^†^†thanks: Corresponding author: junlimath@buaa.edu.cn^†^†thanks: Corresponding author: linchen@buaa.edu.cn

Detection of genuine multipartite entanglement based on local sum uncertainty relations

Jun Li School of Mathematical Sciences, Beihang University, Beijing 100191, China Lin Chen School of Mathematical Sciences, Beihang University, Beijing 100191, China International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China

Abstract

Genuine multipartite entanglement (GME) offers more significant advantages in quantum information compared with entanglement. We propose a sufficient criterion for the detection of GME based on local sum uncertainty relations for chosen observables of subsystems. We apply the criterion to detect the GME properties of noisy $n$ -partite W state when $n=3,4,5$ and $6$ , and find that the criterion can detect more noisy W states when $n$ ranges from 4 to 6. Moreover, the criterion is also used to detect the genuine entanglement of $3$ -qutrit state. The result is stronger than that based on GME concurrence and fisher information.

I INTRODUCTION

Quantum entanglement MAN ; RPMK ; FMA ; KSS is a remarkable resource in the theory of quantum information, which is one of the most distinctive features of quantum theory as compared to classical theory. Entangled states play the essential roles in quantum cryptography AKE , teleportation chbg and dense coding chbs . Genuine multipartite entanglement has more significant advantages compared with entanglement. It is beneficial in various quantum communication protocols, such as secret sharing nggr , extreme spin squeezing assk , quantum computing using cluster states rrhj , high sensitivity in general metrology tasks phwl , and multiparty quantum network mhvb ; vsng . To certify GME, Bell-like inequalities jdbn , various entanglement witness MF1 ; JI2 ; JY3 ; JS4 ; CKM5 ; CEJ6 , and generalized concurrence for multi genuine entanglement ZH11 ; YT12 ; TF13 ; LM14 ; LM15 were derived. Some entanglement criteria for bipartite entangled state and multipartite non fully separable states have been also proposed HPBO ; JIV ; CYSG . In particular, the entanglement criteria based on local sum uncertainty relations (LUR) have been proposed for bipartite systems CJZ and tripartite systems YAMA . Although non-fully separable states contain genuinely entangled states, the criterion of GME based on LUR has not been studied.

In this paper, we study the criterion of GME based on LUR and obtain the sufficient conditions in Theorems 3 and 5. First, for any quantum states, we show that we can always find the lower bound of LUR. Second, we apply the sum of local observables to the multipartite biseparable state, and obtain the lower bound of LUR by using the method in CJZ ; YAMA . The converse negative process is the criterion of detecting GME. When we choose spin observables, the criterion is better than that in RYMD . Third, we use the criterion detect $3$ -qutrit state and noisy W state for $n$ -qubit system ( $n=3,4,5,6$ ) and find that the criterion is strong than the exiting ones LM14 ; QIP2020 . Moreover, it can detect more noisy W states when $n$ changes from 4 to 6, which is consistent with the fully-separability of noisy W states xyc2020 .

In the rest of this paper, we will introduce the criterion of bipartite separability and tripartite fully separability based on LUR in Sec. II. We investigate the GME criterion based on LUR in Sec. III, that is, Theorem 3 and Theorem 5. In Sec. IV, we apply the criterion to noisy W state and $3$ -qutrit state to verify its effectiveness. We conclude in Sec. V.

II PRELIMINARIES

A multipartite state that is not the convex sum of bipartite product states is said to be genuine multipartite entangled OGGT . Take the tripartite system as an example. Let $H^{d}_{A}$ , $H^{d}_{B}$ , $H^{d}_{C}$ denote $\emph{d}-$ dimensional Hilbert spaces of system $A$ , $B$ , $C$ , respectively. A tripartite state $\rho\in\mathfrak{B}(H^{d}_{A}\otimes H^{d}_{B}\otimes H^{d}_{C})$ is biseparable if it can be expressed

\displaystyle\rho_{BS}=P_{1}\sum_{R}\eta_{R}^{(1)}\rho_{1}^{R}\otimes\rho_{23}^{R}+P_{2}\sum_{R^{\prime}}\eta_{R^{\prime}}^{(2)}\rho_{2}^{R^{\prime}}\otimes\rho_{13}^{R^{\prime}}+P_{3}\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(3)}\rho_{3}^{R^{\prime\prime}}\otimes\rho_{12}^{R^{\prime\prime}},

(1)

where $\sum_{k=1}^{3}P_{k}=1,P_{k}\geq 0,$ and $\sum_{R}\eta_{R}^{(k)}=1.$ Here $\rho_{mn}^{R}$ is an arbitrary density operator for the subsystems $m$ and $n$ . Otherwise, $\rho$ is called genuinely tripartite entangled. The definition can be extended to genuine multipartite entangled states. Next, we introduce the criterion of bipartite separability and tripartite fully separability based on local sum uncertainty relations. They can also be detected for the criterion of GME.

Consider the set of local observables $\{A_{k}\}$ and $\{B_{k}\}$ for systems $A$ and $B$ respectively. The sum uncertainty relations for arbitrary state $\rho$ are as follows

\displaystyle\sum_{k}\Delta A^{2}_{k}\geq U_{A},\quad\quad\quad\sum_{k}\Delta B^{2}_{k}\geq U_{B},

(2)

where the non-negative constants $U_{A}$ and $U_{B}$ are independent of $\rho$ and $\Delta O^{2}_{k}=\langle O^{2}_{k}\rangle-\langle O_{k}\rangle^{2}=\mathop{\rm Tr}(O^{2}_{k}\rho)-\mathop{\rm Tr}^{2}(O_{k}\rho)$ with $O\in\{A,B\}$ . An entanglement criterion based on local sum uncertainty relation was introduced for bipartite system $AB$ .

Lemma 1

CJZ For bipartite separable state $\rho_{AB}$ , the following inequality holds,

	$\displaystyle F^{AB}_{\rho_{AB}}$		$\displaystyle:=\sum_{k}\Delta(A_{k}\otimes I+I\otimes B_{k})^{2}-(U_{A}+U_{B}+M^{2}_{AB})$		(3)
			$\displaystyle\geq 0,$		(3)

where $M_{AB}=\sqrt{\sum_{k}\Delta A^{2}_{k}-U_{A}}-\sqrt{\sum_{k}\Delta B^{2}_{k}-U_{B}}$ . The violation of inequality implies entanglement of $\rho_{AB}$ .

For tripartite system, we consider the set of local observables $\{A_{k}\}$ , $\{B_{k}\}$ and $\{C_{k}\}$ for subsystem $A$ , $B$ and $C$ , respectively. Suppose that the sum uncertainty relations for these observables have non-negative constants bounds $U_{A}$ , $U_{B}$ and $U_{C}$ independent of states, i. e.

\displaystyle\sum_{k}\Delta A^{2}_{k}\geq U_{A},\quad\sum_{k}\Delta B^{2}_{k}\geq U_{B},\quad\sum_{k}\Delta C^{2}_{k}\geq U_{C}.

(4)

Recently, the criterion (3) has been extended to a non-fully separability criterion for the tripartite system based on local sum uncertainty relations as follows.

Lemma 2

YAMA For any tripartite fully separable state $\rho_{ABC}$ ,

\displaystyle\rho_{ABC}=\sum_{i}p_{i}\rho^{A}_{i}\otimes\rho^{B}_{i}\otimes\rho^{C}_{i},

(5)

the reduced states $\rho_{AB}$ , $\rho_{AC}$ and $\rho_{BC}$ are also separable. Therefore, $\rho_{AB}$ must satisfy the inequality (3) and also similar statements must hold for $\rho_{AC}$ and $\rho_{BC}$ . That is,

\displaystyle F_{\rho_{AB}}^{AB}\geq 0,\quad F_{\rho_{AC}}^{AC}\geq 0,\quad F_{\rho_{BC}}^{BC}\geq 0,

(6)

where $F_{\rho_{AC}}^{AC}$ and $F_{\rho_{BC}}^{BC}$ have similar definitions with $F_{\rho_{AB}}^{AB}$ . So the following inequalities must hold simultaneously,

\displaystyle F_{\rho_{ABC}}^{AB|C}\geq 0,\quad F_{\rho_{ABC}}^{AC|B}\geq 0,\quad F_{\rho_{ABC}}^{BC|A}\geq 0,

(7)

with

F_{\rho_{ABC}}^{AB|C}=F_{\rho_{ABC}}-(U_{A}+U_{B}+U_{C}+M^{2}_{AB}+M^{2}_{ABC}),

F_{\rho_{ABC}}^{AC|B}=F_{\rho_{ABC}}-(U_{A}+U_{B}+U_{C}+M^{2}_{AC}+M^{2}_{ACB}),

F_{\rho_{ABC}}^{BC|A}=F_{\rho_{ABC}}-(U_{A}+U_{B}+U_{C}+M^{2}_{BC}+M^{2}_{BCA}),

where

\displaystyle F_{\rho_{ABC}}=\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho},

(8)

and

M_{ABC}=\sqrt{F_{\rho_{AB}}^{AB}}-\sqrt{\sum_{k}\Delta C^{2}_{k}-U_{C}},

and $M_{ACB}$ and $M_{BCA}$ have similar definitions. Violation of any inequality in Eqs. (6) and (7) implies non fully separability of $\rho_{ABC}$ .

The method of Lemma 1 and 2 can be used to find the criterion of genuine entanglement in Theorem 3. It may be related to the lower bounds of quantum uncertainty relations for single system and bipartite system. In Eqs. (2) and (4), the lower bound of uncertainty relations $U_{A}$ , $U_{B}$ and $U_{C}$ are also independent of states. Moreover, some lower bound related to states have also been studied. We know some well-known formula of uncertainty relation for two observables MLPA ,

	$\displaystyle(\Delta A)^{2}+(\Delta B)^{2}\geq\pm i\langle\psi\|[A,B]\|\psi\rangle+\|\langle\psi\|A+iB\|\psi^{\bot}\rangle\|^{2},$
	$\displaystyle(\Delta A)^{2}+(\Delta B)^{2}\geq\frac{1}{2}\|\langle\psi_{A+B}^{\bot}\|A+B\|\psi\rangle\|^{2}=\frac{1}{2}[\Delta(A+B)]^{2},$

where $\langle\psi|\psi^{\bot}\rangle=0$ , $|\psi_{A+B}^{\bot}\rangle\varpropto(A-B-\langle A+B\rangle)|\psi\rangle$ , and the sign on the right hand side of the inequality takes $+(-)$ while $i[A,B]$ is positive (negative). Let us mark the right side of the inequality as $U_{\rho}$ . Furthermore, some multiple observables uncertainty relations were proposed BCNP ; BCSM ; QCSL . We consider the local observables $\{A_{k}\}$ and $\{B_{k}\}$ for systems $A$ and $B$ respectively, the multi-observables sum uncertainty relations are as follows

\displaystyle\sum_{k}\Delta A^{2}_{k}\geq U_{\rho_{A}},\quad\quad\sum_{k}\Delta B^{2}_{k}\geq U_{\rho_{B}}.

(9)

Similarly, for bipartite states, the multi-observables sum uncertainty relations are as follows

\displaystyle\sum_{k}\Delta(A_{k}\otimes I+I\otimes B_{k})^{2}\geq U_{\rho_{AB}},

(10)

where $U_{\rho_{A}}$ , $U_{\rho_{B}}$ , and $U_{\rho_{AB}}$ can be obtained by the right side of multi-observables sum uncertainty relations in BCNP ; BCSM ; QCSL . We will use the forementioned notions and facts in the next section.

III MAIN RESULTS

In this section, we investigate the genuine tripartite and multipartite entanglement based on local sum uncertainty relations. We apply the observables in Eq. (8) to the tripartite biseparable state, and obtain the lower bound of inequality by using Eqs. (9) and (10). Thus, we construct a sufficient condition for genuine tripartite entanglement in Theorem 3. Further, we extend this criterion to multipartite system in Theorem 5.

III.1 CRITERIA FOR GENUINE TRIPARTITE ENTANGLEMENT

Theorem 3

For a tripartite quantum state $\rho_{ABC}$ , let Eqs. (9) and (10) be satisfied. If $\rho_{ABC}$ is biseparable, then

$\displaystyle F_{\rho_{ABC}}\geq\min$	$\displaystyle\{U_{\rho_{A}}+U_{\rho_{BC}}+W^{2}_{ABC},$	(11)
	$\displaystyle U_{\rho_{B}}+U_{\rho_{AC}}+W^{2}_{BAC},$
	$\displaystyle U_{\rho_{C}}+U_{\rho_{AB}}+W^{2}_{CAB}\}$

where $F_{\rho_{ABC}}$ is defined in (8), and

	$\displaystyle W_{ABC}=\sqrt{\sum_{k}\Delta(A_{k})^{2}_{\rho_{A}}-U_{\rho_{A}}}$
	$\displaystyle-\sqrt{\sum_{k}\Delta(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})^{2}_{\rho_{BC}}-U_{\rho_{BC}}},$

and $W_{BAC}$ and $W_{CAB}$ can be similarly defined.

Proof.

If $\rho_{ABC}$ is biseparable, it can be written as Eq. (1) JDB ; BJTM ; RYMD ,

	$\displaystyle\rho_{BS}=$		$\displaystyle P_{1}\sum_{R}\eta_{R}^{1}\rho_{1}^{R}\otimes\rho_{23}^{R}+P_{2}\sum_{R^{\prime}}\eta_{R^{\prime}}^{2}\rho_{2}^{R^{\prime}}\otimes\rho_{13}^{R^{\prime}}$
			$\displaystyle+P_{3}\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{3}\rho_{3}^{R^{\prime\prime}}\otimes\rho_{12}^{R^{\prime\prime}}$

with $0\leq P_{k}\leq 1$ , $\sum_{k}P_{k}=1$ and $\sum_{R}\eta_{R}^{k}=1$ .

For any mixture of type $\rho_{mix}=\sum_{R\geq 1}P_{R}\rho^{R}$ , the variance $\Delta^{2}u$ satisfies HFHS

\displaystyle\Delta^{2}u\geq\sum_{R}P_{R}\Delta^{2}_{u_{R}}.

(12)

Hence for the biseparable state,

	$\displaystyle\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{BS}}$
	$\displaystyle\geq P_{1}\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R}}$
	$\displaystyle+P_{2}\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R^{{}^{\prime}}}}$
	$\displaystyle+P_{3}\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R^{{}^{\prime\prime}}}}$
	$\displaystyle\geq\min\{\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R}},$
	$\displaystyle\quad\quad\quad\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R^{{}^{\prime}}}},$
	$\displaystyle\quad\quad\quad\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R^{{}^{\prime\prime}}}}\}.$		(13)

We can always choose as the lower bound the smallest value of $\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{\zeta}}$ in (III.1). So the second inequality can be obtained using the fact that $\sum_{k}P_{k}=1$ .

Then we consider $\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}$ that corresponds to the bipartition $\sum_{R}\eta_{R}^{1}\rho_{1}^{R}\otimes\rho_{23}^{R}$ ,

	$\displaystyle\sum_{k}\Delta(A_{k}\otimes I_{BC}+B_{k}\otimes I_{AC}+I_{AB}\otimes C_{k})^{2}_{\rho_{R}}$
	$\displaystyle=\sum_{k}\{\langle[A_{k}\otimes I_{BC}+I_{A}\otimes(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})]^{2}\rangle$
	$\displaystyle-\langle A_{k}\otimes I_{BC}+I_{A}\otimes(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})\rangle^{2}\}$
	$\displaystyle=\sum_{k}\Delta(A_{k})^{2}_{\rho_{A}}+\sum_{k}\Delta(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})^{2}_{\rho_{BC}}$
	$\displaystyle+2\sum_{k}[\langle A_{k}\otimes(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})\rangle-$
	$\displaystyle\langle A_{k}\otimes I_{BC}\rangle\langle I_{A}\otimes(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})\rangle]$
	$\displaystyle\geq\sum_{k}\Delta(A_{k})^{2}_{\rho_{A}}+\sum_{k}\Delta(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})^{2}_{\rho_{BC}}$
	$\displaystyle-2\sqrt{[\sum_{k}\Delta(A_{k})^{2}_{\rho_{A}}-U_{\rho_{A}}]}\cdot\sqrt{[\sum_{k}\Delta(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})^{2}_{\rho_{BC}}-U_{\rho_{BC}}]}$
	$\displaystyle=U_{\rho_{A}}+U_{\rho_{BC}}+W^{2}_{ABC},$		(14)

where $W_{ABC}=\sqrt{\sum_{k}\Delta(A_{k})^{2}_{\rho_{A}}-U_{\rho_{A}}}-\sqrt{\sum_{k}\Delta(B_{k}\otimes I_{C}+I_{B}\otimes C_{k})^{2}_{\rho_{BC}}-U_{\rho_{BC}}}$ . The inequality is due to Lemma 1 in CJZ .

Combining Eq. (III.1) and Eq. (III.1), we can obtain Eq. (11). In Eq. (11), the first term in the bracket $\{\}$ , namely, $U_{\rho_{A}}+U_{\rho_{BC}}+W^{2}_{ABC}$ is implied by the biseparable state $\sum_{R}\eta_{R}^{1}\rho_{1}^{R}\otimes\rho_{23}^{R}$ . Similarly, the second term is implied by the biseparable state $\sum_{R}\eta_{R^{\prime}}^{2}\rho_{2}^{R^{\prime}}\otimes\rho_{13}^{R^{\prime}}$ , and the third term is implied by the biseparable state $\sum_{R}\eta_{R^{\prime\prime}}^{3}\rho_{3}^{R^{\prime\prime}}\otimes\rho_{12}^{R^{\prime\prime}}$ . Violation of the inequality (11) is sufficient to confirm genuine tripartite entanglement of $\rho_{ABC}$ . $\square$ ∎

When we choose spin observables as the observables $A$ , $B$ , and $C$ , the criteria in Theorem 3 require only the statistics of a set of observables. In this sense, it is state independent, which is similar to RYMD . In order to compare Theorem 3 with criterion 1 in RYMD , we consider the sum of $\Delta^{2}u$ and $\Delta^{2}v$ where

	$\displaystyle u=h_{1}J_{x,1}+h_{2}J_{x,2}+h_{3}J_{x,3}$
	$\displaystyle v=g_{1}J_{y,1}+g_{2}J_{y,2}+g_{3}J_{y,3}$		(15)

and $h_{k}$ and $g_{k}$ $({k=1,2,3})$ are real numbers. Here $J_{x,k}$ , $J_{y,k}$ , $J_{z,k}$ are the spin operators for subsystem $k$ , satisfying $[J_{x,k},J_{y,k}]=J_{z,k}$ . Then $F_{\rho_{ABC}}$ in Eq. (11) is equal to $\Delta^{2}u+\Delta^{2}v$ when $A_{1}=h_{1}J_{x,1}$ , $B_{1}=h_{2}J_{x,2}$ , $C_{1}=h_{3}J_{x,3}$ , $A_{2}=g_{1}J_{y,1}$ , $B_{2}=g_{2}J_{y,2}$ , $C_{2}=g_{3}J_{y,3}$ and $k=2$ . This leads us to the following criterion,

Corollary 4

Violation of the inequality

$\displaystyle\Delta^{2}u+\Delta^{2}v\geq\min$	$\displaystyle\{\|g_{1}h_{1}\langle J_{z,1}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{123},$	(16)
	$\displaystyle\|g_{2}h_{2}\langle J_{z,2}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{213},$
	$\displaystyle\|g_{3}h_{3}\langle J_{z,3}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle\|+W^{2}_{312}\}$

is sufficient to confirm genuine tripartite entanglement. Here

	$\displaystyle W_{123}=\sqrt{\Delta^{2}(h_{1}J_{x,1})+\Delta^{2}(g_{1}J_{y,1})-\|g_{1}h_{1}\langle J_{z,1}\rangle\|}-$
	$\displaystyle\sqrt{\Delta^{2}(h_{2}J_{x,2}+h_{3}J_{x,3})+\Delta^{2}(g_{2}J_{y,2}+g_{3}J_{y,3})-\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|},$

$W_{213}$ and $W_{312}$ can be similarly defined.

If $W_{123}=W_{213}=W_{312}=0$ , Eq. (16) is reduced to the result in RYMD , so the above criterion is better than criterion 1 in RYMD . For the specific spin state, we can choose the optimal values for $h_{k}$ , $g_{k}$ .

III.2 CRITERIA FOR GENUINE MULTIPARTITE ENTANGLEMENT

Now we extend the method in previous section used to derive criteria for genuine tripartite entanglement to $N$ -partite system. One can show that the number of possible bipartition is $2^{N-1}-1$ . In order to investigate the criteria of genuine $N$ -partite entanglement, we should consider every bipartition. Here we generalize the criterion in Eq. (11) and Eq. (16) for $N$ -partite system. We denote every bipartition by $S_{r}-S_{s}$ , where $S_{r}$ and $S_{s}$ are the sets of two part in a specific bipartition.

Theorem 5

If a $N$ -partite quantum state $\rho_{A_{1}A_{2}\ldots A_{N}}$ is biseparable, then

\displaystyle F_{\rho_{A_{1}A_{2}\ldots A_{N}}}\geq\min\{U_{BS}\},

(17)

where $U_{BS}$ is the set of the quantity $U_{\rho_{k_{r}}}+U_{\rho_{k_{s}}}+W^{2}_{\rho_{k_{r}|k_{s}}}$ defined for each partition $S_{r}-S_{s}$ , $\rho_{k_{r}}$ and $\rho_{k_{s}}$ are the states in set $S_{r}$ and $S_{s}$ respectively. Violation of the inequality (17) is sufficient to confirm genuine $N$ -partite entanglement. The proof of the inequality follows from the proof in Eq. (11).

When the observables in Eq. (17) are spin observables, the following criterion can be obtained.

Corollary 6

Violation of the inequality

\displaystyle\Delta^{2}u+\Delta^{2}v\geq\min\{S_{B}\}

(18)

implies genuine $N$ -partite entanglement. Where $S_{B}$ is the set of $|\Sigma_{k_{r}=1}^{m}h_{k_{r}}g_{k_{r}}\langle J_{z,k_{r}}\rangle|+|\Sigma_{k_{s}=1}^{n}h_{k_{s}}g_{k_{s}}\langle J_{z,k_{s}}\rangle|+W^{2}_{\rho_{k_{r}|k_{s}}}$ defined for each partition $S_{r}-S_{s}$ . When every $W_{\rho_{k_{r}|k_{s}}}=0$ , the inequality is reduced to criterion 6 in RYMD .

For $N=4$ , there will be $2^{4-1}-1=7$ bipartition. They are $1-234$ , $2-134$ , $3-124$ , $4-123$ , $12-34$ , $13-24$ , $14-23$ . Using them we obtain the criterion for genuine four-partite entanglement.

Corollary 7

If a four-partite quantum state is biseparable, then

$\displaystyle\Delta^{2}u+\Delta^{2}v\geq\min$	$\displaystyle\{\|g_{1}h_{1}\langle J_{z,1}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{1\|234},$	(19)
	$\displaystyle\|g_{2}h_{2}\langle J_{z,2}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{2\|134},$
	$\displaystyle\|g_{3}h_{3}\langle J_{z,3}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{3\|124},$
	$\displaystyle\|g_{4}h_{4}\langle J_{z,4}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{4\|123},$
	$\displaystyle\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle\|+\|g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{12\|34},$
	$\displaystyle\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{13\|24},$
	$\displaystyle\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{14\|23}\}$

where

	$\displaystyle W_{1\|234}=\sqrt{\Delta^{2}(h_{1}J_{x,1})+\Delta^{2}(g_{1}J_{y,1})-\|g_{1}h_{1}\langle J_{z,1}\rangle\|}-$
	$\displaystyle\sqrt{\Delta^{2}(h_{2}J_{x,2}+h_{3}J_{x,3}+h_{4}J_{x,4})+\Delta^{2}(g_{2}J_{y,2}+g_{3}J_{y,3}+g_{4}J_{y,4})-\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|},$

$W_{2|134}$ , $W_{3|124}$ , $W_{4|123}$ , $W_{12|34}$ , $W_{13|24}$ , and $W_{14|23}$ can be similarly defined.

The violation of the inequality in Eq. (19) implies genuine four-partite entanglement. If $W_{1|234}=W_{2|134}=W_{3|124}=W_{4|123}=W_{12|34}=W_{13|24}=W_{14|23}=0$ , Eq. (19) is reduced to the criterion 8 in RYMD , so the above inequality is better than the result in RYMD .

IV EXAMPLE

In this section, we illustrate the utility of the criteria by a few examples.

Example 8

Consider the $n$ -qubit W state mixed with the white noise,

\rho_{W_{n}}(q)=\frac{1-q}{2^{n}}I+q|W_{n}\rangle\langle W_{n}|,

where $0\leq q\leq 1$ , $|W_{n}\rangle=\frac{1}{\sqrt{2}}(|10\ldots 00\rangle+|01\ldots 00\rangle+\ldots+|00\ldots 01\rangle)$ and $I$ is the $2^{n}\times 2^{n}$ identity matrix.

Set $A_{1}=B_{1}=-C_{1}=\sigma_{x}$ , $A_{2}=B_{2}=-C_{2}=\sigma_{y}$ , and $A_{3}=B_{3}=C_{3}=\sigma_{z}$ . The criterion in Theorem 3 is computed to be $f(q)=-q^{2}-\frac{14}{3}q+\frac{35}{9}-(\sqrt{\frac{1}{9}-\frac{1}{9}q^{2}}-\sqrt{-\frac{4}{9}q^{2}-\frac{10}{3}q+\frac{34}{9}})^{2}$ , which means the left side of (11) minus the right of that, as shown in FIG 1. Comparing with the criterion in LM14 and QIP2020 , Theorem 3 can detect more genuinely tripartite entangled states.

Refer to caption — Figure 1: The abscissa and ordinate represent $q$ and $f(q)$ , respectively. Below the abscissa axis means that Theorem 3 can detect genuinely entangled state for $0.512\leq q\leq 1$ .

Now we consider genuine entanglement of $\rho_{W_{n}}(q)$ for three cases.

1.

When $n=4$ , we set $A_{1}=B_{1}=C_{1}=-D_{1}=\sigma_{x}$ , $A_{2}=B_{2}=C_{2}=-D_{2}=\sigma_{y}$ , and $A_{3}=B_{3}=C_{3}=D_{3}=\sigma_{z}$ in Theorem 5. By calculation, we can obtain $f_{4}(q)=-4q^{2}+3-(\sqrt{-q^{2}+2q}-\sqrt{-q^{2}-2q+3})^{2}$ , which means the left side of (17) minus the right side.
2.

when $n=5$ , we set $A_{1}=B_{1}=C_{1}=-D_{1}=-E_{1}=\sigma_{x}$ , $A_{2}=B_{2}=C_{2}=-D_{2}=-E_{2}=\sigma_{y}$ , and $A_{3}=B_{3}=C_{3}=D_{3}=E_{3}=\sigma_{z}$ . By calculation, we can obtain $f_{5}(q)=-9q^{2}+\frac{4}{5}q+\frac{91}{25}-(\sqrt{-\frac{81}{25}q^{2}-\frac{2}{5}q+\frac{91}{25}}-\sqrt{-\frac{36}{25}q^{2}+2q})^{2}$ .
3.

When $n=6$ , we set $A_{1}=B_{1}=C_{1}=-D_{1}=-E_{1}=-F_{1}=\sigma_{x}$ , $A_{2}=B_{2}=C_{2}=-D_{2}=-E_{2}=-F_{2}=\sigma_{y}$ , and $A_{3}=B_{3}=C_{3}=D_{3}=E_{3}=F_{3}=\sigma_{z}$ . By calculation, we can obtain $f_{6}(q)=-16q^{2}+6q+\frac{77}{9}-(\sqrt{-\frac{100}{9}q^{2}+4q+\frac{64}{9}}-\sqrt{\frac{4}{9}-\frac{4}{9}q^{2}})^{2}$ .

We describe the three cases in FIG 2. The same method can be used when $n\geq 7$ by choosing appropriate observables, but the calculation will become more and more complex.

It is worth mentioning that xyc2020 the noisy $W$ state $\rho_{W_{n}}(q)$ is fully separable if

\displaystyle q\leq\left\{\begin{array}[]{cc}\frac{1}{1+2^{n}\sqrt{\frac{n-1}{2n}}}&if\quad 2\leq n\leq 5;\\ \frac{n}{n+(n-2)2^{n}}&if\quad n\geq 6.\end{array}\right.

(22)

The condition is necessary and sufficient when $n\leq 5$ . This is similar to the genuine entanglement criterion in Theorems 3 and 5, that is, (22) can detect more states with the increase of $n$ . We describe these results in FIG 3.

This criterion can not only detect the GME of qubit states, but also detect that of qurit states. Here is an example of the 3-qutrit state.

Example 9

Consider a $3$ -qutrit state mixed with the white noise LM14 ,

\rho=\frac{1-x}{27}I+x|\varphi\rangle\langle\varphi|,

where $0\leq x\leq 1$ , $|\varphi\rangle=\frac{1}{\sqrt{3}}(|012\rangle+|021\rangle+|111\rangle)$ and $I$ is the $3^{3}\times 3^{3}$ identity matrix.

Set $A_{1}=-B_{1}=-C_{1}=J_{x}$ , $A_{2}=-B_{2}=-C_{2}=J_{y}$ , and $A_{3}=B_{3}=C_{3}=J_{z}$ . Here $J_{x}$ , $J_{y}$ , $J_{z}$ are spin operators. The criterion in Theorem 3 is computed to be $f(x)=\frac{25}{9}-4x-(\sqrt{-\frac{x^{2}}{9}-3x+\frac{28}{9}}-\sqrt{-\frac{x^{2}}{9}+\frac{x}{3}+\frac{7}{3}})^{2}$ , which means the left side of (11) minus the right of that, as shown in FIG 4. The criterion can detect GME better than the criterion in LM14 .

V CONCLUSION

The detection of GME is a basic and important object in quantum theory. In view of the bipartite entanglement and tripartite non-fully separable criteria based on LUR, we have studied the GME based on LUR. We have obtained an effective criterion to detecting the GME for tripartite system, which be extended to multipartite system. Comparing with some existing criteria, the criterion can detect more genuinely entangled states by theoretical analysis and numerical examples. Also, we found the relation of $n$ and genuinely entanglement for $n$ - qubit noisy W state. The method used in this paper can also be generalized to arbitrary multipartite qudit systems. It would be also worthwhile to investigate the $k$ -separability of multipartite systems.

Acknowledgments Authors were supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12080401 and ZG216S1902).

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$\displaystyle\Delta^{2}u+\Delta^{2}v\geq\min$	$\displaystyle\{\|g_{1}h_{1}\langle J_{z,1}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{123},$	(16)
	$\displaystyle\|g_{2}h_{2}\langle J_{z,2}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{213},$
	$\displaystyle\|g_{3}h_{3}\langle J_{z,3}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle\|+W^{2}_{312}\}$

$\displaystyle\Delta^{2}u+\Delta^{2}v\geq\min$	$\displaystyle\{\|g_{1}h_{1}\langle J_{z,1}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{1\|234},$	(19)
	$\displaystyle\|g_{2}h_{2}\langle J_{z,2}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{2\|134},$
	$\displaystyle\|g_{3}h_{3}\langle J_{z,3}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{3\|124},$
	$\displaystyle\|g_{4}h_{4}\langle J_{z,4}\rangle\|+\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{4\|123},$
	$\displaystyle\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{2}h_{2}\langle J_{z,2}\rangle\|+\|g_{3}h_{3}\langle J_{z,3}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{12\|34},$
	$\displaystyle\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+W^{2}_{13\|24},$
	$\displaystyle\|g_{1}h_{1}\langle J_{z,1}\rangle+g_{4}h_{4}\langle J_{z,4}\rangle\|+\|g_{2}h_{2}\langle J_{z,2}\rangle+g_{3}h_{3}\langle J_{z,3}\rangle\|+W^{2}_{14\|23}\}$