Criteria to detect genuine multipartite entanglement using spin measurements

TL;DR

This paper develops inequalities based on spin measurements to detect genuine multipartite entanglement, applicable to various quantum systems, and demonstrates their effectiveness through experimental examples involving atomic ensembles, optical modes, and Bose-Einstein condensates.

Contribution

It introduces new spin-variance inequalities for certifying genuine multipartite entanglement, extending previous work to fixed spin systems and linking to Stokes operator measurements.

Findings

Inequalities can detect multipartite entanglement via local spin measurements.

Violation of inequalities indicates creation of spin squeezing and entanglement.

Experimental demonstrations confirm the inequalities' effectiveness across different quantum platforms.

Abstract

We derive conditions in the form of inequalities to detect the genuine $N$-partite entanglement of $N$ systems. The inequalities are expressed in terms of variances of spin operators, and can be tested by local spin measurements performed on the individual systems. Violation of the inequalities is sufficient (but not necessary) to certify the multipartite entanglement, and occurs when a type of spin squeezing is created. The inequalities are similar to those derived for continuous-variable systems, but instead are based on the Heisenberg spin-uncertainty relation $\Delta J_{x}\Delta J_{y}\geq|\langle J_{z}\rangle|/2$. We also extend previous work to derive spin-variance inequalities that certify the full tripartite inseparability or genuine multi-partite entanglement among systems with fixed spin $J$, as in Greenberger-Horne-Zeilinger (GHZ) states and W states where $J=1/2$. These inequalities are derived from the planar spin-uncertainty relation $(\Delta J_{x})^{2}+(\Delta J_{y})^{2}\geq C_{J}$ where $C_{J}$ is a constant for each $J$. Finally, it is shown how the inequalities detect multipartite entanglement based on Stokes operators. We illustrate with experiments that create entanglement shared among separated atomic ensembles, polarization-entangled optical modes, and the clouds of atoms of an expanding spin-squeezed Bose-Einstein condensate. For each example, we give a criterion to certify the mutual entanglement.

Tables

Table 1. Table 1: Values of the gains g 𝑔 g and h ℎ h that minimize the variance sum and product in Criteria 1 and 3.

r	CV GHZ		CV EPR
	$h$	$g$	$h$	$g$
0	0	0	0	0
0.25	0.36	-0.27	0.33	-0.33
0.50	0.68	-0.40	0.54	-0.54
0.75	0.86	-0.46	0.64	-0.64
1.00	0.95	-0.49	0.68	-0.68
1.50	0.99	-0.50	0.70	-0.70
2.00	1.00	-0.50	0.70	-0.70

Equations

$${\rho }_{BS}$$

$$+P_3\sum _{R^{\prime }}{\eta }_{R^{\prime \prime }}^{(3)}{\rho }_3^{R^{\prime \prime }}{\rho }_{12}^{R^{\prime \prime }},$$

$${\rho }_S$$

$$u$$

$$v$$

$${\rho }_{BS}$$

$$+P_3\sum _{R^{\prime \prime }}{\eta }_{R^{\prime \prime }}^{(3)}{\rho }_{12}^{R^{\prime \prime }}{\rho }_3^{R^{\prime \prime }}$$

$${\Delta }^{2}u$$

$${\Delta }^{2}u+{\Delta }^{2}v$$

$$+P_2\sum _{R^{\prime }}{\eta }_{R^{\prime }}^{(2)}\left[{\Delta }^2{u}_{R^{\prime }}+{\Delta }^2{v}_{R^{\prime }}\right]$$

$$+P_3\sum _{R^{\prime \prime }}{\eta }_{R^{\prime \prime }}^{(3)}\left[{\Delta }^2{u}_{R^{\prime \prime }}+{\Delta }^2{v}_{R^{\prime \prime }}\right].$$

$${\Delta }^2{u}_{\zeta }+{\Delta }^2{v}_{\zeta }$$

$${\Delta }^{2}u+{\Delta }^{2}v$$

$$+P_3\sum _{R^{\prime \prime }}{\eta }_{R^{\prime \prime }}^{(3)}\left|g_3{h}_3⟨J_{z,3}{⟩}_{R^{\prime \prime }}+g_1{h}_1⟨J_{z,1}{⟩}_{R^{\prime \prime }}+g_2{h}_2⟨J_{z,2}{⟩}_{R^{\prime \prime }}\right|$$

$${\Delta }^{2}u+{\Delta }^{2}v$$

$$\left|g_2{h}_2⟨J_{z,2}⟩\right|+\left|g_1{h}_1⟨J_{z,1}⟩+g_3{h}_3⟨J_{z,3}⟩\right|,$$

$$\left|g_3{h}_3⟨J_{z,3}⟩\right|+\left|g_1{h}_1⟨J_{z,1}⟩+g_2{h}_2⟨J_{z,2}⟩\right|\}$$

$${\Delta }^2{J}_{z,k}$$

$$\frac{1}{|⟨J_{z,k}⟩{|}^2}\{P_1⟨(J_{z,k}-⟨J_{z,k}{⟩}_I{)}^2{⟩}_I+P_2⟨(J_{z,k}-⟨J_{z,k}{⟩}_{II}{)}^2{⟩}_{II}+P_3⟨(J_{z,k}-⟨J_{z,k}{⟩}_{III}{)}^2{⟩}_{III}\}$$

$${\Delta }^{2}u+{\Delta }^{2}v$$

$$B_I$$

$$B_{II}$$

$$B_{III}$$

$$B_I$$

$$B_{II}$$

$$B_{III}$$

$${\Delta }^{2}u+{\Delta }^{2}v$$

$$B_I$$

$$B_{II}$$

$$B_{III}$$

$$B_I+B_{II}+B_{III}\ge \left|⟨J_{z,1}⟩\right|+\left|⟨J_{z,2}⟩\right|+\left|⟨J_{z,3}⟩\right|$$

$$B_I$$

$$+P_2\sum _{R^{\prime }}{\eta }_{R^{\prime }}^{(2)}\left[\left|⟨J_{z,1}{⟩}_{R^{\prime }}\right|+{\left|⟨J_{z,2}⟩\right|}_{R^{\prime }}\right]$$

$$\to \left(P_1+P_2\right)\left(\left|⟨J_{z,1}⟩\right|+\left|⟨J_{z,2}⟩\right|\right),$$

$$B_{II}$$

$$\ge P_2\sum _{R^{\prime }}{\eta }_{R^{\prime }}^{(2)}\left[\left|⟨J_{z,2}{⟩}_{R^{\prime }}\right|+{\left|⟨J_{z,3}⟩\right|}_{R^{\prime }}\right]$$

$$+P_3\sum _{R^{\prime \prime }}{\eta }_{R^{\prime \prime }}^{(3)}\left[\left|⟨J_{z,2}{⟩}_{R^{\prime \prime }}\right|+{\left|⟨J_{z,3}⟩\right|}_{R^{\prime \prime }}\right]$$

Full text

I Introduction

Genuine multipartite quantum entanglement is a resource required for many protocols in the field of quantum information and computation Gottesman (1996); Bose et al. (1998); Hillery et al. (1999); van Loock and Braunstein (2000); Raussendorf and Briegel (2001); Jing et al. (2003); Giovannetti et al. (2004); Briegel et al. (2009); Bancal et al. (2011). $N$ systems are said to be genuinely $N$-partite entangled if the systems are mutually entangled in such a way that the entanglement cannot be constructed by mixing entangled states involving fewer than $N$ parties Acín et al. (2001); Jungnitsch et al. (2011); Bancal et al. (2011). Mathematically, a tripartite system is genuinely tripartite entangled if and only if the density operator characterizing the system cannot be represented in the biseparable form Horodecki et al. (1996); Acín et al. (2001); Jungnitsch et al. (2011); Bancal et al. (2011)

[TABLE]

where $\sum_{k=1}^{3}P_{k}=1$, $P_{k}\geq 0$, and $\sum_{R}\eta_{R}^{\left(k\right)}=1$. $\rho_{k}^{R}$ is an arbitrary density operator for the subsystem $k$, while $\rho_{mn}^{R}$ is an arbitrary density operator for the subsystems $m$ and $n$. The definition of genuine $N$-partite entanglement follows similarly.

Criteria to certify genuine $N$-partite entanglement for continuous variable (CV) systems have been derived by Shalm et al. Shalm et al. (2012) and Teh and Reid Teh and Reid (2014). These criteria take the form of variance inequalities, similar to those derived for CV bipartite entanglement Reid (1989); Duan et al. (2000); Giovannetti et al. (2003). The work of Refs. Shalm et al. (2012); Teh and Reid (2014) extended earlier work by van Loock and Furusawa, who developed CV criteria for the related but different concept of full $N$-partite inseparability van Loock and Furusawa (2003); Aoki et al. (2003) (see also Refs. Sperling and Vogel (2013); Gerke et al. (2018)). Full $N$-partite inseparability rules out the possibility that the system can be described by a mixture of states of a given bipartition (so that only one of the above $P_{i}$ can be nonzero), but does not rules out mixtures of different bipartitions. Although genuine $N$-partite entanglement implies full $N$-partite inseparability, the converse is not true, and full $N$-partite inseparability is therefore a weaker form of correlation. Nonetheless, for pure states, full $N$-partite inseparability is sufficient to imply genuine $N$-partite entanglement. Experiments have confirmed both full $N$-partite inseparability Aoki et al. (2003); Coelho et al. (2009); Pysher et al. (2011); Armstrong et al. (2012); Gerke et al. (2015) and genuine $N$-partite entanglement ($N\geq 3$) for CV systems Shalm et al. (2012); Armstrong et al. (2015); Barros et al. (2015); Wu et al. (2016); Sandbo Chang et al. (2018). Here, “continuous variable (CV)” refers to the use of measurements that have continuous-variable outcomes e.g. field quadrature phase amplitudes $X$ and $P$, or position and momentum. The CV criteria are derived from the commutation relation $\left[X,P\right]=2i$ and the associated uncertainty relations.

In this paper, we derive criteria for genuine $N$-partite entanglement and full $N$-partite inseparability that are useful for discrete variable systems involving spin degrees of freedom. In this case, measurements correspond to spin observables, and it is the spin commutation relation $\left[J_{x},J_{y}\right]=iJ_{z}$ and associated spin-uncertainty relations that are relevant. The criteria we derive involve variances and apply to all physical systems, provided the measurements correspond to operators satisfying spin commutation relations. This approach extends to $N$ systems that of Hofmann and Takeuchi Hofmann and Takeuchi (2003) and Raymer et al. Raymer et al. (2003) who used spin-uncertainty relations to derive variance criteria for bipartite entanglement. The question of how to detect genuine $N$-partite entanglement has been studied previously but most work has been in the context of qubit (spin $1/2$) systems Svetlichny (1987); Collins et al. (2002); Bourennane et al. (2004); Tóth and Gühne (2005); Lougovski et al. (2009); Papp et al. (2009); Lavoie et al. (2009); Horodecki et al. (2009); Gühne and Seevinck (2010); Lu et al. (2011); He and Reid (2013) or systems of fixed dimension Huber et al. (2010); Ma et al. (2011); Spengler et al. (2013); Li et al. (2017); Regula et al. (2018); Kłobus et al. (2019).

The development of criteria to certify the genuine multipartite entanglement of discrete systems, as in this paper, is motivated by the increasing number of experiments detecting entanglement with atoms. For example, bipartite entanglement has been created between atomic ensembles and separated atomic modes Julsgaard et al. (2001); Fadel et al. (2018); Lange et al. (2018), and multi-partite entanglement has been created among the separated clouds of a Bose-Einstein condensate (BEC) Kunkel et al. (2018). It is sometimes possible to rewrite the spin commutation relation in a form that resembles the position-momentum commutation relation. This is often true where the spin observables are expressed as Schwinger operators, and justifies the use of CV entanglement criteria for the spin system in that case. For instance, Julsgaard et al. Julsgaard et al. (2001) characterize the entanglement in the collective spins between two atomic ensembles using CV criteria. However, as pointed out by Raymer et al. Raymer et al. (2003), this is only valid in a restricted sense and will not give correct results in general. In other words, the complete spin commutation relation should be used in any derivation of criteria certifying the genuine multipartite entanglement of spin systems.

The program of characterizing entanglement in spin systems has been largely motivated by the observation that a spin-squeezed system exhibits quantum correlations among the spin particles. Sørensen et al. Sørensen et al. (2001) derived an $N$-partite entanglement criterion that implies the presence of an $N$-partite entangled state. Here, an $N$-partite entangled state is a state that cannot be expressed in the form

[TABLE]

where $\sum_{R}P_{R}=1$. A host of criteria Tóth (2004); Tóth et al. (2007); Tóth (2007); Tóth et al. (2009); Vitagliano et al. (2011); He et al. (2011a) were subsequently derived to certify the presence of $N$-partite entanglement in spin systems. However, these criteria only rule out the possibility of $N$-partite separable states of the form Eq. (2) and not the more general $N$-partite biseparable states of the form Eq. (1) (as extended to higher $N$) where all separable bipartitions (and mixtures of them) are considered. Hence they are not criteria for genuine $N$-partite entanglement, where the entanglement is mutually shared among all $N$ parties. An exception is the spin-squeezing criteria of Sørensen and Mølmer (and others like it) which imply a genuine $k$-particle entanglement shared among $k$ particles of an $N$-particle system ($k\leq N$) Sørensen and Mølmer (2001). Such criteria differ from those derived in this paper, however, being based on collective spin measurements made on the composite system, rather than local measurements made on separated subsystems, and thus cannot directly test nonlocal models (as described in Ref. Cavalcanti et al. (2009)).

The task of characterizing genuine multipartite entanglement in spin systems was carried out by Korbicz et al. Korbicz et al. (2005, 2006). Korbicz and co-workers used the positivity of partial transpose (PPT) criterion or the Peres-Horodecki criterion Peres (1996); Horodecki et al. (1996) as the starting point to derive entanglement criteria, and showed genuine tripartite entanglement for symmetric states. The PPT criterion, however, is less useful for $N$-partite separability when $N$ is large Horodecki et al. (1996). In this paper, we derive criteria for genuine multipartite entanglement for spin systems by ruling out the possibility of the state in a biseparable form as in Eq. (1) using the uncertainty relations for spin operators.

The remainder of the paper is structured as follows. In Section II, we derive criteria to detect the full tripartite inseparability and genuine tripartite entanglement of three systems using spin measurements. The generalization to genuine $N$-partite entanglement is given in Section IV. These criteria are derived using methods similar to those developed by van Loock and Fursusawa van Loock and Furusawa (2003), Teh and Reid Teh and Reid (2014) and Shalm et al. Shalm et al. (2012) for CV systems. In Section III, we extend variance criteria derived for Einstein-Podolsky-Rosen (EPR) steering by He and Reid He and Reid (2013), pointing out that these inequalities apply to certify genuine tripartite entanglement as well as steering, which is a form of entanglement closely connected with the EPR paradox Wiseman et al. (2007); Jones et al. (2007); Cavalcanti et al. (2009). The criteria are derived using planar spin-uncertainty relations He et al. (2011b, 2012a); Vitagliano et al. (2017, 2018); Puentes et al. (2013) and apply to subsystems with a fixed spin $J$. We show that the criteria may be used to detect the genuine tripartite entanglement of GHZ states, and the full tripartite inseparability of W states. Finally, in Section V, we explain how to generate genuinely-entangled spin systems based on Stokes operators. We then demonstrate the application of the criteria derived in Sections II and IV to certify full $N$-partite inseparability and genuine $N$-partite entanglement of these systems.

II Criteria for full tripartite inseparability and genuine tripartite

entanglement

The criteria derived in this section involve variances of the sum of spin observables defined for each subsystem. In this work, all the caret symbols that denote the spin operators are dropped, unless specified otherwise, and we use the symbol $\Delta^{2}x$ to denote the variance of $x$.

In this section, we will derive criteria for both full tripartite inseparability and genuine tripartite entanglement. Criteria 1 and 3 will be applicable to full tripartite inseparability, but only apply to genuine tripartite entanglement in a limit of large spin. Criteria 2 and 4 are applicable to both full tripartite inseparability and genuine tripartite entanglement, for states with arbitrary spin.

II.1 The sum inequalities

II.1.1 Sum of two variances

Consider the sum of $\Delta^{2}u$ and $\Delta^{2}v$ where

[TABLE]

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 the commutation relation $\left[J_{x,k},J_{y,k}\right]=iJ_{z,k}$. We derive the bound for $\Delta^{2}u+\Delta^{2}v$ such that the violation of the bound implies the genuine tripartite entanglement in the spin degree of freedom.

This leads us to Criterion 1. Firstly, we assume that the spin state is in a biseparable mixture state

[TABLE]

as in Eq. (1). This implies that the variance of an observable $\Delta^{2}u$ is greater or equal to the sum of the variances of the observable of its component state $\Delta^{2}u_{R}$, i.e. Hofmann and Takeuchi (2003)

[TABLE]

The sum of $\Delta^{2}u$ and $\Delta^{2}v$ is then

[TABLE]

To proceed, we consider $\Delta^{2}u_{\zeta}+\Delta^{2}v_{\zeta}$ that corresponds to an arbitrary bipartition $\rho_{k}^{\zeta}\rho_{lm}^{\zeta}$:

[TABLE]

The lower bound given in this inequality is derived in the Appendix 1, using the uncertainty relations for spin. With these lower bounds for different bipartitions, Eq. (4) has the expression

[TABLE]

Criterion 1.** **Violation of the inequality

[TABLE]

is sufficient to confirm full tripartite inseparability. Genuine tripartite entanglement is confirmed if the inequality is violated in the limit where the mean spins are large, $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$, and where fluctuations satisfy $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$.

Proof. To prove the first statement, we need to rule out all any given fixed bipartition i.e. we need to exclude (1) where only one of the $P_{i}$ can be nonzero. This follows straightforwardly, from (4) on using (5) where only one of the $P_{1}$, $P_{2,}$ or $P_{3}$ can be nonzero. Taking the minimum value (5) over the three possible bipartitions gives the required result.

To prove the second result, we first note that

[TABLE]

where we define $\langle J_{z,k}\rangle_{I}=\sum_{R}\eta_{R}^{(1)}\langle J_{z,k}\rangle_{R}$, $\langle J_{z,k}\rangle_{II}=\sum_{R^{\prime}}\eta_{R^{\prime}}^{(1)}\langle J_{z,k}\rangle_{R^{\prime}}$ and $\langle J_{z,k}\rangle_{III}=\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(1)}\langle J_{z,k}\rangle_{R^{\prime\prime}}$. The subscripts $I$, $II$ and $III$ indicate averages and variances over all the states with the given bipartitions $\rho_{1}\rho_{23}$, $\rho_{2}\rho_{13}$ and $\rho_{3}\rho_{12}$, respectively. We thus obtain

[TABLE]

as $|\langle J_{z,k}\rangle|\rightarrow\infty$. This also implies, from Eq. (8), that $\frac{\langle J_{z,k}\rangle_{I}}{\langle J_{z,k}\rangle}=\frac{\langle J_{z,k}\rangle_{II}}{\langle J_{z,k}\rangle}=\frac{\langle J_{z,k}\rangle_{III}}{\langle J_{z,k}\rangle}\rightarrow 1.$ Continuing from Eq. (6), $\Delta^{2}u+\Delta^{2}v$ satisfies the following inequality:

[TABLE]

We can always choose for the lower bound the smallest value of the terms in bracket in Eq. (9). Hence, Eq. (9) becomes Eq. (7), where we use the fact that $\sum\eta_{R}^{(1)}=1$ and $\sum P_{k}=1$. $\square$

In Eq. (7), the first term in the bracket $\{\}$ is implied by the biseparable state $\rho_{1}\rho_{23}$, the second term is implied by the biseparable state $\rho_{2}\rho_{13}$, and the final term is implied by the biseparable state $\rho_{3}\rho_{12}$. We have taken large mean spins and small fluctuations, such that $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$ and hence $\Delta J_{z,k}/|\langle J_{z,k}\rangle|\rightarrow 0$ as $|\langle J_{z,k}\rangle|\rightarrow\infty$. The optimal values for $g_{k},h_{k}$ depend on the specific spin state. The criterion given by Eq. (7) is thus a general result that allows us to derive a host of other criteria. Examples of optimal choices for different types of spin states will be given in Section V. We derive further criteria for genuine tripartite entanglement without the limiting assumptions in the next subsection.

II.1.2 Van Loock-Furusawa inequalities for spin

We can also derive the spin version of a set of inequalities derived by van Loock and Furusawa van Loock and Furusawa (2003). The quantities $B_{I}$, $B_{II}$ and $B_{III}$ are defined as

[TABLE]

Criteria can be derived both in the limiting case $\Delta J_{z,k}/|\langle J_{z,k}\rangle|\rightarrow 0$ as $|\langle J_{z,k}\rangle|\rightarrow\infty$, considered in Criterion 1, and more generally, as we will show in Criterion 2. In the limiting case considered above, the inequalities can be used to demonstrate tripartite inseparability and genuine tripartite entanglement. We show this next.

Considering $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, the violation of any two of the inequalities in the set of inequalities below:

[TABLE]

implies full tripartite inseparability.

Proof. Note that for a biseparable state $\rho_{1}\rho_{23}=\sum_{R}\eta_{R}^{(1)}\rho_{1}^{R}\rho_{23}^{R}$, the inequality $\Delta^{2}u+\Delta^{2}v\geq\left|g_{1}h_{1}\langle J_{z,1}\rangle_{R}\right|+\left|g_{2}h_{2}\langle J_{z,2}\rangle_{R}+g_{3}h_{3}\langle J_{z,3}\rangle_{R}\right|$ holds for a tripartite separable state, while the inequality $\Delta^{2}u+\Delta^{2}v\geq\left|g_{2}h_{2}\langle J_{z,2}\rangle_{R^{\prime}}\right|+\left|g_{1}h_{1}\langle J_{z,1}\rangle_{R^{\prime}}+g_{3}h_{3}\langle J_{z,3}\rangle_{R^{\prime}}\right|$ is satisfied by the biseparable state $\rho_{2}\rho_{13}=\sum_{R^{\prime}}\eta_{R^{\prime}}^{(2)}\rho_{2}^{R^{\prime}}\rho_{13}^{R^{\prime}}$, and the biseparable state $\rho_{3}\rho_{12}=\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(3)}\rho_{3}^{R^{\prime\prime}}\rho_{12}^{R^{\prime\prime}}$ satisfies $\Delta^{2}u+\Delta^{2}v\geq\left|g_{3}h_{3}\langle J_{z,3}\rangle_{R^{\prime\prime}}\right|+\left|g_{1}h_{1}\langle J_{z,1}\rangle_{R^{\prime\prime}}+g_{2}h_{2}\langle J_{z,2}\rangle_{R^{\prime\prime}}\right|$. In the limit of $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, the derivation as in Eq. (9) leads to measurable bounds. The inequalities that are satisfied by $\rho_{1}\rho_{23}$, $\rho_{2}\rho_{13}$ and $\rho_{3}\rho_{12}$ are respectively

[TABLE]

By choosing the coefficients $g_{k}$ and $h_{k}$ in Eq. (12), we obtain a set of inequalities satisfied by $B_{I}$, $B_{II}$ and $B_{III}$. For example, the left side of the criterion in Eq. (12) is equal to $B_{I}$ when $h_{1}=1,\,h_{2}=-1,\,h_{3}=0$ and $g_{1}=g_{2}=1$. This gives

[TABLE]

which is implied by the biseparable states $\rho_{1}\rho_{23}$ and $\rho_{2}\rho_{13}$. Similarly choosing $h_{2}=g_{2}=g_{3}=-h_{3}=1$ and $h_{1}=0$, we obtain $B_{II}\geq\left(\left|\langle J_{z,2}\rangle\right|+\left|\langle J_{z,3}\rangle\right|\right)$ which is implied by the biseparable states $\rho_{2}\rho_{13}$ and $\rho_{3}\rho_{12}$. Finally, $B_{III}\geq\left(\left|\langle J_{z,1}\rangle\right|+\left|\langle J_{z,3}\rangle\right|\right)$ is satisfied by the biseparable states $\rho_{1}\rho_{23}$ and $\rho_{3}\rho_{12}$. Thus, in the limiting case $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, full tripartite inseparability can be proved if any two of the inequalities (13) are violated. $\square$

The full tripartite inseparability does not rule out the possibility of a biseparable mixture state, and thus, does not signify genuine tripartite entanglement. When the biseparable mixture state is taken into account, the van Loock-Furusawa set of inequalities is given below:

[TABLE]

This leads to a condition for genuine tripartite inseparability. Genuine tripartite entanglement is confirmed if the inequality

[TABLE]

is violated.

*Proof. *It is first useful to prove this in the limiting case where with $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$. By choosing $h_{1}=g_{1}=-h_{2}=g_{2}=1$ and $h_{3}=0$, inequality Eq. (6) becomes

[TABLE]

where the final line is obtained using the same limit as in Eq. (9) with $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$. Similarly choosing $h_{2}=g_{2}=g_{3}=-h_{3}=1$ and $h_{1}=0$

[TABLE]

in the same limit. Finally,

[TABLE]

Using the inequalities in Eq. (13), we can readily prove the criterion for genuine tripartite entanglement in the limiting case. Taking the sum of $B_{I}$, $B_{II}$ and $B_{III}$ in Eqs. (15), (16) and (17) respectively, we obtain

[TABLE]

Continuing, we see that

[TABLE]

$\square$

The condition for genuine tripartite entanglement can be proved for all parameters, without the assumptions $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$. We do this below.

Criterion 2. For arbitrary states, both full tripartite inseparability and genuine tripartite entanglement are observed if the inequality

[TABLE]

is violated.

Proof. By choosing $h_{1}=g_{1}=-h_{2}=g_{2}=1$ and $h_{3}=0$, inequality Eq. (6) becomes

[TABLE]

since the term involving $P_{3}$ is always non-negative. Similarly choosing $h_{2}=g_{2}=g_{3}=-h_{3}=1$ and $h_{1}=0$

[TABLE]

Similarly,

[TABLE]

From these inequalities, we now prove Criterion 2 for genuine tripartite entanglement:

[TABLE]

where we use that $\langle J_{z,k}\rangle=P_{1}\sum_{R}\eta_{R}^{(1)}\langle J_{z,k}\rangle_{R}+P_{2}\sum_{R^{\prime}}\eta_{R^{\prime}}^{(2)}\langle J_{z,k}\rangle_{R^{\prime}}+P_{3}\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(3)}\langle J_{z,k}\rangle_{R^{\prime\prime}}$ and hence $|\langle J_{z,k}\rangle|\leq P_{1}\sum\eta_{R}^{(1)}|\langle J_{z,k}\rangle_{R}|+P_{2}\sum_{R^{\prime}}\eta_{R^{\prime}}^{(2)}|\langle J_{z,k}\rangle_{R^{\prime}}|+P_{3}\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(3)}|\langle J_{z,k}\rangle_{R^{\prime\prime}}|$.

It follows that the criterion is also for full tripartite inseparability, since full tripartite inseparability is a special case of genuine tripartite entanglement. $\square$

The number of moment measurements in the criterion given by Eq. (18) can be reduced by using a criterion that only involves two of the three quantities $B_{I}$, $B_{II}$ and $B_{III}$. For the limiting case of $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, we can use the inequalities (13) to deduce that

[TABLE]

is satisfied by any mixture of all tripartite biseparable states. The violation of the criterion in Eq. (23) then implies genuine tripartite entanglement. This is also true for other combinations $B_{I}+B_{III}\geq 2\left|\langle J_{z,1}\rangle\right|+\left|\langle J_{z,2}\rangle\right|+\left|\langle J_{z,3}\rangle\right|$ and $B_{II}+B_{III}\geq\left|\langle J_{z,1}\rangle\right|+\left|\langle J_{z,2}\rangle\right|+2\left|\langle J_{z,3}\rangle\right|$.

However, as before, we can derive an inequality that is valid generally, without the assumption $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$. Setting $g_{1}=g_{2}=g_{3}=1$, we see from Eqs. (19), (20) and (21) that

[TABLE]

Continuing, we find

[TABLE]

The triangle inequality is used to obtain the last line. This inequality is thus satisfied by any mixture of all tripartite biseparable states. Thus, we have proved the following criterion, which holds true for an arbitrary state.

Criterion 2b: The violation of the inequality $B_{I}+B_{II}\geq\left|\langle J_{z,2}\rangle\right|$ in Eq. (24) implies full tripartite inseparability and genuine tripartite entanglement. This is also true for other combinations $B_{I}+B_{III}\geq\left|\langle J_{z,1}\rangle\right|$ and $B_{II}+B_{III}\geq\left|\langle J_{z,3}\rangle\right|$.

II.2 The product inequalities

II.2.1 Product of two variances

Criteria involving products rather than sums can also be derived. Again, we consider the two quantities $\Delta^{2}u=\Delta^{2}\left(h_{1}J_{x,1}+h_{2}J_{x,2}+h_{3}J_{x,3}\right)$ and $\Delta^{2}v=\Delta^{2}\left(g_{1}J_{y,1}+g_{2}J_{y,2}+g_{3}J_{y,3}\right)$.The product of two variances $\Delta^{2}u$ and $\Delta^{2}v$ satisfies the inequality

[TABLE]

where the Cauchy-Schwarz inequality is used. For an arbitrary bipartition $\rho_{k}^{\zeta}\rho_{lm}^{\zeta}$, $\Delta^{2}u_{\zeta}\Delta^{2}v_{\zeta}$ satisfies the inequality (see Appendix 2):

[TABLE]

Again, provided that $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $(\Delta J_{z,k})^{2}/\left|\langle J_{z,k}\rangle\right|\leq 1$, the following criterion can be obtained along similar lines to the proof given in Section II.A above. We omit the proof and only present the result here.

Criterion 3. Full tripartite inseparability is observed if the inequality

[TABLE]

is violated. Genuine tripartite entanglement is confirmed if the inequality is violated in the limit where the mean spins are large, $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$, and where fluctuations satisfy $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$.

II.2.2 Van Loock-Furusawa product inequalities

The product version of the van Loock-Furusawa inequalities can be obtained, with proofs that proceed in the same way as the corresponding van Loock-Furusawa sum inequalities. The quantities involved are $S_{I}$, $S_{II}$, and $S_{III}$, as defined below:

[TABLE]

First, we look at the case with $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $(\Delta J_{z,k})^{2}/\left|\langle J_{z,k}\rangle\right|\leq 1$. By choosing the coefficients $g_{i}$ and $h_{i}$ in Eq. (27), we obtain a set of inequalities satisfied by $S_{I}$, $S_{II}$ and $S_{III}$. For example, the left side of the criterion in Eq. (27) is equal to $S_{I}$ when $h_{1}=1,\,h_{2}=-1,\,h_{3}=0$ and $g_{1}=g_{2}=1$. Similar to Eq. (11), the product version of the set of van Loock-Furusawa inequalities $S_{I}$, $S_{II}$ and $S_{III}$ satisfy the following inequalities*:*

[TABLE]

and the violation of any two of these inequalities implies full tripartite inseparability. Again, following the same derivation as in the sum version of van Loock-Furusawa set of inequalities, when the biseparable mixture state is taken into account, the product version of the set of inequalities is given below:**

[TABLE]

Genuine tripartite entanglement is observed if the inequality $S_{I}+S_{II}+S_{III}\geq\frac{1}{2}\left(\left|\langle J_{z,1}\rangle\right|+\left|\langle J_{z,2}\rangle\right|+\left|\langle J_{z,3}\rangle\right|\right)\;$ is violated. This is first proved in the limiting case $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $(\Delta J_{z,k})^{2}/\left|\langle J_{z,k}\rangle\right|\leq 1$ below.

Proof. Taking the sum of $S_{I}$, $S_{II}$ and $S_{III}$ in Eq. (30), we obtain

[TABLE]

Continuing we see that

[TABLE]

as required. $\square$

Next, we derive the van-Loock Furusawa type product inequalities in the general case, without the assumption $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $(\Delta J_{z,k})^{2}/\left|\langle J_{z,k}\rangle\right|\leq 1$.

**Criterion 4. **Genuine tripartite entanglement and full tripartite inseparability are observed if the following inequality is violated:

[TABLE]

Proof. We need only prove for genuine tripartite entanglement, since full tripartite inseparability follows as a special case. By choosing $h_{1}=g_{1}=-h_{2}=g_{2}=1$ and $h_{3}=0$, inequality Eq. (25) has the expression

[TABLE]

Similarly choosing $h_{2}=g_{2}=g_{3}=-h_{3}=1$ and $h_{1}=0$

[TABLE]

Similarly,

[TABLE]

To determine the criterion for genuine tripartite entanglement, we then see that

[TABLE]

The number of moment measurements in the criterion can be reduced by using a criterion that only involves two of the three quantities $S_{I}$, $S_{II}$ and $S_{III}$. Following the proof of Criterion 2b, we obtain the following Criterion which hold true for all states.

Criterion 4b: The violation of the inequality $S_{I}+S_{II}\geq\left|\langle J_{z,2}\rangle\right|/2$ implies genuine tripartite entanglement. This is also true for other combinations $S_{I}+S_{III}\geq\left|\langle J_{z,1}\rangle\right|/2$ and $S_{II}+S_{III}\geq\left|\langle J_{z,3}\rangle\right|/2$.

III Inequalities involving planar spin uncertainty relations

The inequalities in the previous two sections used the canonical spin uncertainty relations. For certain quantum states such as the multipartite spin GHZ state, the right side of these inequalities might be zero, giving the trivial relation that a sum or product of variances should be positive. Here, we consider the planar uncertainty relation, where the sum of uncertainties in two of the orthogonal spin observables has a lower bound that is a function of the spin value of the state. The planar uncertainty relation was obtained for spin $J=1/2$ Finkel (1987) and $J=1$ Hofmann and Takeuchi (2003), and was later calculated for an arbitrary spin $J$ by He et al. He et al. (2011b). In that work, they minimized $\Delta^{2}J_{x}+\Delta^{2}J_{y}$ for a general quantum state written in the spin-$z$ basis as

[TABLE]

Here $R_{m},\phi_{m}$ are real numbers characterizing the amplitude and phase of the basis state $|J,m\rangle$, while $n$ is the normalization factor given by $n=\sum_{m=-J}^{J}R_{m}^{2}$. He et al. found the lower bound $C_{J}$ ($C_{J}>0$) such that for a given $J$

[TABLE]

Also in that work He et al. (2011b), a criterion that verifies the $N$-partite inseparability was derived. Since the total $N$-partite separable state is a probabilistic sum of tensor product of $N$ density operators, the planar uncertainty relation can be used. This is not the case for genuine multipartite entanglement where a biseparable state contains partitions that cannot be expressed as a product state of those particles/ modes in those partitions.

Nevertheless, the planar uncertainty relation can be used to detect genuine tripartite entanglement, if we use an inference variance method Reid (1989); Reid et al. (2009).

Criterion 5. Consider the inequality given by

[TABLE]

where

[TABLE]

and $O_{lm}^{(k)}$, $P_{lm}^{(k)}$ are observables defining measurements that can be made on the combined subsystems that we denote by $l$ and $m$. The violation of this inequality suffices to confirm genuine tripartite entanglement of the three systems denoted $1$, $2$ and $3$. Full tripartite inseparability is observed if

[TABLE]

for each $k=1,2,3$.

Proof. Consider $\Delta^{2}\left(J_{x,1}-O_{23}^{(1)}\right)$ and $\Delta^{2}\left(J_{y,1}-P_{23}^{(1)}\right)$ where $O_{23}^{(1)}$ and $P_{23}^{(1)}$ are operators for systems $2$ and $3$. We derive the following inequality that holds for an arbitrary pure state with a separable bipartition $\rho_{1}^{\zeta}\rho_{23}^{\zeta}$.

[TABLE]

This holds also for all mixtures of separable bipartitions $\rho_{1}^{\zeta}\rho_{23}^{\zeta}$. Similarly, the inequalities

[TABLE]

and

[TABLE]

follow from the separable bipartitions $\rho_{2}^{\zeta}\rho_{13}^{\zeta}$ and $\rho_{3}^{\zeta}\rho_{12}^{\zeta}$ respectively. For a pure state, if all three inequalities are violated, we can conclude that the three systems are genuinely tripartite entangled. For a mixed state the conditions change. We require to falsify an arbitrary biseparable mixed state given by $\rho_{BS}=P_{1}\sum_{R}\eta_{R}^{\left(1\right)}\rho_{1}^{R}\rho_{23}^{R}+P_{2}\sum_{R^{\prime}}\eta_{R^{\prime}}^{\left(2\right)}\rho_{2}^{R^{\prime}}\rho_{13}^{R^{\prime}}+P_{3}\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(3)}\rho_{3}^{R^{\prime\prime}}\rho_{12}^{R^{\prime\prime}}$, as defined by Eq. (1). We give a proof similar to those given for Criteria 2 and 4. For brevity, we index the biseparable states $\sum_{R}\eta_{R}^{\left(1\right)}\rho_{1}^{R}\rho_{23}^{R}$, $\sum_{R^{\prime}}\eta_{R^{\prime}}^{\left(2\right)}\rho_{2}^{R^{\prime}}\rho_{13}^{R^{\prime}}$ and $\sum_{R^{\prime\prime}}\eta_{R^{\prime\prime}}^{(3)}\rho_{3}^{R^{\prime\prime}}\rho_{12}^{R^{\prime\prime}}$ by $k=1,2,3$, respectively. Thus, we denote $B_{1,1}$ to be the quantity $B_{1}$ that is evaluated using the biseparable state $\sum_{R}\eta_{R}^{\left(1\right)}\rho_{1}^{R}\rho_{23}^{R}$. Then, for the biseparable mixture,

[TABLE]

Similarly, for a biseparable mixture, $B_{2}\geq P_{2}C_{J}$ and $B_{3}\geq P_{3}C_{J}$. In order to include all possible biseparable mixtures, we consider

[TABLE]

using $\sum_{k}P_{k}=1$. Thus, all biseparable mixtures are excluded when this inequality is violated. $\square$

This inequality has been derived in Ref. He and Reid (2013) in a similar context, to give a condition for genuine tripartite steering. Steering is a form of entanglement linked to the Einstein-Podolsky-Rosen paradox, and hence a steering criterion will also be a criterion for entanglement Wiseman et al. (2007). The entanglement criterion might be made stronger, if one can make use of uncertainty relations for the operators $O_{lm}^{(k)}$ and $P_{lm}^{(k)}$ once these are established for a given scenario.

It is straightforward to see that the inequality is violated for the GHZ state Greenberger et al. (1990), defined as

[TABLE]

where $|\uparrow\uparrow\uparrow\rangle$ ($|\downarrow\downarrow\downarrow\rangle$) is the state with $z$-spins up (down) for all subsystems $k=1,2,3$. This is because, as is well-known for the GHZ state, the $z$-spin, $x$-spin and the $y$-spin of any of the three subsystems can be inferred by joint measurements made on the other two subsystems. This result is clear for inferring the value of $J_{z,k}$. The inequality (38) applies for all spin pairs, and if we replace $J_{y,i}$ with $J_{z,i}$, it is clear that by taking $P_{lm}^{(k)}=J_{z}^{(l)}$, one can achieve $\Delta^{2}\left(J_{z,k}-P_{lm}^{(k)}\right)=0$ for each $k$. For inferring $J_{x,k}$, it is also clear, since the GHZ state is an eigenstate of $J_{x,1}J_{x,2}J_{x,3}$ with eigenvalue $-1$. Thus, $O_{lm}^{(k)}$ is the measurement given as follows: Measure the spin $J_{x}$ of each of the other subsystems $l$ and $m$, and assign the value of the measurement by multiplying the spins values together. If the product is $+1$, then the outcome of $O_{lm}^{(k)}$ is $-1$. If the product is $-1$, then the outcome of $O_{lm}^{(k)}$ is $+1$. In this way, we see that $\Delta^{2}\left(J_{x,k}-O_{lm}^{(k)}\right)=0$, for each $k=1,2,3$ with $l\neq m\neq k$. Hence, the inequality (38) is violated, giving a simple method to detect the genuine tripartite entanglement of GHZ states (or approximate GHZ states) in an experiment.

We may ask whether the inequality is also violated for the $W$ state Dür et al. (2000) given by

[TABLE]

Here we will use the criterion expressed in Pauli spins, so that $B_{i}=\Delta^{2}\left(\sigma_{z,i}-O_{jk}^{(1)}\right)+\Delta^{2}\left(\sigma_{x,i}-P_{jk}^{(1)}\right)$ where $i\neq j\neq k$. The conditions then utilize $C_{J}=1$ since $J=1/2$ Finkel (1987). The spin $\sigma_{z}$ of system $1$ can be inferred by measuring the spin product of $2$ and $3$. We find that $\Delta^{2}(\sigma_{z,1}-O_{23}^{(1)})=0$. Now consider that the spins $\sigma_{x}$ of systems $2$ and $3$ are simultaneously measured. We consider the measurement $P_{23}^{(1)}$ to have an outcome of $1$ if both spins are measured as $+1$; an outcome $-1$ if the spins are measured as $-1$; and zero otherwise. Simple calculation tells us that $\Delta^{2}(\sigma_{x,1}-P_{23}^{(1)})=\frac{1}{2}$. By symmetry of the W state, this result holds for all permutations of the subsystems. Thus we see that we are able to confirm entanglement across each bipartition, since the condition (40) for Pauli spins reduces to $B_{1}\geq 1$. Since we find $B_{1}=B_{2}=B_{3}=\frac{1}{2}$, the condition for tripartite inseparability is satisfied. If in an experiment we are able to verify a pure state, then this implies genuine tripartite entanglement. We note the above condition for mixed states, $B_{1}+B_{2}+B_{3}<1$ is not satisfied. The $W$ state (44) is genuinely tripartite entangled. That the condition is not satisfied merely reflects that the criteria we derive are sufficient, but not necessary, to certify genuine tripartite entanglement. Svetlichny derived conditions to detect the genuine tripartite entanglement of three spin $1/2$ systems in the form of Bell inequalities. Further criteria for the certification of the genuine tripartite entanglement of GHZ, W and cluster states have been derived* in Refs. Tóth and Gühne (2005); Korbicz et al. (2006); Sperling and Vogel (2013). The method given above is not necessarily advantageous over these earlier methods. It can be readily extended (by applying uncertainty relation (37)) however to conditions for higher $J$.*

IV Criteria for genuine $N$-partite entanglement

The method used in Section II to derive criteria for full $N$-partite inseparability and genuine tripartite entanglement can be extended to $N$-partite systems. The complication arises in that the set of possible bipartitions scales as $\left(2^{N-1}-1\right)$, and every bipartition has to be taken into account in the derivation of these criteria that certify genuine $N$-partite entanglement.

Here, we generalize the criterion in Eq. (7) for $N$-partite spin systems. First, we present criteria that are valid when $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$. These criteria are Criteria 6, 7, 8, and 9. Then we present Criterion 10, which is a criterion for genuine $4$-partite entanglement without the extra assumptions. The derivations can be readily extended to a larger number of systems $N$.

Criterion 6. We denote each bipartition by $S_{r}-S_{s}$, where $S_{r}$ and $S_{s}$ are two sets of modes in the partitions in a specific bipartition. Then, the violation of the inequality

[TABLE]

implies full $N$-partite inseparability, where $S_{B}$ is $\left(\left|\sum_{k_{r}=1}^{m}h_{k_{r}}g_{k_{r}}\langle J_{z,k_{r}}\rangle\right|+\left|\sum_{k_{s}=1}^{n}h_{k_{s}}g_{k_{s}}\langle J_{z,k_{s}}\rangle\right|\right)$. In the limit of $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, the violation implies genuine $N$-partite entanglement. The proof for this inequality follows similarly to the proof for the inequality in Eq. (7).

Criterion 7. Similarly, the Criterion 6 holds, with the sum inequality replaced by the corresponding product inequality given by

[TABLE]

IV.1 Criteria for genuine four-partite entanglement

IV.1.1 Sum and product inequalities

Criterion 8. For $N=4$, there will be $2^{4-1}-1=7$ bipartitions. They are, using the $S_{r}-S_{s}$ notation, $1-234$, $2-134$, $3-124$, $4-123$, $12-34$, $13-24$ and $14-23$. The Criterion 6 becomes this Criterion, where the sum inequality given by Eq. (45) is

[TABLE]

Criterion 9. Similarly, the product inequality of Criterion 7 is given by

[TABLE]

where $S_{B,4}$ is defined in Eq. (47). The violation of inequality in Eq. (47) or Eq. (48) implies the presence of full $4$-partite inseparability, and genuine four-partite entanglement in the limit given by Criterion 6 and 7.

IV.1.2 Criteria involving van Loock-Furusawa inequalities

Next, we derive the van Loock-Furusawa type inequality for $N$-partite without the assumptions of large mean spins and small fluctuations. Van Loock and Furusawa van Loock and Furusawa (2003) derived a set of six inequalities to rule out four-partite inseparability. In the limiting case where $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, we can show that

[TABLE]

Following the same proof for Eq. (11) for the tripartite case, $B_{I}$ is implied by biseparable states $\rho_{1}\rho_{234}$, $\rho_{2}\rho_{134}$, $\rho_{13}\rho_{24}$ and $\rho_{14}\rho_{23}$, $B_{II}$ is implied by $\rho_{2}\rho_{134}$, $\rho_{3}\rho_{124}$, $\rho_{12}\rho_{34}$ and $\rho_{24}\rho_{13}$, $B_{III}$ is implied by $\rho_{1}\rho_{234}$, $\rho_{3}\rho_{124}$, $\rho_{12}\rho_{34}$ and $\rho_{14}\rho_{23}$, $B_{IV}$ is implied by $\rho_{3}\rho_{124}$, $\rho_{4}\rho_{123}$, $\rho_{13}\rho_{24}$ and $\rho_{23}\rho_{14}$, $B_{V}$ is implied by $\rho_{2}\rho_{134}$, $\rho_{4}\rho_{123}$, $\rho_{12}\rho_{34}$ and $\rho_{23}\rho_{14}$, and $B_{VI}$ is implied by $\rho_{1}\rho_{234}$, $\rho_{4}\rho_{123}$, $\rho_{12}\rho_{34}$ and $\rho_{13}\rho_{24}$. We note that the violation of any three of these inequalities implies that the system cannot be in any of the biseparable state, thus impying full 4-partite inseparability.

We can derive similar inequalities to certify genuine four-partite entanglement. The six spin inequalities are given by

[TABLE]

[TABLE]

Here, for brevity, we denote the biseparable states $\rho_{1}\rho_{234}$, $\rho_{2}\rho_{134}$, $\rho_{3}\rho_{124}$, $\rho_{4}\rho_{123}$, $\rho_{12}\rho_{34}$, $\rho_{13}\rho_{24}$ and $\rho_{14}\rho_{23}$ by $\rho_{I}$, $\rho_{II}$, $\rho_{III}$, $\rho_{IV}$, $\rho_{V}$, $\rho_{VI}$, and $\rho_{VII}$ respectively. In the limit of $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$, and using the derivation as in Eq. (9), Eq. (50) is then given by

[TABLE]

We then see that the violation of the inequality

[TABLE]

implies genuine 4-partite entanglement.

*Proof. *Taking the sum of the set of inequalities in Eq. (51), we obtain

[TABLE]

where $\sum_{k}P_{k}=1$. $\square$

In fact, the criterion for genuine 4-partite entanglement can be proved for the general case, without the limiting assumption, as below.

Criterion 10**. **Genuine four-partite entanglement is verified if the inequality

[TABLE]

is violated. These criteria are sufficient but not necessary conditions for four-partite inseparability, or genuine four-partite entanglement.

Proof. From the bounds given in Eq. (50), we get

[TABLE]

where $\langle J_{z,k}\rangle=P_{1}\langle J_{z,k}\rangle_{\rho_{I}}+P_{2}\langle J_{z,k}\rangle_{\rho_{II}}+P_{3}\langle J_{z,k}\rangle_{\rho_{III}}+P_{4}\langle J_{z,k}\rangle_{\rho_{IV}}+P_{5}\langle J_{z,k}\rangle_{\rho_{V}}+P_{6}\langle J_{z,k}\rangle_{\rho_{VI}}+P_{7}\langle J_{z,k}\rangle_{\rho_{VII}}$. $\square$

V Applications

We now show how one may create $N$-partite entangled states satisfying the criteria derived in Sections II and IV of this paper. In Section V.1, we outline optical experiments involving polarization entanglement, where the measured observables at each site are the Stokes operators for two polarization modes. We then consider, in Section V.2, experiments that entangle spatially-separated atomic ensembles. In Section V.3, we analyze recent experiments that generate entanglement between spatially-separated clouds of atoms formed from a spin-squeezed Bose-Einstein condensate. Here, for each separated subsystem, the measured observable is a Schwinger operator involving two internal atomic levels. The Schwinger and Stokes operators satisfy the same commutation relation as spin operators, and hence all the criteria derived in Sections II-IV are applicable.

V.1 Polarization entanglement

The polarization of a quantum state can be characterized by the Stokes operators defined as Bowen et al. (2002)

[TABLE]

where $a_{H}$ and $a_{V}$ are the annihilation operators of the horizontal and vertical polarization modes respectively, and $\theta$ is the phase difference between these polarization modes. In the work of Bowen et al. Bowen et al. (2002), bipartite polarization entanglement was created by first generating CV bipartite entanglement in the quadrature degree of freedom, and then transferring the entanglement into the polarization degree of freedom.

This scheme can be extended to generate genuine tripartite polarization entanglement. Genuine CV tripartite entanglement in the quadratures is first created in an optical setup involving squeezed vacuums and beam splitters, as shown in Figs. 1 and 2. The three entangled modes from the outputs of these beam splitters are horizontally polarized. Each of these modes is subsequently mixed with a bright coherent beam with vertical polarization using a polarizing beam splitter. At each site $i=1,2,3$ prior to mixing, one can define pairs of orthogonally polarized modes (with annihilation operators $a_{Hi}$, $a_{V,i}$). The choice of polarizer angle determines which Stokes observable is measured, after a number difference is taken. The final readout is given as a difference current. After the mixing, the genuine CV entanglement has been transformed into genuine tripartite polarization entanglement, as illustrated in Figure 3.

To verify the tripartite polarization entanglement, we consider the sum inequality

[TABLE]

where $\alpha_{v}$ is the coherent amplitude of the vertically polarized coherent beam. Here, we assume the limit of large coherent amplitude $\alpha_{v}$ and Poissonian fluctuations, which allows the linearization of the mode amplitude $a_{V}$.

The variances are

[TABLE]

Here, $S_{x,k}$, $S_{y,k}$ and $S_{z,k}$ are the Stokes operators defined in (55) for each mode pair at site $k$. $X_{H,k}$ and $P_{H,k}$ are the $X$ and $P$ quadratures for beam $k$: $P_{H,k}=a_{H,k}^{\dagger}e^{i\theta}+a_{H,k}e^{-i\theta}$ and $X_{H,k}=ia_{H,k}e^{-i\theta}-ia_{H,k}^{\dagger}e^{i\theta}$. The $h$ and $g$ are gain factors defined in the criteria, where we take $h_{1}=1$, $h_{2}=h_{3}=h$ and $g_{1}=1$, $g_{2}=g_{3}=g$. Note that the commutation relations satisfied by these Stokes operators are $\left[S_{x},S_{y}\right]=2iS_{z}$, which differ from the spin commutation relations by a factor of $2$. As a result, the sum and product inequalities below have an extra factor of $2$ compared to the sum and product inequalities in Eqs. (7) and (27) respectively. With these variances, the sum inequality Eq. (7) and the product inequality Eq. (27) are respectively transformed into a continuous-variable genuine tripartite entanglement sum and product criterion, according to

[TABLE]

and

[TABLE]

Hence, any CV genuine tripartite quadrature entanglement then implies genuine tripartite polarization entanglement.

There are two types of states that show genuine tripartite entanglement in the quadratures. These are the CV GHZ and CV EPR-type states, defined in Refs. van Loock and Furusawa (2003) and Teh and Reid (2014), and illustrated in Figs. 1 and 2 respectively. It has been shown previously that these two states violate both the quadrature sum inequality in Eq. (58) and the product inequality in Eq. (59) with specific values for the gains, $g_{1}=h_{1}=1$ and $g_{i>1}=g,\,h_{i>1}=h$ Teh and Reid (2014). The gains $g,h$ are chosen such that the variance sum and product are minima, and are given in Table 1. With these gain values, as shown in Ref. Teh and Reid (2014), Criteria 1 and 3 are always violated for any nonzero squeezing of the squeezed vacuum inputs, implying the presence of genuine tripartite entanglement. The inequalities of Criteria 1 and 3 are also useful in showing genuine tripartite entanglement. The optimal gains for these inequalities can be found in Ref. Teh and Reid (2014).

Genuine tripartite entanglement is also created using a third configuration involving only one squeezed input, shown in Fig. 4. Normally, two squeezed vacuum inputs are combined across a beam splitter to create strong EPR-correlations between the output modes Reid et al. (2009); Aoki et al. (2003). It is also possible to create EPR-entangled modes, using only one squeezed vacuum input Reid (1989). While the EPR correlations are weaker, the entanglement is sufficiently strong that a subsequent beam-splitter interaction with a non-squeezed vacuum input can create genuine tripartite entanglement. A summary of this calculation is given in the Appendix 3, where we show how the entanglement that is generated can be detected by Criterion 5 of Ref. Teh and Reid (2014) with the gains $h=-1/2$ and $g=1/2$. This tripartite entanglement is not sufficiently strong to generate tripartite EPR-steering correlations, but can be transformed into genuine tripartite polarization-entanglement using the configuration of Fig. 3. The spin sum-inequality given by Criterion 3 is then useful to detect the genuine tripartite entanglement.

V.1.1 Validity of van Loock-Furusawa type inequalities

Criteria 2, 4 and 10 can be used for any value of $\langle J_{z,k}\rangle$ and $\Delta^{2}J_{z,k}$ i.e. without the assumption $\left|\langle J_{z,k}\rangle\right|\rightarrow\infty$ and $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$. Reference Teh and Reid (2014) derives continuous-variable van Loock-Furusawa Criteria that are useful to demonstrate the genuine tripartite entanglement of the CV GHZ and EPR states discussed in Sections V.A and depicted in Figs. 1 and 2. These are given as Criteria 1 and 2 (Eqs. 14 and 15 respectively) in Ref. Teh and Reid (2014). In Section V.A, a correspondence between the spin and continuous variable cases is given. Based on that correspondence, we see that Criteria 2 and 4 are useful to demonstrate the genuine tripartite entanglement of the spin systems depicted in Fig. 3, without extra assumptions.

V.2 Tripartite entanglement of atomic

ensembles

Tripartite entanglement can also be created among three atomic ensembles by successively passing polarized light through the ensembles. Here, we outline a generalization of the scheme of Julsgaard et al. that creates bipartite entanglement between two atomic ensembles Julsgaard et al. (2001). The observables for the atomic ensembles are the collective Schwinger spins defined by the operators:

[TABLE]

which satisfy the commutation relation $\left[\hat{J}_{x},\hat{J}_{y}\right]=i\hat{J}_{z}$. Here, $a_{+},a_{-}$ are the operators for spin-up and spin-down along the spin-$x$ axis, respectively. We label the operators for each ensemble by the subscript $k$ ($k=1,2,3$).

Firstly, three atomic ensembles are prepared such that the mean collective spins for these atomic ensembles are pointing along the $x$-axis: $J_{x1}=-2J_{x2}=-2J_{x3}=J_{x}$. A linearly polarized light along the $x$-axis is then applied to the ensembles. The light-spin interaction is given by the Hamiltonian $H_{\text{int}}=\omega\hat{S}_{z}\hat{J}_{z123}$, where $\hat{J}_{z123}=\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3}$. The light variable then evolves in terms of the inputs to give an output of

[TABLE]

while the spin variables evolve as

[TABLE]

By measuring $\hat{S}_{y}^{\text{out}}$, $\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3}$ can be inferred. Also, $\hat{J}_{y1}+\hat{J}_{y2}+\hat{J}_{y3}$ can be measured using another light pulse without affecting the measured value of $\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3}$. This is possible because $\left[\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3},\hat{J}_{y1}+\hat{J}_{y2}+\hat{J}_{y3}\right]=0$. Hence, the quantity $\Delta^{2}\left(\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3}\right)+\Delta^{2}\left(\hat{J}_{y1}+\hat{J}_{y2}+\hat{J}_{y3}\right)$ can be arbitrarily small. Using the sum inequality Eq. (7) and product inequality Eq. (27) with gain values $g_{i}=h_{i}=1$, $\left(i=1,2,3\right)$, a genuine tripartite entanglement is certified among the atomic ensembles if $\Delta^{2}\left(\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3}\right)+\Delta^{2}\left(\hat{J}_{y1}+\hat{J}_{y2}+\hat{J}_{y3}\right)<2J_{x}$ for the sum inequality and $\Delta\left(\hat{J}_{z1}+\hat{J}_{z2}+\hat{J}_{z3}\right)\Delta\left(\hat{J}_{y1}+\hat{J}_{y2}+\hat{J}_{y3}\right)<J_{x}$ for the product inequality.

V.3 Entangled Bose-Einstein condensate clouds

In the experiment of Kunkel et al. Kunkel et al. (2018), a ${}^{87}\text{Rb}$ Bose-Einstein condensate is first generated in the magnetic substate $m_{F}=0$ of the $F=1$ hyperfine manifold, before a spin-squeezing operation coherently populates the $m_{F}=\pm 1$ states and entangles all the atoms in the condensate. The condensate is then allowed to expand into three spatially separated partitions. The tripartite entanglement among these partitions is verified by measuring $\hat{F}_{0,k}$ and $\hat{F}_{\pi/2,k}$ for each partition $k$, where $\hat{F}_{\phi,k}=\left[\left(\hat{a}_{+1}^{\dagger}+\hat{a}_{-1}^{\dagger}\right)\hat{a}_{0}e^{i\phi}+h.c.\right]/\sqrt{2}$, $\hat{a}_{j}^{\dagger}$ is the creation operator for a state $m_{F}=j$. These operators satisfy the commutation relation $\left[\hat{F}_{0,k},\hat{F}_{\pi/2,k}\right]=2i\hat{N}_{k}$, where $\hat{N}_{k}$ is the number operator for the partition $k$. By applying $\pi/2$ pulses and rotations, these observables are measured by reading out the population difference between the states $m_{F}=\pm 1$. If the number of atoms in group $m_{F}=0$ is large, then the measurement becomes similar to a homodyne detection of the amplitudes ($\left(\hat{a}_{+1}^{\dagger}+\hat{a}_{-1}^{\dagger}\right)e^{i\phi}+h.c.$) associated with the atoms of each of the partitions, carried out with the second larger group of atoms (given by $\hat{a}_{0}$) acting as the local oscillator, as explained in Refs. Ferris et al. (2008); Gross et al. (2011). More generally, spin relations must be used. In the atomic experiment of Kunkel et al., the genuine $N$-partite entanglement (up to $N=5$) mutually shared among the clouds is certified using criteria similar to that derived in Ref. Teh and Reid (2014), for quadrature phase amplitudes, but properly accounting for the spin and number operators that apply in this case.

In another experiment based on the two hyperfine states $|1\rangle=|F=1,m_{F}=-1\rangle$ and $|2\rangle=|F=2,m_{F}=1\rangle$ of a ${}^{87}\text{Rb}$ BEC, Fadel et al. Fadel et al. (2018) prepare the system in an atomic spin-squeezed state, and allow the condensate to expand into two separated partitions (which we denote $A$ and $B$). This creates a bipartite entanglement between the two clouds, which is detected using the entanglement criterion Giovannetti et al. (2003); Fadel et al. (2018)

[TABLE]

Here, $S_{z,A/B}$ and $S_{y,A/B}$ are the collective Schwinger spin operators Hu et al. (2015); Pezzè et al. (2018) along the $z$- axis and $y$-axis respectively, for partition $A/B$. Explicitly, the collective spin operator $S_{z,A/B}$ is given as the number difference

[TABLE]

where $N_{z,A/B}^{1}$ and $N_{z,A/B}^{2}$ are the number of atoms in the internal spin states $|1\rangle$ and $|2\rangle$ respectively, along the spin $z$-axis, for partition $A/B$. The collective spin operators along the $y$-axis $S_{y,A/B}$ are defined accordingly following Eq. (60), but noting the switching of the labels $x,y,z$. Other proposals exist to create a similar bipartite entanglement that can be detected using a similar spin criterion He et al. (2011c, 2012b); Opanchuk et al. (2012).

The experiment of Fadel et al. observed bipartite entanglement and EPR steering, but did not investigate tripartite entanglement. It is likely however that one could detect a genuine tripartite entanglement for clouds generated by further splitting the BEC. This would seem possible, given the result obtained in the Appendix 3 and depicted in Fig. 4, where tripartite entanglement is generated using only one squeezed input, followed by a sequence of splitting of the modes using beam splitter interactions. This works, because entangled modes can be created from a beam splitter with only one squeezed vacuum input Reid (1989). The tripartite entanglement created in the three modes of Fig. 4 can be detected using the Criterion 5 of Ref. Teh and Reid (2014) with the gains $h=-1/2$ and $g=1/2$. If one considers transforming into an equivalent tripartite entanglement in the Schwinger operators, then the suitable criterion would be Criterion 3 in Eq. (27) with the gains $h=-1/2$ and $g=1/2$.

A realization of a beam splitter interaction for the BEC can be obtained in several ways. An analogy of optical beam splitters with the splitting of a condensate (which is envisaged to be a realization of the final beam splitter of Fig. 4) is explained in Ref Killoran et al. (2016). The splitting into two modes is described by the interaction Hamiltonian

[TABLE]

where $a_{+},a_{+0}$ are the annihilation operators for modes labelled $A_{+}$ and $A_{+,0}$ respectively, and $\phi$ is the phase difference between these two modes. The transformation is equivalent to the beam splitter relations

[TABLE]

where $\tau$ is the interaction time and $a_{+,\text{out}}=a_{+}\left(\tau\right)\,,a_{+0,\text{out}}=a_{+0}\left(\tau\right)$. One can adjust the effective transmission to reflection ratio by adjusting the interaction time between the two modes.

We thus consider two separated clouds $A$ and $B$ that show spin entanglement with respect to the difference operators $g_{z}S_{z,A}+S_{z,B}$ and $g_{y}S_{y,A}+S_{y,B}$ so that the criterion of Eq. (63) is satisfied. These two clouds are analogous to the entangled outputs after the first beam splitter $BS$ of the configuration shown in Fig. 4. Each cloud is identified with Schwinger spin observables. For example, $S_{z,A}$ and $S_{y,A}$ are measurements that can be made on cloud $A$, where $S_{z,A}=\frac{1}{2}\left(a_{+}^{\dagger}a_{+}-a_{-}^{\dagger}a_{-}\right)$ and $a_{+}$, $a_{-}$ correspond to the two atomic levels. To generate the tripartite entanglement, the system $A$ is transformed according to a beam splitter interaction (splitting) modeled as Eq. (65). Since the splitting is insensitive to the internal spin degrees of freedom, there is a similar independent interaction for $a_{-}$. The outputs of $a_{\pm 1}$ and $a_{\pm 2}$ are then spatially separated, so that three separate clouds are created, labelled $A1$, $A2$ and $B$, these being analogous to the three outputs of the configuration of Fig. 4. The final Schwinger operators at $A_{1}$ and $A_{2}$ are defined by the $a_{\pm 1}$ at $A_{1}$, and the $a_{\pm 2}$ at $A_{2}$. The different Schwinger components can be measured using Rabi rotations or equivalent Gross et al. (2011); Fadel et al. (2018). The calculation carried out in Appendix 5 predicts a tripartite entanglement between the three clouds that could be detected by Criteria 1 and 3. Using Eqs. (81) and (82) in Appendix 5, the inequality of Criterion 3 is then

[TABLE]

The violation of this inequality implies genuine tripartite entanglement. We show in Appendix 5 that, assuming the number of atoms is large, $S_{z,A1}+S_{z,A2}\approx S_{z,A}$, $S_{y,A1}+S_{y,A2}\approx S_{y,A}$, and $S_{x,A1}+S_{x,A2}\approx S_{x,A}$. The criterion for genuine tripartite entanglement will therefore be satisfied if there is sufficient entanglement as measured by the bipartite criterion given in Eq. (63). Assuming $S_{x,A}$ and $S_{x,B}$ correspond to the Bloch vectors, with the directions of axes being chosen to ensure $\langle S_{x,A}\rangle$ and $\langle S_{x,B}\rangle$ are positive, we see that the beam splitter transformation (refer Appendix 5) ensures the signs of $S_{x,A1}$ and $S_{x,A2}$ are also positive. The right-side of the inequality is then either precisely that given by the right-side of Eq. (63) (if $g_{z}g_{y}>0$), or is less than this value (if $g_{z}g_{y}<0$).

We note from the results reported in Refs. van Loock and Furusawa (2003); Armstrong et al. (2012); Teh and Reid (2014) that we can generate $N$-partite entangled states ($N>3)$ by successive use of beam splitters with vacuum inputs, once an initial entangled state is created from two squeezed inputs or some other means. This has been implemented for a BEC by Ref. Kunkel et al. (2018) (for $N=5$). We show in the Appendix 4 that we can also create genuinely $4-$partite entangled states from a single squeezed input (refer Fig. 5), followed by multiple beam splitter combinations and vacuum inputs (with no squeezing). This may provide an avenue (using successive splittings) for the generation of multi-partite entanglement in experiments such as Ref. Fadel et al. (2018).

V.4 Remarks on the applicability and validity of genuine

multipartite entanglement criteria

In this subsection, we discuss the applicability and validity of the genuine multipartite entanglement criteria in the limit of large mean spins and small fluctuations where $\Delta^{2}J_{z,k}/\left|\langle J_{z,k}\rangle\right|\leq 1$ and hence $\Delta J_{z,k}/|\langle J_{z,k}\rangle|\rightarrow 0$ as $|\langle J_{z,k}\rangle|\rightarrow\infty$.

This limiting case justifies the application of Criterion 1 in Section V.A, because in that system the fields denoted $a_{V,i}$ are assumed to be intense coherent fields (see below). It also justifies application of the continuous variable Criteria derived in Ref. Teh and Reid (2014) to experiments where the quadrature phase amplitude of each mode is measured via homodyne detection. This includes the approach summarized at the beginning of Section V.C. The local oscillators used in the homodyne detection correspond to second field modes $a_{V,i}$, assumed to be intense classical fields with fixed amplitude. To prove these cases directly, we note that the measurement of the number difference at the beam splitter of each homodyne detection corresponds to that of a Schwinger operator: We define

[TABLE]

Taking the limit where field $a_{V,k}$ is the coherent intense local oscillator field, approximated by a classical amplitude $\alpha e^{-i\theta}$ and so that $\langle a_{H,k}^{\dagger}a_{H,k}\rangle\ll\langle a_{V,k}^{\dagger}a_{V,k}\rangle$, we define

[TABLE]

[TABLE]

Replacing $a_{V,k}$ with a classical amplitude gives

[TABLE]

which defines $X_{k}$ and $P_{k}$ as quadrature phase amplitudes. Hence, the fluctuation in $X_{k}$ ($P_{k}$) is due to $a_{H,k}$ only. Also, $[J_{x},J_{y}]=iJ_{z}$ ($\hbar=1$), and hence

[TABLE]

and similarly for bipartitions of type $II$ and $III.$ This then can be used to give the proof of Criterion 1.

[TABLE]

The Criteria 1 and 3 are also applicable in Section V.B, where the ensemble of atoms is considered to be large, and a large spin $J_{x}$ with minimal relative fluctuation ($\Delta^{2}J_{x}/\left|\langle J_{x}\rangle\right|\leq 1$) is assumed. In Section V.C, the bipartite analysis of Eq. (63) is unchanged. The use of Criterion 1 or 3 as in Eq. (67) is justified if $\Delta^{2}S_{x,Ak}/\left|\langle S_{x,Ak}\rangle\right|\leq 1$, $\Delta^{2}S_{x,B}/\left|\langle S_{x,B}\rangle\right|\leq 1$, and $\left|\langle S_{x,Ak}\rangle\right|$ and $\left|\langle S_{x,B}\rangle\right|$ are large.

VI Conclusion

In summary, we have derived several different criteria for certifying full $N$-partite inseparability and genuine $N$-partite entanglement using spin measurements. The criteria are inequalities expressed in terms of variances of spin observables measured at each of the $N$ sites.

In Sections II and IV, we derive criteria based on the standard spin uncertainty relation, involving $|\langle J_{z}\rangle|$. These criteria are valid for any systems, provided at each site the outcomes are reported faithfully, as results of accurately calibrated quantum measurements Bancal et al. (2011); Opanchuk et al. (2014) . We present in Section V three examples of application of these criteria. In these examples, entanglement is created that can be detected using Stokes or Schwinger operators defined at each site. These observables arise naturally in atomic ensembles, where the creation and detection of multi-partite entanglement is important for testing the quantum mechanics of massive systems. The criteria we develop may be useful for this purpose. In particular we specifically propose how to extend the experiments of Julsgaard et al. Julsgaard et al. (2001) and Fadel et al. Fadel et al. (2018), to generate three or more genuinely-entangled spatially-separated ensembles of atoms. The experiment of Kunkel et al. Kunkel et al. (2018) succeeded in generating genuine $5$-partite entanglement.

Where Stokes operators are defined for atomic systems, it is possible to introduce a normalization with respect to total atom number. This concept was introduced by He et al. He et al. (2011c, 2012a) and Żukowski et al. Żukowski et al. (2016, 2016, 2017); Ryu et al. (2019). These authors show how the detection of entanglement and nonlocality can be enhanced using such a normalization. It is likely that the criteria derived in Sections II and IV may also be further improved using this technique.

In Section III, we have outlined criteria derived from the planar spin uncertainty relation $\Delta^{2}J_{x}+\Delta^{2}J_{y}\geq C_{J}$ valid for a system of fixed spin $J$. This is useful for states where $\langle J_{z}\rangle=0$, such as the GHZ states. Such criteria were developed previously for genuine tripartite steering. Although genuine tripartite steering implies genuine tripartite entanglement, we have extended the results of the earlier work by giving details of the application of these criteria to certify the genuine tripartite entanglement and the full tripartite inseparability of the GHZ and W states respectively. While other methods exist to detect the genuine tripartite entanglement of these states (for example Collins et al. (2002); Tóth and Gühne (2005); Korbicz et al. (2006)), the criteria we present in Section III are readily extended to higher spin $J$.

Acknowledgements.

This research has been supported by the Australian Research Council Discovery Project Grants schemes under Grant DP180102470. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. We thank Manuel Gessner and Matteo Fadel for invaluable discussions and for pointing out the problems with Eq. (74) in the initial version of our manuscript.

Appendix

1. Lower bound of the sum inequality for an arbitrary bipartition

We derive the inequality in Eq. (5) for an arbitrary pure state bipartition $\rho_{k}^{\zeta}\rho_{lm}^{\zeta}$.

[TABLE]

Here the subscript $\zeta$ denotes that the statistical moments are calculated with respect to the particular state with bipartition $\rho_{k}^{\zeta}\rho_{lm}^{\zeta}$. The uncertainty relation $\Delta^{2}\left(hJ_{x}\right)_{\zeta}+\Delta^{2}\left(gJ_{y}\right)_{\zeta}\geq|\langle gh\left[J_{x},J_{y}\right]\rangle|_{\zeta}$ is used to obtain the first inequality in Eq. (74). The spin commutation relation $\left[J_{x},J_{y}\right]=iJ_{z}$ is used in the last line.

2. Lower bound of the product inequality for an arbitrary bipartition

We derive the inequality in Eq. (26) for an arbitrary bipartition $\rho_{k}^{\zeta}\rho_{lm}^{\zeta}$.

[TABLE]

In going from the second equality to the first inequality, the inequality for two real numbers $x$ and $y$, $x^{2}+y^{2}\geq 2xy$, is used. The uncertainty relation in the final line is $\Delta\left(hJ_{x}\right)_{\zeta}\Delta\left(gJ_{y}\right)_{\zeta}\geq\langle\left|gh\left[J_{x},J_{y}\right]\right|\rangle_{\zeta}/2$.

3. Generating genuine tripartite entangled states using 3 beam splitters

and one squeezed input

Here we consider the configuration of Fig. 4. The output mode operators $a_{\text{out}}$, $b_{\text{out}}$ and $c_{\text{out}}$ are

[TABLE]

Now, we consider $X_{a,\text{out}}-1/2\left(X_{b,\text{out}}+X_{c,\text{out}}\right)$ and $P_{a,\text{out}}+1/2\left(P_{b,\text{out}}+P_{c,\text{out}}\right)$. Their variances are then

[TABLE]

and their sum is

[TABLE]

giving a minimum of $10/6=1.6667$ for large squeezing parameter $r$. The sum inequality for those variances is $\Delta^{2}\left[X_{a,\text{out}}-1/2\left(X_{b,\text{out}}+X_{c,\text{out}}\right)\right]+\Delta^{2}\left[P_{a,\text{out}}+1/2\left(P_{b,\text{out}}+P_{c,\text{out}}\right)\right]\geq 2$ , as shown in Criterion 5 of Ref. Teh and Reid (2014) with the gains $h=-1/2$ and $g=1/2$. This inequality is violated and hence the final output state in Fig. 4 is genuinely tripartite entangled. We can also consider the input to be squeezed along $X$, in which case the gains $g$ and $h$ will have opposite sign.

4. Generating genuine four-partite entangled states using $4$ beam

splitters and one squeezed input

Here we consider the configuration of Fig. 5. The output mode operators $a_{\text{out}}$, $b_{\text{out}}$ , $c_{\text{out}}$ and $d_{\text{out}}$ are

[TABLE]

Now, we consider $X_{a,\text{out}}-1/3\left(X_{b,\text{out}}+X_{c,\text{out}}+X_{d,\text{out}}\right)$ and $P_{a,\text{out}}+1/3\left(P_{b,\text{out}}+P_{c,\text{out}}+P_{d,\text{out}}\right)$. Their variances are then

[TABLE]

and their sum is $5/3+e^{-2r}$, giving a minimum of $5/3=1.6667$ for large squeezing parameter $r$. The sum inequality for those variances is $\Delta^{2}\left[X_{a,\text{out}}-1/3\left(X_{b,\text{out}}+X_{c,\text{out}}+X_{d,\text{out}}\right)\right]+\Delta^{2}\left[P_{a,\text{out}}+1/3\left(P_{b,\text{out}}+P_{c,\text{out}}+P_{d,\text{out}}\right)\right]\geq 16/9$, as shown in Criterion 8 of Ref. Teh and Reid (2014) for $N=4$ and with the gains $h=-1/3$ and $g=1/3$. This inequality is violated and hence the final output state in Fig. 5 is genuinely four-partite entangled. We note we can also consider the input to be squeezed along $X$, in which case the gains $g$ and $h$ will have opposite sign.

5. Beam splitter operation as a model for splitting BEC clouds

We define the mode operators $a_{+}=\left(a_{+1}-ia_{+2}\right)/\sqrt{2}$ and $a_{-}=\left(a_{-1}-ia_{-2}\right)/\sqrt{2}$, and their corresponding auxiliary mode operators $a_{\text{vac}+}=\left(a_{+1}-ia_{+2}\right)/\sqrt{2}$ and $a_{\text{vac}-}=\left(a_{+1}-ia_{+2}\right)/\sqrt{2}$. This allows us to model the splitting of a BEC cloud with the beam splitter operations where the mode operators $a_{+},a_{\text{vac}+}$ are the inputs of a beam splitter. Since the different spin species does not interact, the mode operators $a_{-},a_{\text{vac}-}$ are also the inputs of a beam splitter and are split independently of the other spin species. With these mode operators, the Schwinger spin operators after splitting are

[TABLE]

Here we take the orientation of $x,y,z$ so that $S_{z}$ corresponds to the number difference. $S_{z,A1}$ and $S_{z,A2}$ are the Schwinger spin operators along the $z$-axis for clouds $A1$ and $A2$ respectively, $F$ and $G$ are terms containing $a_{\text{vac}+},a_{\text{vac}+}^{\dagger},a_{\text{vac}-},a_{\text{vac}-}^{\dagger}$. Similar Schwinger spin operators along the $x$ and $y$-axes have the same expressions as Eqs. (81) and (82) but the spin up and down are relative to their respective axis. From Eqs. (81) and (82), we see that $S_{z,A1}+S_{z,A2}=S_{z,A}+F+G\approx S_{z,A}$. Here we assume the terms $F$ and $G$ involving the incoming unoccupied modes can be neglected in the calculation of the variances, relative to the leading terms which come from the incoming modes with a high occupation (the number of atoms being assumed large). Using a similar argument, we consider $S_{\theta}=\frac{1}{2}\left(a_{+}^{\dagger}a_{-}e^{i\theta}+a_{-}^{\dagger}a_{+}e^{-i\theta}\right)\,.$

[TABLE]

So, for large numbers of atoms, $S_{y,A1}+S_{y,A2}=S_{y,A}+F+G\approx S_{y,A}$ and similarly, $S_{x,A1}+S_{x,A2}\approx S_{x,A}$.