\big(\Delta(H^{(i)})+M(H^{(i)})\big)$.
Since $\rho_+(H^{(i)})$ are all density matrices,
\begin{eqnarray}
&&\rho(G_1\otimes G_2\otimes G_3\otimes G_4) \nonumber \\
&=& \displaystyle\frac{1}{8}[\rho_+(G_{1})\otimes
\rho_+(G_{2})\otimes \rho_+(G_{3})\otimes \rho(G_{4})
+\rho_+(G_{1})\otimes \rho_+(G_{2})\otimes \rho(G_{3})\otimes \rho_+(G_{4}) \nonumber \\
&&+\rho_+(G_{1})\otimes \rho(G_{2})\otimes \rho_+(G_{3})\otimes
\rho_+(G_{4})
+\rho_+(G_{1})\otimes \rho(G_{2})\otimes \rho(G_{3})\otimes \rho(G_{4}) \nonumber \\
&&+\rho(G_{1})\otimes \rho_+(G_{2})\otimes \rho_+(G_{3})\otimes
\rho_+(G_{4})
+\rho(G_{1})\otimes \rho_+(G_{2})\otimes \rho(G_{3})\otimes \rho(G_{4}) \nonumber \\
&&+\rho(G_{1})\otimes \rho(G_{2})\otimes \rho_+(G_{3})\otimes
\rho(G_{4})+\rho(G_{1})\otimes \rho(G_{2})\otimes \rho(G_{3})\otimes
\rho_+(G_{4})].
\end{eqnarray}
Therefore $\rho(G)$ is $4$-separable.
\end{example}

\begin{lemma}
For the matrix
$\sigma=\frac{1}{5}P[\frac{1}{\sqrt{2}}(|ijkp\ra-|rstq\ra
)]+\frac{1}{5}P[\frac{1}{\sqrt{2}}(|ijkq\ra -|rstp\ra
)]+\frac{1}{5}P[\frac{1}{\sqrt{2}}(|ijtp\ra -|rskq\ra
)]+\frac{1}{5}P[\frac{1}{\sqrt{2}}(|iskp\ra -|rjtq\ra )]
+\frac{1}{5}P[\frac{1}{\sqrt{2}}(|rjkp\ra -|istq\ra )]$, the following conlusions hold:

(i) It is a density matrix and $4$-separable.

(ii) The density matrix of complete graph $\rho(K_n)$ is also
$4$-separable.
\end{lemma}

\begin{proof}
(i) Since the project operator is semi-positive,
$\sigma$ is semi-positive. Moreover, the computation shows that $tr(\sigma)=1$,
so $\sigma$ is a density matrix. Let
$$|u^{\pm}\ra=\frac{1}{\sqrt{2}}(|i\ra\pm|r\ra),\ \
|v^{\pm}\ra=\frac{1}{\sqrt{2}}(|j\ra\pm|s\ra),\ \
|w^{\pm}\ra=\frac{1}{\sqrt{2}}(|k\ra\pm|t\ra),\ \
|\varphi^{\pm}\ra=\frac{1}{\sqrt{2}}(|p\ra\pm|q\ra).$$
Then $\sigma$ can be written as the linear combination of tensor
products, thus $\sigma$ is $4$-separable.

(ii) Since $M(K_n)=J_n-I_n$, where $J_n$ is the $n\times n$ all-ones
matrix and $I_n$ is the $n\times n$ identity matrix, whenever there
is an edge $\{u_{i}v_{j}w_{k}\varphi_p,\ u_{r}v_{s}w_{t}\varphi_q\}$,
there must be entangled edges
$\{u_{r}v_{j}w_{k}\varphi_p,\ u_{i}v_{s}w_{t}\varphi_p\},$
$\{u_{i}v_{s}w_{k}\varphi_p,\ u_{r}v_{j}w_{t}\varphi_p\},$
$\{u_{i}v_{j}w_{t}\varphi_p,\ u_{r}v_{s}w_{k}\varphi_p\}$ and
$\{u_{i}v_{j}w_{k} \varphi_q,\ u_{r}v_{s}w_{t}\varphi_q\},$
so the density matrix $\rho(K_n)$ is $4$-separable.
\end{proof}

\begin{theorem}
Given a graph $G_1\otimes G_2\otimes G_3\otimes
G_4,$ the density matrix $\rho(G_1\otimes G_2\otimes G_3\otimes
G_4)$ is $4$-separable. However if a density matrix $\rho(G)$ is
$4$-separable, it does not necessarily mean that $G=G_1\otimes G_2
\otimes G_3 \otimes G_4$ for some graphs $G_1,\ G_2 ,\ G_3 ,\ G_4.$
\end{theorem}

\begin{proof}
The first result follows from Example 10. On the other side, from Lemma 11, the density matrix of the complete graph
$\rho(K_n)$ is $4$-separable. Because of the complete graph is not a
tensor product of three graphs \cite{10}, it is not a tensor product
of four graphs.
\end{proof}

\section{Separability of Some Special Graphs}

In this section, we will derive the results of some special graphs.

\begin{theorem}
The density matrix of star graphs $\rho(K_{1,
n-1})$ is $N$-partite entangled for $n=n_{1}n_{2}\cdots n_{N}\geq
2^{N}.$
\end{theorem}

\begin{proof}
For a star graph $G=K_{1, n-1}$ on $n=n_{1}\cdots
n_{N}$ vertices with orthonormal basis $|\alpha_1\ra,\
|\alpha_2\ra,\ \cdots,\ |\alpha_n\ra,$ we have
\begin{eqnarray}
\rho(G)=\frac{1}{n-1}\sum_{k=2}^n\rho(H_{1k})=
\frac{1}{n-1}\sum_{k=2}^nP[\frac{1}{\sqrt{2}}(|\alpha_1\ra-|\alpha_k\ra)].
\end{eqnarray}

Consider $\rho(G)$ in $C^{n_1}_{A_1}\otimes
C^{n_2}_{A_2}\otimes\cdots\otimes C^{n_N}_{A_N}$, where
$C^{n_i}_{A_i}$ are associated to ${\cal H}_{A_i}$ , $i=1,\cdots,N$
respectively. Let $\{|v_1^i\ra,\ |v_2^i\ra,\ \cdots,\
|v_{n_i}^i\ra\}$ be the orthonormal basis of $C^{n_i}_{A_i}$
respectively. So
\begin{eqnarray}
\rho(G)=\frac{1}{n-1}\sum_{k=2}^{n}P[\frac{1}{\sqrt{2}}(|v_1^1\cdots
v_1^N\ra-|v_{r_k}^1\cdots v_{r_k}^N\ra)],
\end{eqnarray}
it follows that
\begin{eqnarray}
&&\rho(G) \nonumber \\
&=&\displaystyle\frac{1}{n-1}\Big\{\sum_{i_1=2}^{n_1}P[\frac{1}{\sqrt{2}}(|v_1^1\ra-|v_{i_1}^1\ra)|v_1^2\cdots v_1^N\ra]+\cdots+\sum_{i_N=2}^{n_N}P[\frac{1}{\sqrt{2}}|v_1^1\cdots v_1^{n-1}\ra(|v_1^N\rangle-v_{i_N}^N\ra)] \nonumber \\
&&+\cdots\cdots+\displaystyle\sum_{i_1,\cdots,i_N=2}^{n_1,\cdots,n_N}P[\frac{1}{\sqrt{2}}
(|v_1^1\cdots v_1^N\ra-|v_{i_1}^1\cdots v_{i_N}^N \ra)]\Big\}.
\end{eqnarray}
Consider the following projectors $P_i=|v_1^i\ra\la
v_1^i|+|v_2^i\ra\la v_2^i|$, $i=1,\cdots,N$, then
\begin{eqnarray}
&&(P_1\otimes\cdots\otimes P_N)\rho(G)(P_1\otimes\cdots\otimes P_N) \nonumber \\
&=&\frac{1}{n-1}\Big\{\frac{n-2^N}{2}P[|v_1^1\cdots v_1^N\ra]+P[\frac{1}{\sqrt{2}}(|v_1^1 v_1^2\cdots v_1^N\ra-|v_2^1 v_1^2\cdots v_1^N\ra)] \nonumber \\
&&+\cdots+P[\frac{1}{\sqrt{2}}(|v_1^1\cdots v_1^{N-1} v_1^N\ra-|v_1^1\cdots v_1^{N-1}v_2^N\ra)] \nonumber \\
&&+\cdots\cdots+P[\frac{1}{\sqrt{2}}(|v_1^1\cdots
v_1^N\ra-|v_2^1\cdots v_2^N\ra)]\Big\}.
\end{eqnarray}
So in the basis$|v_1^1\cdots v_1^N\ra, |v_2^1\cdots v_1^N\ra
,\cdots, |v_2^1\cdots v_2^N\ra$, we get a $2^N\times2^N$ matrix
\begin{eqnarray}
(P_1\otimes\cdots\otimes P_N)\rho(G)(P_1\otimes\cdots\otimes P_N)
=\frac{1}{n-1} \left(
\begin{array}{cccccccccccc}
\frac{n-1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\cdots&-\frac{1}{2}\\[3mm]
-\frac{1}{2}&\frac{1}{2}&0&0&\cdots&0\\[3mm]
-\frac{1}{2}&0&\frac{1}{2}&0&\cdots&0\\[3mm]
-\frac{1}{2}&0&0&\frac{1}{2}&\cdots&0\\[3mm]
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\[3mm]
-\frac{1}{2}&0&0&0&\cdots&\frac{1}{2}\\[3mm]
\end{array}
\right).
\end{eqnarray}

The eigenvalues of the partially transposition of
$(P_1\otimes\cdots\otimes P_N)\rho(G)(P_1\otimes\cdots\otimes P_N)$
are $\frac{1}{2(n-1)}$ (with multiplicity $2^N-3$),
$\frac{1}{(n-1)}$ and the roots of the polynomial
$(n-1)^{2}\lambda^{2}-\frac{(n-1)^{2}}{2}\lambda-\frac{1}{2}$. Here
the matrix has a negative eigenvalue, therefore $\rho(G)$ is
$N$-partite entangled.
\end{proof}

Next we give an example about $4$-partite entangled states.

\begin{example}
For a star graph $G=K_{1,\ n-1}$ on
$n=n_1n_2n_3n_4$ vertices, the density matrix $\rho(G)$ is 4-partite
entangled. From theorem 13, we can get that the partially
transposition of $(P_1\otimes P_2\otimes P_3\otimes
P_4)\rho(G)(P_1\otimes P_2\otimes P_3\otimes P_4)$ has negative
eigenvalue, so $\rho(G)$ is 4-partite entangled.
\end{example}

Now, let us consider the nearest point graph.

\begin{theorem}
Let $G$ be a nearest point graph on $n=n_1 n_2
n_3 n_4$ vertices, then the density matrix $\rho(G)$ is 4-separable
in $C^m_A\otimes C^p_B\otimes C^q_C\otimes C^q_D$ if and only if
it satisfies the degree condition $\Delta(G)=\Delta(G^{\Gamma_A})=\Delta(G^{\Gamma_B})=\Delta(G^{\Gamma_C})=\Delta(G^{\Gamma_D}).$
\end{theorem}

\begin{proof}
We first prove that the condition is sufficient.
Let $\rho(G)$ be the density matrix of a graph on
$n=n_1n_2n_3n_4$ vertices and
$\Delta(G)=\Delta(G^{\Gamma_A})=\Delta(G^{\Gamma_B})=\Delta(G^{\Gamma_C})=\Delta(G^{\Gamma_D})$
as the degree condition. The degree condition is equivalent to PPT
criterion \cite{16}. If $\rho(G)$ is separable in $C^{n_1}_A\otimes
C^{n_2}_B\otimes C^{n_3}_C\otimes C^{n_4}_D$, then
$\Delta(G)=\Delta(G^{\Gamma_A})=\Delta(G^{\Gamma_B})=\Delta(G^{\Gamma_C})=\Delta(G^{\Gamma_D}).$

In the following, we prove that for the nearest graphs, the
condition is also necessary. Let $G$ be a nearest point graph on
$n=n_1 n_2 n_3 n_4$ vertices and $f$ edges. We associate to $G$ the
orthonormal basis $\{|\alpha_{l}\ra:l=1,\ 2,\ \cdots,\ n\}
=\{|u_{i}\ra\otimes|v_{j}\ra\otimes |w_{k}\ra\otimes
|\varphi_{p}\ra:\ i=1,\ \cdots,\ n_1;\ j=1,\cdots,\ n_2;\ k=1,\
\cdots, \ n_3; \ p=1,\ \cdots, \ n_4\},$ where $|u_{i}\ra$ ,
$|v_{j}\ra$ , $|w_{k}\ra$ and $|\varphi_{p}\ra$ are the orthonormal
basis of $C^{n_1}_A$, $C^{n_2}_B$, $C^{n_3}_C$, $C^{n_4}_D$
respectively. Let $i,\ r\in\{1,\ \cdots,\ n_1\},\ j,\ s\in\{1,\
\cdots,\ n_2\},\ k,\ t\in\{1, \ \cdots,\ n_3\},\ p,\ q\in\{1, \
\cdots,\ n_4\}.\ \lambda_{ijkp,\ rstq}\in \{0,\ 1\}$ is defined as
\begin{eqnarray}
\lambda_{ijkp,\ rstq}=\left\{
\begin{array}{ll}
1, & \hbox { if\ $(u_{i}v_{j}w_{k}\varphi_p,\ u_{r}v_{s}w_{t}\varphi_q)\in E(G);$}\\
0, & \hbox { if\ $(u_{i}v_{j}w_{k}\varphi_p,\ u_{r}v_{s}w_{t}\varphi_q)\notin E(G),$}
\end{array}
\right.
\end{eqnarray}
where $i,\ j,\ k,\ r,\ s,\ t$ satisfy the following $14$ conditions:
\begin{itemize}
\item $i=r,\ j=s,\ k=t,\ p=q+1;\ \
i=r,\ j=s,\ k=t+1,\ p=q;$
\item $i=r,\ j=s+1,\ k=t,\ p=q;\ \
i=r+1,\ j=s,\ k=t,\ p=q;$
\item $i=r+1,\ j=s+1,\ k=t,\ p=q;\ \
i=r+1,\ j=s,\ k=t+1,\ p=q;$
\item $i=r+1,\ j=s,\ k=t,\ p=q+1;\ \
i=r,\ j=s+1,\ k=t+1,\ p=q;$
\item $i=r,\ j=s+1,\ k=t,\ p=q+1;\ \
i=r,\ j=s,\ k=t+1,\ p=q+1;$
\item $i=r+1,\ j=s+1,\ k=t+1,\ p=q;\ \
i=r+1,\ j=s+1,\ k=t,\ p=q+1;$
\item $i=r+1,\ j=s,\ k=t+1,\ p=q+1;\ \
i=r,\ j=s+1,\ k=t+1,\ p=q+1$.
\end{itemize}

Let $\rho(G),\ \rho(G^{\Gamma_A}),\ \rho(G^{\Gamma_B}),\ \rho(G^{\Gamma_C})$ and $\rho(G^{\Gamma_D})$
be the density matrices corresponding to the graph
$G,\ G^{\Gamma_A},\ G^{\Gamma_B},\ G^{\Gamma_C}$ and $G^{\Gamma_D}$ respectively.
Then
\begin{eqnarray}
\rho(G)=&\frac{1}{2f}(\Delta(G)-M(G)),\ &\rho(G^{\Gamma_{A}})=\frac{1}{2f}(\Delta(G^{\Gamma_{A}})-M(G^{\Gamma_{A}})), \nonumber \\
\rho(G^{\Gamma_{B}})=&\frac{1}{2f}(\Delta(G^{\Gamma_{B}})-M(G^{\Gamma_{B}})),\ & \rho(G^{\Gamma_{C}})=\frac{1}{2f}(\Delta(G^{\Gamma_{C}})-M(G^{\Gamma_{C}})), \nonumber \\
\rho(G^{\Gamma_{D}})=&\frac{1}{2f}(\Delta(G^{\Gamma_{D}})-M(G^{\Gamma_{D}})).
\end{eqnarray}
Let $G_1$ be the subgraph of $G$ whose edges are all the entangled
edges of $G$. An edge $\{u_iv_jw_k,\ u_rv_sw_t\}$ is entangled if
$i\neq r,\ j\neq s,\ k\neq t$.
Let $G^A_1,\ \ G^B_1,\ \ G^C_1,\ \ G^D_1$ be the subgraph of $G^{\Gamma_A}$, $G^{\Gamma_B}$, $G^{\Gamma_C}$, $G^{\Gamma_D}$
corresponding to all the entangled edges.
Obviously,
$G^A_1=(G_1)^{\Gamma_A},\ G^B_1=(G_1)^{\Gamma_B},\ G^C_1=(G_1)^{\Gamma_C},\ G^D_1=(G_1)^{\Gamma_D}.$ We have
\begin{eqnarray}
\rho(G_1)=\frac{1}{f}\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}\sum_{k=1}^{n_{3}}\sum_{P=1}^{n_{4}}\lambda_{ijkp,\ rstq}P[\frac{1}{\sqrt{2}}(|u_iv_jw_k\varphi_p\ra-|u_rv_sw_t\varphi_q\ra)],
\end{eqnarray}
where
$i,\ j,\ k,\ p;\ r,\ s,\ t,\ q$ must satisfy the 14 conditions. We can get $\rho(G^A_1),\ \rho(G^B_1),\ \rho(G^C_1)$ and $\rho(G^D_1)$
by commuting the index of $u,\ v,\ w$ in the above equation respectively.
Also we have
\begin{eqnarray}
\Delta(G_1)=\frac{1}{2f}\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}\sum_{k=1}^{n_{3}}\sum_{P=1}^{n_{4}}
\lambda_{ijkp,\ rstq}P[|u_iv_jw_k\varphi_p\ra],
\end{eqnarray}
where $i,\ j,\ k,\ p;\ r,\ s,\ t,\ q$
must satisfy either the 14 conditions.
We can get $\Delta(G^A_1),\ \Delta(G^B_1)$, $\Delta(G^C_1)$
and $\Delta(G^D_1)$ by commuting the
index of $\lambda$ . Let $G_2,\ G^A_2,\ G^B_2$, $G^C_2$
and $G^D_2$ be the subgraph of $G,\ G^A,\ G^B$ , $G^C$ and $G^D$ containing  all the unentangled edges, respectively.
It is obvious that
$\Delta(G_2)=\Delta(G^{\Gamma_A}_2)=\Delta(G^{\Gamma_B}_2)=\Delta(G^{\Gamma_C}_2)=\Delta(G^{\Gamma_D}_2).$  So
$\Delta(G)=\Delta(G^{\Gamma_A})=\Delta(G^{\Gamma_B})=\Delta(G^{\Gamma_C})=\Delta(G^{\Gamma_D})$
if and only if
$\Delta(G_1)=\Delta(G^{\Gamma_A}_1)=\Delta(G^{\Gamma_B}_1)=\Delta(G^{\Gamma_C}_1)=\Delta(G^{\Gamma_D}_1).$
The condition implies that
\begin{eqnarray}
\lambda_{ijkp,\ rstq}=\lambda_{rjkp,\
istq}=\lambda_{iskp,\ rjtq}=\lambda_{ijtp ,\ rskq}=\lambda_{ijkq,\ rstp},
\end{eqnarray}
for $i,\
r\in\{1,\ 2,\ \cdots,\ n_1\},\ j,\ s \in\{1,\ 2,\ \cdots,\ n_2\},\ \ k,\
t \in\{1,\ 2,\ \cdots,\ n_3 \,\ \ p,\
q \in\{1,\ 2,\ \cdots,\ n_4 \}.$
The above equations show that
whenever there is an entangled edge $\{u_{i}v_{j}w_{k}\varphi_p,\
u_{r}v_{s}w_{t}\varphi_q\}$ in $G$ (here $i\neq r,\ j\neq s,\
k\neq t,\ p\neq q$), there must be the entangled edges $\{u_{r}v_{j}w_{k}\varphi_p,\
u_{i}v_{s}w_{t}\varphi_q\},\ \{u_{i}v_{s}w_{k}\varphi_p,\ u_{r}v_{j}w_{t}\varphi_q\}$, $\{u_{i}v_{j}w_{t}\varphi_p,\ u_{r}v_{s}w_{k}\varphi_q\}$
and
$\{u_{i}v_{j}w_{k}\varphi_q,\ u_{r}v_{s}w_{t}\}\varphi_p$ in $G$.
Let
\begin{eqnarray}
&&\rho(i,\ j,\ k,\ p;\ r,\ s,\ t,\ q) \nonumber \\
&=&\frac{1}{4}(P[\frac{1}{\sqrt{2}}(|u_{i}v_{j}w_{k}\varphi_p\ra-|u_{r}v_{s}w_{t}\varphi_q\ra)]
+P[\frac{1}{\sqrt{2}}(|u_{r}v_{j}w_{k}\varphi_p\ra-|u_{i}v_{s}w_{t}\varphi_q\ra)] \nonumber \\
&&+P[\frac{1}{\sqrt{2}}(|u_{i}v_{s}w_{k}\varphi_p\ra-|u_{r}v_{j}w_{t}\varphi_q\ra)]
+P[\frac{1}{\sqrt{2}}(|u_{i}v_{j}w_{t}\varphi_p\ra-|u_{r}v_{s}w_{k}\varphi_q\ra)]) \nonumber \\
&&+P[\frac{1}{\sqrt{2}}(|u_{i}v_{j}w_{k}\varphi_q\ra-|u_{r}v_{s}w_{t}\varphi_p\ra)]).
\end{eqnarray}
By Lemma 11, we know that $\rho(i,\ j,\ k,\ p;\ r,\ s,\ t,\ q)$ is
4-separable. By using Theorem 3 in Ref. \cite{6} we can easily get
$\rho(G_2)$ is 4-separable. Therefore the density matrix $\rho(G)$
is 4-separable in $C^m_A\otimes C^p_B\otimes C^q_C\otimes C^q_D$
\end{proof}

Similarly, the result can be generalized  to multipartite systems.

\begin{theorem}
Let $G$ be a nearest point graph on $n=n_1 n_2
\cdots n_N$ vertices. The density matrix $\rho(G)$ is
$N$-separable in $C^m_{A_1}\otimes C^p_{A_2}\otimes \cdots \otimes
C^q_{A_N}$ if and only if
$\Delta(G)=\Delta(G^{\Gamma_{A_1}})=\Delta(G^{\Gamma_{A_2}})=\cdots=\Delta(G^{\Gamma_{A_N}}).$
\end{theorem}

\section{Comments and Conclusions}

We have studied the separability of Laplacian matrices of
graphs. Using the matrix realignment, a entanglement criterion for
graphs has been presented. We have proved that if the density matrix
$\rho$ is the tensor product of $N$ graphs, then it is
$N$-separable. But if a density matrix is $4$-separable, it can not
necessarily be written as tensor product of four graphs.
Furthermore, we studied the entanglement of star graphs and nearest
point graphs. We proved that the star graph $\rho(K_{1,\ n-1})$ is
$N$-partite entangled for $n=n_{1}n_{2}\cdots n_{N}\geq 2^{N}.$ And
the density matrix $\rho(G)$ of nearest point graph is 4-separable
if and only if it satisfies the degree condition. These results are
useful to better understand the physical characteristics and
mathematical structures of entangled states.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}

We greatly appreciate the referees$'$ valuable suggestions for improving the original version of this paper. We also greatly
thank Tao Li for her advice.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}
