1 Introduction

Correlations among the results of space-like separated measurements on composite quantum systems can be incompatible with a local model [1]. Such phenomenon, known as quantum nonlocality, is an intrinsic quantum feature and lies behind several applications in quantum information theory [2,3,4,5,6]. By performing local measurements on an n-partite entangled state one obtains outcomes that may be nonlocal, in the sense that they violate a Bell inequality [7]. Since the seminal work of Bell, nonlocality has been a central subject of study in the foundations of quantum theory and has been supported by many experiments [8, 9]. More recently, it has also been realized that it plays a key role in various quantum information applications [10, 11], where it represents a resource different from entanglement. For instance, the security of device independent quantum key distribution requires the existence of nonlocal correlations between the honest parties, very much in the spirit of Ekert’s protocol [2, 3, 12, 13], and the only entanglement witnesses that do not rely on assumptions on the dimension of the Hilbert spaces are Bell inequalities, i.e. witnesses of nonlocality [3]. While nonlocality has been extensively studied in the bipartite (n = 2) and to a lesser extent in the tripartite (n = 3) case, the general n-partite case remains much unexplored, their characterization remains a general unsolved problem. The physics of many-particle systems, however, is well known to differ fundamentally from the one of a few particles and to give rise to new interesting phenomena, such as phase transitions or quantum computing. Entanglement theory, in particular, appears to have a much more complex and richer structure in the n-partite case than it has in the bipartite setting [14, 15]. This is reflected by the fact that multipartite entanglement is a very active field of research that has led to important insights into our understanding of many-particle physics [16, 17]. In this point of view, it seems worthy to investigate how nonlocality manifests itself in a multipartite scenario. Generalized Bell inequalities have been reported for n-particle systems which show that quantum mechanics violates local realism in these situations [18,19,20]. However such results are insufficient to show that all of the particles in a system are acting nonlocality, it is possible to imagine a nonlocal many-particle system as consisting of a finite number of nonlocal subsystems, but with only local correlations present between these subsystems. For example a state of three particles |ψ〉123 which can be decomposed as |ψ〉1|ψ〉23 only exhibits nonlocal correlations between particles 2 and 3. Hence, it is necessary for us to extend the concept of locality of all particles in a multipartite system to the locality of groups of subsystems.

In this paper, we introduce Λ-locality and Λ-nonlocality of multipartite states and prove the related nonlocality inequalities. The remain of this paper are organized as follows. In Section 2, we introduce the Λ-locality of multipartite states and discuss the related properties. In Section 3, we establish some nonlocality inequalities, which are necessary conditions for a Λ-local state. In Section 4, we give a class of 2-separable nonlocal states, which shows that a 2-separable tripartite state is not necessarily local.

2 Λ-Nonlocality of Multipartite States

We consider the composite system \(\mathcal {H}^{(n)}:=\mathcal {H}_{1} \otimes \mathcal {H}_{2}\otimes \ldots \otimes \mathcal {H}_{n}\), and use \(D(\mathcal {H}^{(n)})\) to denote the set of all mixed states of the system \(\mathcal {H}^{(n)}\) and I k to denote the identity operator on \(\mathcal {H}_{k}\).

To describe different local cases of the composite system \(\mathcal {H}^{(n)}\), we use Ω = {1,2,…,n} to denote the set of all indices of the subsystems. And for a subset {i1,i2,…,i k } of Ω with i0 + 1 = 1 ≤ i1 < i2 < … < i k = n, put

$$ {\Lambda}=\{i_{0}+ 1,\ldots,i_{1};i_{1}+ 1,\ldots,i_{2};\ldots;i_{k-1}+ 1,\ldots,i_{k}\}, $$
(2.1)

called a local pattern.

For a local pattern (2.1), put

$$A_{1}=\{i_{0}+ 1,\ldots,i_{1}\},A_{2}=\{i_{1}+ 1,\ldots,i_{2}\},\ldots,$$
$$A_{k}=\{i_{k-1}+ 1,\ldots,i_{k}\}.$$

We denote Λ simply by Λ = (A1,A2,…,A k ). Put \(\mathcal {H}_{A_{s}}=\mathcal {H}_{i_{s-1}+ 1}\otimes \ldots \otimes \mathcal {H}_{i_{s}}\), then

$$\mathcal{H}^{(n)}=\mathcal{H}_{1} \otimes \mathcal{H}_{2}\otimes\ldots\otimes \mathcal{H}_{n}=\mathcal{H}_{A_{1}} \otimes \mathcal{H}_{A_{2}}\otimes\ldots\otimes \mathcal{H}_{A_{k}}.$$

Thus, every n-partite state ρ of \(\mathcal {H}^{(n)}\) can be viewed as a k-partite state. In this case, we use tr\(_{A_{i}}(\rho )\) to denote the reduced state of ρ with respect to i th subsystem \(\mathcal {H}_{A_{i}}\), which is a state of \(\mathcal {H}_{A_{1}} \otimes \ldots \otimes \mathcal {H}_{A_{i-1}}\otimes \mathcal {H}_{A_{i + 1}}\otimes \ldots \otimes \mathcal {H}_{A_{k}}\).

Definition 2.1

Let Λ be a local pattern given by (2.1).

  1. (1)

    A state \(\rho \in D(\mathcal {H}^{(n)})\) is said to be Λ-local if for every measurement assemblage

    $$\mathcal{M}=\{M^{x_{1},x_{2},\ldots,x_{k}}: x_{j}= 1,2,\ldots,m_{j}(1\le j\le k)\}\equiv \{M^{x_{1},x_{2},\ldots,x_{k}}\}_{x_{1},x_{2},\ldots,x_{k}}$$

    of local POVMs:

    $$M^{x_{1},x_{2},\ldots,x_{k}}=\{M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k}}_{b_{k}}: b_{i}\in N_{i}(1\le i\le k)\}$$

    on \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \ldots \otimes \mathcal {H}_{A_{k}}\), there exists a probability distribution π = {π λ }λ∈Γ such that

    $$ \text{tr}(M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k}}_{b_{k}})\rho =\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(b_{1}|x_{1},\lambda)P_{A_{2}} (b_{2}|x_{2},\lambda){\ldots} P_{A_{k}}(b_{k}|x_{k},\lambda) $$
    (2.2)

    for all x i ,b i ,where \(P_{A_{i}}(b_{i}|x_{i},\lambda )\geq 0, {\sum }_{b_{i}}P_{A_{i}}(b_{i}|x_{i},\lambda )= 1(i = 1,2,\ldots ,k)\). Otherwise, ρ is said to be Λ-nonlocal.

  2. (2)

    A mixed state \(\rho \in D(\mathcal {H}^{(n)})\) is said to be genuinely nonlocal if it is Λ-nonlocal for every Λ.

Remark 2.1

By definition, ρ is Λ-local if and only if for every \(\mathcal {M}\), there exists a PD π such that (2.2) holds; ρ is Λ-nonlocal if and only if there exists an \(\mathcal {M}\), the PD π satisfying (2.2) does not exits.

Remark 2.2

By definition above, we see that when a state \(\rho \in D(\mathcal {H}^{(n)})\) is Λ-local, then for every measurement assemblage \(\mathcal {M}\), there exists a probability distribution π = {π λ }λ∈Γ such that

$$\text{tr}(M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k}}_{b_{k}})\rho =\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(b_{1}|x_{1},\lambda)P_{A_{2}} (b_{2}|x_{2},\lambda){\ldots} P_{A_{k}}(b_{k}|x_{k},\lambda)$$

for all x i ,b i . Finding the sums of two sides for b j yields that

$$\begin{array}{@{}rcl@{}} &&\text{tr}(M^{x_{1}}_{b_{1}}\otimes\ldots\otimes M^{x_{j-1}}_{b_{j-1}} \otimes I_{A_{j}}\otimes M^{x_{j + 1}}_{b_{j + 1}}\otimes\ldots\otimes M^{x_{k}}_{b_{k}})\rho\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(b_{1}|x_{1},\lambda){\ldots} P_{A_{j-1}}(b_{j-1}|x_{j-1},\lambda)P_{A_{j + 1}} (b_{j + 1}|x_{j + 1},\lambda){\ldots} P_{A_{k}}(b_{k}|x_{k},\lambda). \end{array} $$

This shows that the measurement results of the other subsystems except the subsystem \(\mathcal {H}_{A_{j}}\) are independent of the measurements of the subsystem \(\mathcal {H}_{A_{j}}\).

Remark 2.3

When a state \(\rho \in D(\mathcal {H}^{(n)})\) can be written as

$$\rho=\sum\limits_{\lambda= 1}^{m}p_{\lambda} \rho^{\lambda}_{1{\ldots} i_{1}}\otimes\rho^{\lambda}_{(i_{1}+ 1){\ldots} i_{2}}\otimes\ldots\otimes\rho^{\lambda}_{(i_{k-1}+ 1){\ldots} i_{k}},$$

where \(\rho ^{\lambda }_{(i_{j-1}+ 1){\ldots } i_{j}}\in D(\mathcal {H}_{A_{j}})\), and \(\{p_{\lambda }\}_{\lambda = 1}^{m}\) is a probability distribution, for every measurement assemblage \(\mathcal {M}\), we compute that (2.2) holds for

$${\Pi}_{\lambda}=p_{\lambda}, P_{A_{j}}(b_{j}|x_{j},\lambda)=\text{tr}(M^{x_{j}}_{b_{j}}\rho^{\lambda}_{(i_{j-1}+ 1){\ldots} i_{j}})(j = 1,2,\ldots,k).$$

This shows that ρ is Λ-local.

Theorem 2.1

Suppose that ρ is a(A1,A2,…,A k )-localstate of\(\mathcal {H}^{(n)}\),then\(\rho _{12\ldots (k-1)}:=\text {tr}_{A_{k}}(\rho )\)is a(A1,A2,…,Ak− 1)-localstate of\(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \ldots \otimes \mathcal {H}_{A_{k-1}}\).

Proof

Suppose that ρ is (A1,A2,…,A k )-local. For every measurement assemblage

$$\mathcal{N}=\{M^{x_{1},x_{2},\ldots,x_{k-1}}: x_{j}= 1,2,\ldots,m_{j}(j = 1,2,\ldots,k-1)\}$$

of local POVMs:

$$M^{x_{1},x_{2},\ldots,x_{k-1}}=\{M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k-1}}_{b_{k-1}}: b_{i}\in N_{i}(1\le i\le k-1)\}$$

on \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \ldots \otimes \mathcal {H}_{A_{k-1}}\), by letting \(M^{x_{k}}_{b_{k}}=I_{A_{k}}(x_{k}= 1,b_{k}= 1)\), the identity operator on \( \mathcal {H}_{A_{k}}\), we obtain a measurement assemblage \(\mathcal {M}=\{M^{x_{1},x_{2},\ldots ,x_{k}}: x_{j}= 1,2,\ldots ,m_{j}(j = 1,2,\ldots ,k)\}\) of local POVMs:

$$M^{x_{1},x_{2},\ldots,x_{k}}=\{M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k}}_{b_{k}}: b_{i}\in N_{i}(1\le i\le k)\}$$

on \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \ldots \otimes \mathcal {H}_{A_{k}}\) with m k = 1,N k = {1}. By Definition 2.1, we know that there exists a probability distribution {π λ }λ∈Γ such that ∀x i ,b i , it holds that

$$\begin{array}{@{}rcl@{}} &&\text{tr}(M^{x_{1}}_{b_{1}}\otimes{\ldots} \otimes M^{x_{k-1}}_{b_{k-1}} \otimes M^{x_{k}}_{b_{k}})\rho\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(b_{1}|x_{1},\lambda){\ldots} P_{A_{k-1}} (b_{k-1}|x_{k-1},\lambda)P_{A_{k}}(b_{k}|x_{k},\lambda),\end{array} $$

where \(P_{A_{i}}(b_{i}|x_{i},\lambda )\geq 0, {\sum }_{b_{i}}P_{A_{i}}(b_{i}|x_{i},\lambda )= 1(i = 1,2,\ldots ,k)\). Since x k = 1,b k = 1 and \({\sum }_{b_{k}}P_{A_{k}}(b_{k}|x_{k},\lambda )= 1\), we get that \(P_{A_{k}}(b_{k}|x_{k},\lambda )= 1\). Consequently,

$$\begin{array}{@{}rcl@{}} &&\text{tr}(M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k-1}}_{b_{k-1}})\rho_{12\ldots(k-1)}\\ &=&\text{tr}(M^{x_{1}}_{b_{1}}\otimes M^{x_{2}}_{b_{2}} \otimes\ldots\otimes M^{x_{k-1}}_{b_{k-1}}\otimes M^{x_{k}}_{b_{k}})\rho\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(b_{1}|x_{1},\lambda){\ldots} P_{A_{k-1}}(b_{k-1}|x_{k-1},\lambda), \end{array} $$

for all x i = 1,2,…,m i ,b i ∈ N i (i = 1,2,…,k − 1). By using Definition 2.1 again, we conclude that ρ12…(k− 1) is a (A1,A2,…,Ak− 1)-local state of \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \ldots \otimes \mathcal {H}_{A_{k-1}}\). The proof is completed. □

3 Λ-Nonlocality Inequalities

Suppose that \(\rho \in D(\mathbb {C}^{2}\otimes \mathbb {C}^{2} \otimes \mathbb {C}^{2})\). If ρ is {1,2;3}-local, then the following three Clauser-Horne-Shimony-Holt Bell inequalities hold, which were mentioned in [21](5a-5c):

$$\begin{array}{@{}rcl@{}} &&|\langle \sigma_{z}\otimes\sigma_{z}\otimes P\rangle_{\rho}\,+\,\langle \sigma_{z}\otimes \sigma_{z}\otimes Q\rangle_{\rho}\,+\,\langle \sigma_{z}\otimes\sigma_{x}\otimes P\rangle_{\rho} \,-\,\langle \sigma_{z}\otimes\sigma_{x}\otimes Q\rangle_{\rho}|\!\leq\! 2, \end{array} $$
(3.1)
$$\begin{array}{@{}rcl@{}} &&|\langle \sigma_{x}\otimes\sigma_{z}\otimes P\rangle_{\rho}\,+\,\langle \sigma_{x} \otimes\sigma_{z}\otimes Q\rangle_{\rho}\,+\,\langle \sigma_{x}\otimes\sigma_{x}\otimes P\rangle_{\rho}\,-\,\langle \sigma_{x}\otimes\sigma_{x}\otimes Q\rangle_{\rho}|\!\leq\! 2, \end{array} $$
(3.2)
$$\begin{array}{@{}rcl@{}} &&|\langle I\otimes\sigma_{z}\otimes P\rangle_{\rho}\,+\,\langle I\otimes\sigma_{z} \otimes Q\rangle_{\rho}\,+\,\langle I\otimes\sigma_{x}\otimes P\rangle_{\rho} \,-\,\langle I\otimes\sigma_{x}\otimes Q\rangle_{\rho}|\!\leq\! 2, \end{array} $$
(3.3)

where P,Q are ± 1-valued observables. Next, we will generalize the above inequalities and obtain the following theorem.

Theorem 3.1

Suppose that\(\rho \in D(\mathcal {H}_{1} \otimes \mathcal {H}_{2}\otimes \mathcal {H}_{3})\).Ifρis{1,2;3}-local,then the inequality

$$ |\langle CP\rangle_{\rho}+\langle CQ\rangle_{\rho}+\langle D P\rangle_{\rho} -\langle DQ\rangle_{\rho}|\leq 2 $$
(3.4)

holds for all ± 1-valuedobservables C,D on \(\mathcal {H}_{A_{1}}: =\mathcal {H}_{1}\otimes \mathcal {H}_{2}\), and± 1-valuedobservables P,Q on \(\mathcal {H}_{A_{2}}:=\mathcal {H}_{3}\),where CP = (C ⊗ I3)(I1 ⊗ I2 ⊗ P)and so on.

Proof

Because that C,D,P and Q are ± 1-valued observables, they have their spectral decompositions:

$$C=C^{+}-C^{-}, D=D^{+}-D^{-},P=P^{+}-P^{-}, Q=Q^{+}-Q^{-}.$$

By taking

$$M^{x_{1}}_{+}=C^{+}, M^{x_{1}}_{-}=C^{-},M^{x_{2}}_{+}=P^{+}, M^{x_{2}}_{-}=P^{-};$$
$$M^{y_{1}}_{+}=D^{+}, M^{y_{1}}_{-}=D^{-},M^{y_{2}}_{+}=Q^{+}, M^{y_{2}}_{-}=Q^{-},$$

we obtain POVMs \(M^{x_{1}}=\{M^{x_{1}}_{+}, M^{x_{1}}_{-}\}\) and \(M^{y_{1}}=\{M^{y_{1}}_{+}, M^{y_{1}}_{-}\}\) on \(\mathcal {H}_{A_{1}}\), and \(M^{x_{2}}=\{M^{x_{2}}_{+}, M^{x_{2}}_{-}\}\) and \(M^{y_{2}}=\{M^{y_{2}}_{+}, M^{y_{2}}_{-}\}\) on \(\mathcal {H}_{A_{2}}\). Thus, we obtain a measurement assemblage:

$$\mathcal{M}=\{M^{x_{1}}\otimes M^{x_{2}},\ M^{x_{1}}\otimes M^{y_{2}},\ M^{y_{1}}\otimes M^{x_{2}},\ M^{y_{1}}\otimes M^{y_{2}}\}$$

of four local POVMs on \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\). Explicitly,

$$M^{x_{1}}\otimes M^{x_{2}}=\{C^{+}\otimes P^{+}, C^{+}\otimes P^{-}, C^{-}\otimes P^{+}, C^{-}\otimes P^{-}\},$$
$$M^{x_{1}}\otimes M^{y_{2}}=\{C^{+}\otimes Q^{+}, C^{+}\otimes Q^{-}, C^{-}\otimes Q^{+}, C^{-}\otimes Q^{-}\},$$
$$M^{y_{1}}\otimes M^{x_{2}}=\{D^{+}\otimes P^{+}, D^{+}\otimes P^{-}, D^{-}\otimes P^{+}, C^{-}\otimes P^{-}\},$$
$$M^{y_{1}}\otimes M^{y_{2}}=\{D^{+}\otimes Q^{+}, D^{+}\otimes Q^{-}, D^{-}\otimes Q^{+}, D^{-}\otimes Q^{-}\}.$$

Since ρ is {1,2;3}-local, by Definition 2.1, for this \(\mathcal {M}\), there exists a probability distribution {π λ }λ∈Γ such that (2.2) holds. Hence,

$$\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{+})\rho=\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}}(+|x_{2},\lambda),$$

and so on. Thus,

$$\begin{array}{@{}rcl@{}} \langle CP\rangle_{\rho} &=&\text{tr}(C^{+}\otimes P^{+})\rho-\text{tr}(C^{+}\otimes P^{-})\rho- \text{tr}(C^{-}\otimes P^{+})\rho+\text{tr}(C^{-}\otimes P^{-})\rho\\ &=&\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{+})\rho-\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{-})\rho -\text{tr}(M^{x_{1}}_{-}\otimes M^{x_{2}}_{+})\rho+\text{tr}(M^{x_{1}}_{-}\otimes M^{x_{2}}_{-})\rho\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} \left\{P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)-P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (-|x_{2},\lambda)\right.\\ &&\left.-P_{A_{1}}(-|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)+P_{A_{1}}(-|x_{1},\lambda)P_{A_{2}}(-|x_{2},\lambda)\right\}\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda}(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda)). \end{array} $$

Similarly,

$$\langle CQ\rangle_{\rho} =\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda}(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda)), $$
$$\langle DQ\rangle_{\rho} =\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda}(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda)), $$
$$\langle DP\rangle_{\rho} =\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda}(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda)). $$

Therefore

$${{\Delta}:} =\langle CP\rangle_{\rho}+\langle CQ\rangle_{\rho}+\langle DP\rangle_{\rho} -\langle DQ\rangle_{\rho}=\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} \widetilde{p_{2}}(\lambda),$$

where

$$\widetilde{p_{2}}(\lambda) =(\alpha_{1}-\alpha_{2})(a-b+x-y)+(\beta_{1}-\beta_{2})(a-b-x+y),$$

where

$$P_{A_{1}}(+|x_{1},\lambda)=\alpha_{1},P_{A_{1}}(-|x_{1},\lambda))=\alpha_{2}, P_{A_{1}}(+|y_{1},\lambda)=\beta_{1},P_{A_{1}}(-|y_{1},\lambda)=\beta_{2},$$
$$P_{A_{2}}(+|x_{2},\lambda)=a,P_{A_{2}}(-|x_{2},\lambda)=b, P_{A_{2}}(+|y_{2},\lambda)=x,P_{A_{2}}(-|y_{2},\lambda)=y.$$

Clearly, we get that

$$\alpha_{1}+\alpha_{2}= 1,\ \ \alpha_{1},\alpha_{2}\geq 0;$$
$$\beta_{1}+\beta_{2}= 1,\ \ \beta_{1},\beta_{2}\geq 0;$$
$$a+b = 1,\ \ a,b\geq 0;$$
$$x+y = 1,\ \ x,y\geq 0.$$

Put a − b = m1,x − y = m2. Clearly, − 1 ≤ m i ≤ 1(i = 1,2) and

$$\widetilde{p_{2}}(\lambda)= (\alpha_{1}-\alpha_{2})(m_{1}+m_{2})+(\beta_{1}-\beta_{2})(m_{1}-m_{2}).$$

Therefore

$$|\widetilde{p_{2}}(\lambda)|^{2}\leq(|m_{1}+m_{2}|+|m_{1}-m_{2}|)^{2}= 4\max\{{m_{1}^{2}},{m_{2}^{2}}\} \le4.$$

So, \(|\widetilde {p_{2}}(\lambda )|\leq 2\) for all λ and thus |Δ|≤ 2. i.e.

$$|\langle CP\rangle_{\rho}+\langle CQ\rangle_{\rho}+\langle DP\rangle_{\rho} -\langle DQ\rangle_{\rho}|\leq 2.$$

The proof is completed. □

Remark 3.1

From Theorem 3.1 we know that every {1,2;3}-local state satisfies inequality (3.4). Thus, if there exist C,D,P,Q such that the inequality (3.4) is not satisfied, then ρ is {1,2;3}-nonlocal.

Remark 3.2

It is easy to see that \(\widetilde {p_{2}}(\lambda )\) is obtained by changing C to \(P_{A_{1}}(+|x_{1},\lambda ) -P_{A_{1}}(-|x_{1},\lambda )\), D to \(P_{A_{1}}(+|y_{1},\lambda ) -P_{A_{1}}(-|y_{1},\lambda )\), P to \(P_{A_{2}}(+|x_{2},\lambda ) -P_{A_{2}}(-|x_{2},\lambda )\), and Q to \(P_{A_{2}}(+|y_{2},\lambda ) -P_{A_{2}}(-|y_{2},\lambda )\) in the expression of CP + CQ + DP − DQ.

Specially, when C = σ z ⊗ σ z ,D = σ z ⊗ σ x and C = σ x ⊗ σ z ,D = σ x ⊗ σ x and C = I ⊗ σ z ,D = I ⊗ σ x in Theorem 3.1, respectively, we get (3.1)-(3.3).

Similar to the proof of Theorem 3.1, one can prove the following conclusion.

Theorem 3.2

Suppose that\(\rho \in D(\mathcal {H}_{1} \otimes \ldots \otimes \mathcal {H}_{n})\).Ifρis(A1,A2)-local,then the inequality

$$ |\langle CP\rangle_{\rho}+\langle CQ\rangle_{\rho}+\langle DP\rangle_{\rho} -\langle DQ\rangle_{\rho}|\leq 2 $$
(3.5)

holds for all ± 1-valuedobservables C,D on \(\mathcal {H}_{A_{1}}: =\mathcal {H}_{1}\otimes \ldots \otimes \mathcal {H}_{j}\), and± 1-valuedobservables P,Q on \(\mathcal {H}_{A_{2}}:=\mathcal {H}_{j + 1}\otimes \ldots \otimes \mathcal {H}_{n}\).

To describe more general locality (k > 2), we suppose that Λ is given by (2.1) and \(\{A^{x_{s}}_{a_{s}}\}(s = 1,2,\ldots ,k)\) is ± 1-valued observables on \(\mathcal {H}_{A_{s}}\). Let the two party Mermin polynomial [20] be

$$ M_{2}=\frac{1}{2}(A^{0}_{a_{1}}A^{0}_{a_{2}}+ A^{0}_{a_{1}}A^{1}_{a_{2}}+A^{1}_{a_{1}}A^{0}_{a_{2}}-A^{1}_{a_{1}}A^{1}_{a_{2}}), $$
(3.6)
figure a

Then M m and \(M^{\prime }_{m}\) are generated from Mm− 1 by recursion relation:

$$ M_{m}=\frac{1}{2}[M_{m-1}(A^{0}_{a_{m}}+A^{1}_{a_{m}}) +M^{\prime}_{m-1}(A^{0}_{a_{m}}-A^{1}_{a_{m}})]. $$
(3.7)
figure b

Following [22] we define the Svetlichny polynomials as

$$S_{m}=\left\{ \begin{array}{ll} M_{m}, & {m = 2n;} \\ \frac{1}{2}(M_{m}+M^{\prime}_{m}), & {m = 2n + 1.} \end{array} \right. $$
(3.8)
figure c

By these definitions, we see that

$$M_{m}+M^{\prime}_{m}=S_{m}+S^{\prime}_{m},\ \ \forall m = 2,3,\ldots,k.$$

Remark 3.3

Let ρ be Λ = (A1,A2)-local. Then we can get

$$|\langle S_{2}\rangle_{\rho}|=\frac{1}{2}\left|\langle A^{0}_{a_{1}}A^{0}_{a_{2}}\rangle_{\rho}+ \langle A^{0}_{a_{1}}A^{1}_{a_{2}}\rangle_{\rho}+\langle A^{1}_{a_{1}}A^{0}_{a_{2}}\rangle_{\rho} -\langle A^{1}_{a_{1}}A^{1}_{a_{2}}\rangle_{\rho}\right|\le1,$$

by using (3.5) for \(C=A^{0}_{a_{1}}, D=A^{1}_{a_{1}}, P=A^{0}_{a_{2}}\) and \(Q=A^{1}_{a_{2}}\).

Similarly,

$$|\langle S^{\prime}_{2}\rangle_{\rho}|=\frac{1}{2}\left|\langle A^{1}_{a_{1}}A^{1}_{a_{2}}\rangle_{\rho} +\langle A^{1}_{a_{1}}A^{0}_{a_{2}}\rangle_{\rho}+ \langle A^{0}_{a_{1}}A^{1}_{a_{2}}\rangle_{\rho} -\langle A^{0}_{a_{1}}A^{0}_{a_{2}}\rangle_{\rho}\right|\le 1.$$

Furthermore, since ρ is Λ = (A1,A2)-local, by Definition 2.1, there exists a probability distribution {π λ }λ∈Γ such that \(\langle S_{2}\rangle _{\rho }={\sum }_{\lambda \in {\Gamma }}{\Pi }_{\lambda } p_{2}(\lambda )\) and \(\langle S^{\prime }_{2}\rangle _{\rho }={\sum }_{\lambda \in {\Gamma }}{\Pi }_{\lambda } p^{\prime }_{2}(\lambda )\), where

$$\begin{array}{@{}rcl@{}} p_{2}(\lambda) &=&\frac{1}{2}\left\{(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\right.\\ &&\left.+(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\right.\\ &&\left.+(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\right.\\ &&\left.-(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\right\}, \end{array} $$
$$\begin{array}{@{}rcl@{}} p^{\prime}_{2}(\lambda) &=&\frac{1}{2}\left\{(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\right.\\ &&\left.+(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\right.\\ &&\left.+(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\right.\\ &&\left.-(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\right\}. \end{array} $$

Clearly, p2(λ) and \(p^{\prime }_{2}(\lambda )\) are obtained by changing \(A^{0}_{a_{1}}\) to \(P_{A_{1}}(+|x_{1},\lambda )-P_{A_{1}}(-|x_{1},\lambda )\), \(A^{1}_{a_{1}}\) to \(P_{A_{1}}(+|y_{1},\lambda )-P_{A_{1}}(-|y_{1},\lambda )\), \(A^{0}_{a_{2}}\) to \(P_{A_{2}}(+|x_{2},\lambda )-P_{A_{2}}(-|x_{2},\lambda )\), and \(A^{1}_{a_{2}}\) to \(P_{A_{2}}(+|y_{2},\lambda ) -P_{A_{2}}(-|y_{2},\lambda )\) in the definitions of S2 and \(S^{\prime }_{2}\), respectively. It is easy to see that \(p_{2}(\lambda )=\frac {1}{2}\widetilde {p_{2}}(\lambda )\). From the proof of Theorem 3.1, we can get |p2(λ)|≤ 1 for all λ. Similarly, we can prove \(|p^{\prime }_{2}(\lambda )|\leq 1\) for all λ.

Theorem 3.3

Suppose thatρisΛ = (A1,A2,A3)-local,then

$$ |\langle S_{3}\rangle_{\rho}|=|\langle S^{\prime}_{3}\rangle_{\rho}|\leq 1. $$
(3.9)

Proof

Using (3.6)-(3.8) yield

$$\begin{array}{@{}rcl@{}} {S_{3}}&=&\frac{1}{2}(M_{3}+M^{\prime}_{3})\\ &=&\frac{1}{2}\left\{\frac{1}{2}[M_{2}(A^{0}_{a_{3}}+A^{1}_{a_{3}})+M^{\prime}_{2} (A^{0}_{a_{3}}-A^{1}_{a_{3}})]\right.\\ &&\left.+\frac{1}{2}[M^{\prime}_{2}(A^{1}_{a_{3}}+A^{0}_{a_{3}})+M_{2} (A^{1}_{a_{3}}-A^{0}_{a_{3}})]\right\}\\ &=&\frac{1}{2}(M_{2}A^{1}_{a_{3}}+M^{\prime}_{2}A^{0}_{a_{3}})\\ &=&\frac{1}{4}\left\{(A^{0}_{a_{1}}A^{0}_{a_{2}}+A^{0}_{a_{1}}A^{1}_{a_{2}}+ A^{1}_{a_{1}}A^{0}_{a_{2}}-A^{1}_{a_{1}}A^{1}_{a_{2}})A^{1}_{a_{3}}\right.\\ &&\left.+(A^{1}_{a_{1}}A^{1}_{a_{2}}+A^{1}_{a_{1}}A^{0}_{a_{2}}+ A^{0}_{a_{1}}A^{1}_{a_{2}}-A^{0}_{a_{1}}A^{0}_{a_{2}})A^{0}_{a_{3}}\right\}. \end{array} $$

Because that \(A^{0}_{a_{1}},A^{1}_{a_{1}},A^{0}_{a_{2}},A^{1}_{a_{2}},A^{0}_{a_{3}},A^{1}_{a_{3}}\) are ± 1-valued observables, they have their spectral decompositions:

$$A^{0}_{a_{i}}=(A^{0}_{a_{i}})^{+}-(A^{0}_{a_{i}})^{-}, A^{1}_{a_{i}}=(A^{1}_{a_{i}})^{+}-(A^{1}_{a_{i}})^{-},i = 1,2,3. $$

By taking

$$M^{x_{1}}_{+}=(A^{0}_{a_{1}})^{+}, M^{x_{1}}_{-}=(A^{0}_{a_{1}})^{-}, $$
$$ M^{x_{2}}_{+}=(A^{0}_{a_{2}})^{+}, M^{x_{2}}_{-}=(A^{0}_{a_{i}})^{-}, $$
$$ M^{x_{3}}_{+}=(A^{0}_{a_{3}})^{+}, M^{x_{3}}_{-}=(A^{0}_{a_{3}})^{-}, $$
$$M^{y_{1}}_{+}=(A^{1}_{a_{1}})^{+}, M^{y_{1}}_{-}=(A^{1}_{a_{1}})^{-}, $$
$$M^{y_{2}}_{+}=(A^{1}_{a_{2}})^{+}, M^{y_{2}}_{-}=(A^{1}_{a_{2}})^{-}, $$
$$M^{y_{3}}_{+}=(A^{1}_{a_{3}})^{+}, M^{y_{3}}_{-}=(A^{1}_{a_{3}})^{-}, $$

we obtain POVMs \(M^{x_{1}}\,=\, \{M^{x_{1}}_{+}, M^{x_{1}}_{-}\}\) and \(M^{y_{1}}\,=\,\{M^{y_{1}}_{+}, M^{y_{1}}_{-}\}\) on \(\mathcal {H}_{A_{1}}\), and \(M^{x_{2}}\,=\,\{M^{x_{2}}_{+},\)\(M^{x_{2}}_{-}\}\) and \(M^{y_{2}}=\{M^{y_{2}}_{+}, M^{y_{2}}_{-}\}\) on \(\mathcal {H}_{A_{2}}\) and \(M^{x_{3}}=\{M^{x_{3}}_{+}, M^{x_{3}}_{-}\}\) and \(M^{y_{3}}=\{M^{y_{3}}_{+}, M^{y_{3}}_{-}\}\) on \(\mathcal {H}_{A_{3}}\). Thus, we obtain a measurement assemblage \(\mathcal {M}\) consisting of the following eight local POVMs on \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \mathcal {H}_{A_{3}}\):

$$M^{x_{1}}\otimes M^{x_{2}}\otimes M^{y_{3}},\ M^{x_{1}}\otimes M^{y_{2}}\otimes M^{y_{3}},\ M^{y_{1}}\otimes M^{x_{2}}\otimes M^{y_{3}}, M^{y_{1}}\otimes M^{y_{2}}\otimes M^{y_{3}},$$
$$M^{y_{1}}\otimes M^{y_{2}}\otimes M^{x_{3}},\ M^{y_{1}}\otimes M^{x_{2}}\otimes M^{x_{3}},\ M^{x_{1}}\otimes M^{y_{2}}\otimes M^{x_{3}}, M^{x_{1}}\otimes M^{x_{2}}\otimes M^{x_{3}}.$$

Since ρ is (A1,A2,A3)-local, by Definition 2.1, for this \(\mathcal {M}\), there exists a probability distribution {π λ }λ∈Γ such that (2.2) holds. Thus, we have

$$\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{+}\otimes M^{y_{3}}_{+})\rho= \sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)P_{A_{3}}(+|y_{3},\lambda),$$

and so on. Hence,

$$\begin{array}{@{}rcl@{}} &&\langle A^{0}_{a_{1}}A^{0}_{a_{2}}A^{1}_{a_{3}}\rangle_{\rho}\\ &=&\text{tr}((A^{0}_{a_{1}})^{+}\otimes (A^{0}_{a_{2}})^{+}\otimes (A^{1}_{a_{3}})^{+})\rho-\text{tr}((A^{0}_{a_{1}})^{+}\otimes (A^{0}_{a_{2}})^{-}\otimes (A^{1}_{a_{3}})^{+})\rho\\ &&-\text{tr}((A^{0}_{a_{1}})^{-}\otimes (A^{0}_{a_{2}})^{+}\otimes (A^{1}_{a_{3}})^{+})\rho+\text{tr}((A^{0}_{a_{1}})^{-}\otimes (A^{0}_{a_{2}})^{-}\otimes (A^{1}_{a_{3}})^{+})\rho\\ &&-\text{tr}((A^{0}_{a_{1}})^{+}\otimes (A^{0}_{a_{2}})^{+}\otimes (A^{1}_{a_{3}})^{-})\rho+\text{tr}((A^{0}_{a_{1}})^{+}\otimes (A^{0}_{a_{2}})^{-}\otimes (A^{1}_{a_{3}})^{-})\rho\\ &&+\text{tr}((A^{0}_{a_{1}})^{-}\otimes (A^{0}_{a_{2}})^{+}\otimes (A^{1}_{a_{3}})^{-})\rho-\text{tr}((A^{0}_{a_{1}})^{-}\otimes (A^{0}_{a_{2}})^{-}\otimes (A^{1}_{a_{3}})^{-})\rho\\ &=&\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{+}\otimes M^{y_{3}}_{+})\rho-\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{-}\otimes M^{y_{3}}_{+})\rho -\text{tr}(M^{x_{1}}_{-}\otimes M^{x_{2}}_{+}\otimes M^{y_{3}}_{+})\rho\\ &&+\text{tr}(M^{x_{1}}_{-}\otimes M^{x_{2}}_{-}\otimes M^{y_{3}}_{+})\rho-\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{+}\otimes M^{y_{3}}_{-})\rho+\text{tr}(M^{x_{1}}_{+}\otimes M^{x_{2}}_{-}\otimes M^{y_{3}}_{-})\rho\\ &&+\text{tr}(M^{x_{1}}_{-}\otimes M^{x_{2}}_{+}\otimes M^{y_{3}}_{-})\rho-\text{tr}(M^{x_{1}}_{-}\otimes M^{x_{2}}_{-}\otimes M^{y_{3}}_{-})\rho\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} \left\{P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)P_{A_{3}}(+|y_{3},\lambda)\right.\\ &&\left.-P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (-|x_{2},\lambda)P_{A_{3}}(+|y_{3},\lambda)-P_{A_{1}}(-|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)P_{A_{3}}(+|y_{3},\lambda)\right.\\ &&\left. +P_{A_{1}}(-|x_{1},\lambda)P_{A_{2}}(-|x_{2},\lambda)P_{A_{3}}(+|y_{3},\lambda)-P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)P_{A_{3}}(-|y_{3},\lambda)\right.\\ &&\left.+P_{A_{1}}(+|x_{1},\lambda)P_{A_{2}} (-|x_{2},\lambda)P_{A_{3}}(-|y_{3},\lambda)+P_{A_{1}}(-|x_{1},\lambda)P_{A_{2}} (+|x_{2},\lambda)P_{A_{3}}(-|y_{3},\lambda) \right.\\ &&\left.-P_{A_{1}}(-|x_{1},\lambda)P_{A_{2}}(-|x_{2},\lambda)P_{A_{3}}(-|y_{3},\lambda)\right\}\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\\ &&\times(P_{A_{3}}(+|y_{3},\lambda)-P_{A_{3}}(-|y_{3},\lambda)). \end{array} $$

Similarly,

$$\begin{array}{@{}rcl@{}} \langle A^{1}_{a_{1}}A^{0}_{a_{2}}A^{1}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\\ &&\times(P_{A_{3}}(+|y_{3},\lambda)-P_{A_{3}}(-|y_{3},\lambda)), \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle A^{0}_{a_{1}}A^{1}_{a_{2}}A^{1}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\\ &&\times(P_{A_{3}}(+|y_{3},\lambda)-P_{A_{3}}(-|y_{3},\lambda)), \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle A^{1}_{a_{1}}A^{1}_{a_{2}}A^{1}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\\ &&\times(P_{A_{3}}(+|y_{3},\lambda)-P_{A_{3}}(-|y_{3},\lambda)), \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle A^{1}_{a_{1}}A^{1}_{a_{2}}A^{0}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\\ &&\times(P_{A_{3}}(+|x_{3},\lambda)-P_{A_{3}}(-|x_{3},\lambda)), \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle A^{0}_{a_{1}}A^{1}_{a_{2}}A^{0}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\\ &&\times(P_{A_{3}}(+|x_{3},\lambda)-P_{A_{3}}(-|x_{3},\lambda)), \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle A^{1}_{a_{1}}A^{0}_{a_{2}}A^{0}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\\ &&\times(P_{A_{3}}(+|x_{3},\lambda)-P_{A_{3}}(-|x_{3},\lambda)) \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle A^{0}_{a_{1}}A^{0}_{a_{2}}A^{0}_{a_{3}}\rangle_{\rho} &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} (P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\\ &&\times(P_{A_{3}}(+|x_{3},\lambda)-P_{A_{3}}(-|x_{3},\lambda)) \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} \langle S_{3}\rangle_{\rho} &=&\frac{1}{4}\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} \left\{\{(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\right.\\ &&\left.+(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\right.\\ &&\left.+(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\right.\\ &&\left.-(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda)) (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))\}\right.\\ &&\left.\times (P_{A_{3}}(+|y_{3},\lambda)-P_{A_{3}}(-|y_{3},\lambda))+\{(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda))\right.\\ &&\left.\times(P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))+(P_{A_{1}}(+|y_{1},\lambda)-P_{A_{1}}(-|y_{1},\lambda))\right.\\ &&\left.\times(P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))+(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda))\right.\\ &&\left.\times (P_{A_{2}}(+|y_{2},\lambda)-P_{A_{2}}(-|y_{2},\lambda))-(P_{A_{1}}(+|x_{1},\lambda)-P_{A_{1}}(-|x_{1},\lambda))\right.\\ &&\left.\times(P_{A_{2}}(+|x_{2},\lambda)-P_{A_{2}}(-|x_{2},\lambda))\} (P_{A_{3}}(+|x_{3},\lambda)-P_{A_{3}}(-|x_{3},\lambda))\right\}\\ &=&\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} p_{3}(\lambda), \end{array} $$

where

$$p_{3}(\lambda)=\frac{1}{2}\left\{p_{2}(\lambda)(P_{A_{3}}(+|y_{3},\lambda)-P_{A_{3}}(-|y_{3},\lambda)) +p^{\prime}_{2}(\lambda)(P_{A_{3}}(+|x_{3},\lambda)-P_{A_{3}}(-|x_{3},\lambda))\right\},$$

and p2(λ) and \(p^{\prime }_{2}(\lambda )\) are defined in Remark 3.3. Since |p2(λ)|≤ 1 and \(|p^{\prime }_{2}(\lambda )|\le 1\), we obtain that

$$\begin{array}{@{}rcl@{}} |p_{3}(\lambda)| &\leq&\frac{1}{2}\left\{|p_{2}(\lambda)|\cdot|P_{A_{3}}(+|y_{3},\lambda) -P_{A_{3}}(-|y_{3},\lambda)|+|p^{\prime}_{2}(\lambda)|\cdot |P_{A_{3}}(+|x_{3},\lambda)\right.\\ &&~\quad\left.-P_{A_{3}}(-|x_{3},\lambda)|\right\}\\ &\leq&\frac{1}{2}(|p_{2}(\lambda)|+|p^{\prime}_{2}(\lambda)|)\\ &\leq& 1. \end{array} $$

Thus,

$$|\langle S_{3}\rangle_{\rho}|=\left|\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} p_{3}(\lambda)\right|\leq 1.$$

Since \(S_{3}=S^{\prime }_{3}\), so \(|\langle S^{\prime }_{3}\rangle _{\rho }|=|\langle S_{3}\rangle _{\rho }|\leq 1\). The proof is completed. □

Remark 3.4

Combining Theorem 2.1 with Remark 3.3, we can get that if ρ is Λ = (A1,A2,A3)-local, then \(|\langle S_{2}\rangle _{\rho _{12}}|\leq 1\) and \(|\langle S^{\prime }_{2}\rangle _{\rho _{12}}|\leq 1\).

Now let’s generalize above results to the general case.

Theorem 3.4

Suppose thatρisΛ = (A1,A2,…,A k )-local,then

$$ |\langle S_{k}\rangle_{\rho}|\leq 2^{k-\lceil\frac{k}{2}\rceil-1}, $$
(3.10)
$$|\langle S^{\prime}_{k}\rangle_{\rho}|\leq 2^{k-\lceil\frac{k}{2}\rceil-1}, $$
(3.11)

where \(\lceil \frac {k}{2}\rceil \)indicates rounding up to the next nearest integer.

Proof

Because that \(A^{0}_{a_{i}},A^{1}_{a_{i}}(i = 1,2,\ldots ,k)\) are ± 1-valued observables, they have their spectral decompositions:

$$A^{0}_{a_{i}}=(A^{0}_{a_{i}})^{+}-(A^{0}_{a_{i}})^{-}, A^{1}_{a_{i}}=(A^{1}_{a_{i}})^{+}-(A^{1}_{a_{i}})^{-},i = 1,2,\ldots,k.$$

For every i = 1,2,…,k, by taking

$$M^{x_{i}}_{+}=(A^{0}_{a_{i}})^{+}, M^{x_{i}}_{-}=(A^{0}_{a_{i}})^{-}, M^{y_{i}}_{+}=(A^{1}_{a_{i}})^{+}, M^{y_{i}}_{-}=(A^{1}_{a_{i}})^{-},$$

we obtain POVMs \(M^{x_{i}}=\{M^{x_{i}}_{+}, M^{x_{i}}_{-}\}\) and \(M^{y_{i}}=\{M^{y_{i}}_{+}, M^{y_{i}}_{-}\}\) on \(\mathcal {H}_{A_{i}}(i = 1,2,\ldots ,k)\). Thus, we obtain a measurement assemblage

$$\mathcal{M}=\{M^{t_{1}}\otimes M^{t_{2}}\otimes\ldots\otimes M^{t_{k}}: t_{i}=x_{i},y_{i}(i = 1,2,\ldots,k)\}$$

of 2k local POVMs on \(\mathcal {H}_{A_{1}}\otimes \mathcal {H}_{A_{2}}\otimes \ldots \otimes \mathcal {H}_{A_{k}}\).

Since ρ is Λ = (A1,A2,…,A k )-local, by Definition 2.1, for this \(\mathcal {M}\), there exists a probability distribution {π λ }λ∈Γ such that (2.2) holds. Thus, similar to the proofs of Theorem 3.1 and Theorem 3.2, we obtain that

$$\langle S_{k}\rangle_{\rho}=\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} p_{k}(\lambda),\ \ \langle S^{\prime}_{k}\rangle_{\rho}=\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} p^{\prime}_{k}(\lambda),$$

where p k (λ) and \(p^{\prime }_{k}(\lambda )\) are obtained by changing \(A^{0}_{a_{i}}\) to \(P_{A_{i}}(+|x_{i},\lambda )-P_{A_{i}}(-|x_{i},\lambda )\), \(A^{1}_{a_{i}}\) to \(P_{A_{i}}(+|y_{i},\lambda )-P_{A_{i}}(-|y_{i},\lambda )(i = 1,2,\ldots ,k)\) in the definitions of S k and \(S^{\prime }_{k}\), respectively.

Let λ ∈Γ be fixed. Next, we prove that |p2m(λ)|≤ 2m− 1 and \(|p^{\prime }_{2m}(\lambda )|\leq 2^{m-1}\) hold for \(1\le m\le \lfloor \frac {k}{2}\rfloor \), where \(\lfloor \frac {k}{2}\rfloor \) indicates rounding down to the next nearest integer.

When m = 1, by Remark 3.3, we know that |p2(λ)|≤ 1 and \(|p^{\prime }_{2}(\lambda )|\leq 1\) since \(\rho _{12}=\text {tr}_{A_{3}A_{4}{\ldots } A_{k}}(\rho )\) is (A1,A2)-local(Theorem 2.1). Suppose that |p2m(λ)|≤ 2m− 1 and \(|p^{\prime }_{2m}(\lambda )|\leq 2^{m-1}\) hold. Using (3.6)-(3.8) yield that

$$\begin{array}{@{}rcl@{}} S_{2m + 2} \!\!&=&\!\!\frac{1}{2}[M_{2m + 1}(A^{0}_{a_{2m + 2}}+A^{1}_{a_{2m + 2}})+M^{\prime}_{2m + 1} (A^{0}_{a_{2m + 2}}-A^{1}_{a_{2m + 2}})]\\ \!\!&=&\!\! \frac{1}{2}\left\{\frac{1}{2}[M_{2m}(A^{0}_{a_{2m + 1}}+A^{1}_{a_{2m + 1}})+M^{\prime}_{2m} (A^{0}_{a_{2m + 1}}-A^{1}_{a_{2m + 1}})](A^{0}_{a_{2m + 2}}+A^{1}_{a_{2m + 2}})\right.\\ &&\left.+\frac{1}{2}[M^{\prime}_{2m}(A^{1}_{a_{2m + 1}}+A^{0}_{a_{2m + 1}})+M_{2m} (A^{1}_{a_{2m + 1}}-A^{0}_{a_{2m + 1}})] (A^{0}_{a_{2m + 2}}-A^{1}_{a_{2m + 2}})\right\}\\ \!\!&=&\!\! \frac{1}{2}\left\{M_{2m}(A^{0}_{a_{2m + 1}}A^{1}_{a_{2m + 2}}+A^{1}_{a_{2m + 1}}A^{0}_{a_{2m + 2}}) +M^{\prime}_{2m}(A^{0}_{a_{2m + 1}}A^{0}_{a_{2m + 2}}-A^{1}_{a_{2m + 1}}A^{1}_{a_{2m + 2}})\right\}\\ \!\!&=&\!\! \frac{1}{2}\left\{S_{2m}(A^{0}_{a_{2m + 1}}A^{1}_{a_{2m + 2}}+A^{1}_{a_{2m + 1}}A^{0}_{a_{2m + 2}}) +S^{\prime}_{2m}(A^{0}_{a_{2m + 1}}A^{0}_{a_{2m + 2}}-A^{1}_{a_{2m + 1}}A^{1}_{a_{2m + 2}})\right\}. \end{array} $$

Thus,

$$\begin{array}{@{}rcl@{}} p_{2m + 2}(\lambda) &=&\frac{1}{2}\left\{p_{2m}(\lambda)\{(P_{A_{2m + 1}}(+|x_{2m + 1},\lambda) -P_{A_{2m + 1}}(-|x_{2m + 1},\lambda))\right.\\ &&\left.\times(P_{A_{2m + 2}}(+|y_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|y_{2m + 2},\lambda))\right.\\ &&\left.+(P_{A_{2m + 1}}(+|y_{2m + 1},\lambda)-P_{A_{2m + 1}}(-|y_{2m + 1},\lambda))\right.\\ &&\left.\times(P_{A_{2m + 2}}(+|x_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|x_{2m + 2},\lambda))\}\right.\\ &&\left.+p^{\prime}_{2m}(\lambda)\{(P_{A_{2m + 1}}(+|x_{2m + 1},\lambda)-P_{A_{2m + 1}}(-|x_{2m + 1},\lambda))\right.\\ &&\left.\times(P_{A_{2m + 2}}(+|x_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|x_{2m + 2},\lambda))\right.\\ &&\left.-(P_{A_{2m + 1}}(+|y_{2m + 1},\lambda)-P_{A_{2m + 1}}(-|y_{2m + 1},\lambda))\right.\\ &&\left.\times(P_{A_{2m + 2}}(+|y_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|y_{2m + 2},\lambda))\}\right\}, \end{array} $$

therefore,

$$\begin{array}{@{}rcl@{}} |p_{2m + 2}(\lambda)| &\leq&\frac{1}{2}\left\{|p_{2m}(\lambda)|\cdot\{|P_{A_{2m + 1}}(+|x_{2m + 1},\lambda) -P_{A_{2m + 1}}(-|x_{2m + 1},\lambda)|\right.\\ &&\left.\cdot|P_{A_{2m + 2}}(+|y_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|y_{2m + 2},\lambda)|\right.\\ &&\left.+|P_{A_{2m + 1}}(+|y_{2m + 1},\lambda)-P_{A_{2m + 1}}(-|y_{2m + 1},\lambda)|\right.\\ &&\left.\cdot|P_{A_{2m + 2}}(+|x_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|x_{2m + 2},\lambda)|\}\right.\\ &&\left.+|p^{\prime}_{2m}(\lambda)|\cdot\{|P_{A_{2m + 1}}(+|x_{2m + 1},\lambda)-P_{A_{2m + 1}} (-|x_{2m + 1},\lambda)|\right.\\ &&\left.\cdot|P_{A_{2m + 2}}(+|x_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|x_{2m + 2},\lambda)|\right.\\ &&\left.+|P_{A_{2m + 1}}(+|y_{2m + 1},\lambda)-P_{A_{2m + 1}}(-|y_{2m + 1},\lambda)|\right.\\ &&\left.\cdot|P_{A_{2m + 2}}(+|y_{2m + 2},\lambda)-P_{A_{2m + 2}}(-|y_{2m + 2},\lambda)|\}\right\}\\ &\leq& |p_{2m}(\lambda)|+|p^{\prime}_{2m}(\lambda)|\\ &\leq& 2^{m}. \end{array} $$

\(S^{\prime }_{2m + 2}\) can be obtained from S2m+ 2:

$$S^{\prime}_{2m + 2}=\frac{1}{2}[S^{\prime}_{2m}(A^{1}_{a_{2m + 1}}A^{0}_{a_{2m + 2}}+A^{0}_{a_{2m + 1}} A^{1}_{a_{2m + 2}})+S_{2m}(A^{1}_{a_{2m + 1}}A^{1}_{a_{2m + 2}}-A^{0}_{a_{2m + 1}} A^{0}_{a_{2m + 2}})].$$

Similarly, we can get

$$|p^{\prime}_{2m + 2}(\lambda)|\leq 2^{m}.$$

Therefore, by induction, we have |p2m(λ)|≤ 2m− 1 and \(|p^{\prime }_{2m}(\lambda )|\leq 2^{m-1}\) hold for all \(m = 1,2,\ldots ,\lfloor \frac {k}{2}\rfloor \) and all λ ∈Γ.

Using (3.6)-(3.8) yield that

$$\begin{array}{@{}rcl@{}} {S_{2m + 1}}&=&\frac{1}{2}(M_{2m + 1}+M^{\prime}_{2m + 1})\\ &=&\frac{1}{2}\left\{\frac{1}{2}[M_{2m}(A^{0}_{a_{2m + 1}}+A^{1}_{a_{2m + 1}})+M^{\prime}_{2m} (A^{0}_{a_{2m + 1}}-A^{1}_{a_{2m + 1}})]\right.\\ &&\left.+\frac{1}{2}[M^{\prime}_{2m}(A^{1}_{a_{2m + 1}}+A^{0}_{a_{2m + 1}})+M_{2m} (A^{1}_{a_{2m + 1}}-A^{0}_{a_{2m + 1}})]\right\}\\ &=&\frac{1}{2}(M_{2m}A^{1}_{a_{2m + 1}}+M^{\prime}_{2m}A^{0}_{a_{2m + 1}})\\ &=&\frac{1}{2}(S_{2m}A^{1}_{a_{2m + 1}}+S^{\prime}_{2m}A^{0}_{a_{2m + 1}}), \end{array} $$

then

$$\begin{array}{@{}rcl@{}} p_{2m + 1}(\lambda) &=&\frac{1}{2}\left\{p_{2m}(\lambda)[P_{A_{2m + 1}}(+|y_{2m + 1},\lambda) -P_{A_{2m + 1}}(-|y_{2m + 1},\lambda)]\right.\\ &&\left. +p^{\prime}_{2m}(\lambda)[P_{A_{2m + 1}}(+|x_{2m + 1},\lambda)-P_{A_{2m + 1}}(-|x_{2m + 1},\lambda)]\right\}. \end{array} $$

By using the fact that |p2m(λ)|≤ 2m− 1 and \(|p^{\prime }_{2m}(\lambda )|\leq 2^{m-1}\) for all λ, we get that

$$\begin{array}{@{}rcl@{}} |p_{2m + 1}(\lambda)| &\leq&\frac{1}{2}\left\{|p_{2m}(\lambda)|\cdot|P_{A_{2m + 1}}(+|y_{2m + 1},\lambda) -P_{A_{2m + 1}}(-|y_{2m + 1},\lambda)|\right.\\ &&\left.+|p^{\prime}_{2m}(\lambda)|\cdot|P_{A_{2m + 1}}(+|x_{2m + 1},\lambda) -P_{A_{2m + 1}}(-|x_{2m + 1},\lambda)|\right\}\\ &\leq&\frac{1}{2}(|p_{2m}(\lambda)|+|p^{\prime}_{2m}(\lambda)|)\\ &\leq& 2^{m-1}. \end{array} $$

Since \(S^{\prime }_{2m + 1}=\frac {1}{2}(S^{\prime }_{2m}A^{0}_{a_{2m + 1}}+S_{2m}A^{1}_{a_{2m + 1}})=S_{2m + 1}\), we have

$$|p^{\prime}_{2m + 1}(\lambda)|=|p_{2m + 1}(\lambda)|\leq 2^{m-1}.$$

Therefore, we have |p2m+ 1(λ)|≤ 2m− 1 and \(|p^{\prime }_{2m + 1}(\lambda )|\leq 2^{m-1}\) hold for all \(m = 1,2,\ldots ,\lfloor \frac {k-1}{2}\rfloor \) and all λ ∈Γ.

From what has been discussed above, we can get that

$$|p_{k}(\lambda)|\leq 2^{k-\lceil\frac{k}{2}\rceil-1},\ |p^{\prime}_{k}(\lambda)|\leq 2^{k-\lceil\frac{k}{2}\rceil-1}.$$

Hence, we have

$$|\langle S_{k}\rangle_{\rho}|=\left|\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} p_{k}(\lambda)\right|\leq 2^{k-\lceil\frac{k}{2}\rceil-1},\ |\langle S^{\prime}_{k}\rangle_{\rho}|=\left|\sum\limits_{\lambda\in{\Gamma}}{\Pi}_{\lambda} p^{\prime}_{k}(\lambda)\right|\leq 2^{k-\lceil\frac{k}{2}\rceil-1}.$$

The proof is completed. □

Example 3.1

The 3-qubit state \(|\psi \rangle =\frac {1}{\sqrt {2}} (|001\rangle -|110\rangle )\), i.e. ρ = |ψ〉〈ψ| is genuinely nonlocality, i.e. it is not Λ-local for every Λ.

Proof

By computing, we get

$$\rho=|\psi\rangle\langle \psi|=\frac{1}{2}(|001\rangle\langle001| -|001\rangle\langle110|-|110\rangle\langle001|+|110\rangle\langle110|).$$

Generally, for all real unit vectors:

$$\boldsymbol{a}=(a_{x},a_{y},a_{z}), \boldsymbol{b}=(b_{x},b_{y},b_{z}), \boldsymbol{c}=(c_{x},c_{y},c_{z}),$$

and the Pauli operator vector σ = (σ x ,σ y ,σ z ),we have

$$\begin{array}{@{}rcl@{}} &&\boldsymbol{a}\cdot\boldsymbol{\sigma}\otimes\boldsymbol{b}\cdot\boldsymbol{\sigma}\otimes\boldsymbol{c}\cdot \boldsymbol{\sigma}\\ &=&(a_{x}\sigma_{x}+a_{y}\sigma_{y}+a_{z}\sigma_{z})\otimes (b_{x}\sigma_{x}+b_{y}\sigma_{y}+b_{z}\sigma_{z})\otimes (c_{x}\sigma_{x}+c_{y}\sigma_{y}+c_{z}\sigma_{z})\\ &=&{\sum}_{i,j,k}a_{i}b_{j}c_{k}\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}, \end{array} $$

with 〈σ i ⊗ σ j ⊗ σ k 〉 ρ = 0 for all i,j,k ∈{x,y,z} except for the following four cases:

$$\langle\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{x}\rangle_{\rho}=-1, \langle\sigma_{x}\otimes\sigma_{y}\otimes\sigma_{y}\rangle_{\rho}=-1, \langle\sigma_{y}\otimes\sigma_{x}\otimes\sigma_{y}\rangle_{\rho}=-1, \langle\sigma_{y}\otimes\sigma_{y}\otimes\sigma_{x}\rangle_{\rho}= 1.$$

Thus

$$ \langle\boldsymbol{a}\cdot\boldsymbol{\sigma}\otimes\boldsymbol{b}\cdot\boldsymbol{\sigma}\otimes\boldsymbol{c}\cdot \boldsymbol{\sigma}\rangle_{\rho}=-(a_{x}b_{x}-a_{y}b_{y})c_{x}-(a_{x}b_{y}+a_{y}b_{x})c_{y}. $$
(3.12)

Especially, (3.12) holds for

$$\boldsymbol{a}=(a_{x},a_{y},a_{z})=(\cos\alpha,\sin\alpha,0),$$
$$\boldsymbol{b}=(b_{x},b_{y},b_{z})=(\cos\beta,\sin\beta,0),$$
$$\boldsymbol{c}=(c_{x},c_{y},c_{z})=(\cos\gamma,\sin\gamma,0),$$

in this case, we have

$$ \langle\boldsymbol{a}\cdot\boldsymbol{\sigma}\otimes\textbf{b}\cdot\boldsymbol{\sigma}\otimes\boldsymbol{c}\cdot \boldsymbol{\sigma}\rangle_{\rho}=-\cos(\alpha+\beta-\gamma). $$
(3.13)

Firstly, we show that ρ is not {1;2;3}-local. Suppose that this is not the case, then Theorem 3.3 yields that |〈S3〉 ρ |≤ 1, where

$$\begin{array}{@{}rcl@{}} S_{3}&=&\frac{1}{4} \left( A^{0}_{a_{1}}A^{0}_{a_{2}}A^{1}_{a_{3}} +A^{0}_{a_{1}}A^{1}_{a_{2}}A^{1}_{a_{3}} +A^{1}_{a_{1}}A^{0}_{a_{2}}A^{1}_{a_{3}} -A^{1}_{a_{1}}A^{1}_{a_{2}}A^{1}_{a_{3}} \right.\\&&\left.+A^{1}_{a_{1}}A^{1}_{a_{2}}A^{0}_{a_{3}} +A^{1}_{a_{1}}A^{0}_{a_{2}}A^{0}_{a_{3}} +A^{0}_{a_{1}}A^{1}_{a_{2}}A^{0}_{a_{3}} -A^{0}_{a_{1}}A^{0}_{a_{2}}A^{0}_{a_{3}}\right),\end{array} $$

and \(A^{x_{s}}_{a_{s}}(s = 1,2,3)\) are any ± 1-valued observables of \(\mathcal {H}_{A_{s}}\). But for the ± 1-valued observables

$$A^{0}_{a_{1}}=\boldsymbol{a}_{1}^{0}\cdot\boldsymbol{\sigma},\ \ \boldsymbol{a}_{1}^{0}= (\cos\alpha_{0},\sin\alpha_{0},0),$$
$$A^{1}_{a_{1}}=\boldsymbol{a}_{1}^{1}\cdot\boldsymbol{\sigma},\ \ \boldsymbol{a}_{1}^{1}= (\cos\alpha_{1},\sin\alpha_{1},0),$$
$$A^{0}_{a_{2}}=\boldsymbol{a}_{2}^{0}\cdot\boldsymbol{\sigma},\ \ \boldsymbol{a}_{2}^{0}= (\cos\beta_{0},\sin\beta_{0},0),$$
$$A^{1}_{a_{2}}=\boldsymbol{a}_{2}^{1}\cdot\boldsymbol{\sigma},\ \ \boldsymbol{a}_{2}^{1}= (\cos\beta_{1},\sin\beta_{1},0),$$
$$A^{0}_{a_{3}}=\boldsymbol{a}_{3}^{0}\cdot\boldsymbol{\sigma},\ \ \boldsymbol{a}_{3}^{0}= (\cos\gamma_{0},\sin\gamma_{0},0),$$
$$A^{1}_{a_{3}}=\boldsymbol{a}_{3}^{1}\cdot\boldsymbol{\sigma},\ \ \boldsymbol{a}_{3}^{1}= (\cos\gamma_{1},\sin\gamma_{1},0),$$

Equation (3.13) implies that

$$\begin{array}{@{}rcl@{}} \langle S_{3}\rangle_{\rho} &=&\frac{1}{4}[-\cos(\alpha_{0}+\beta_{0}-\gamma_{1})-\cos(\alpha_{0}+\beta_{1}-\gamma_{1}) \\&&-\cos(\alpha_{1}+\beta_{0}-\gamma_{1}) +\cos(\alpha_{1}+\beta_{1}-\gamma_{1})\\ &&-\cos(\alpha_{1}+\beta_{1}-\gamma_{0})-\cos(\alpha_{1}+\beta_{0}-\gamma_{0}) \\&&-\cos(\alpha_{0}+\beta_{1}-\gamma_{0}) +\cos(\alpha_{0}+\beta_{0}-\gamma_{0})]. \end{array} $$

Especially, we take

$$\alpha_{0}= 0, \alpha_{1}=-\frac{\pi}{2}, \beta_{0}=\frac{3\pi}{4}, \beta_{1}=\frac{\pi}{4}, \gamma_{0}= 0, \gamma_{1}=\frac{\pi}{2},$$

and get

$$\cos(\alpha_{0}+\beta_{0}-\gamma_{1})=\frac{\sqrt{2}}{2}, \cos(\alpha_{0}+\beta_{1}-\gamma_{1})=\frac{\sqrt{2}}{2}, $$
$$\cos(\alpha_{1}+\beta_{0}-\gamma_{1})=\frac{\sqrt{2}}{2}, \cos(\alpha_{1}+\beta_{1}-\gamma_{1})=-\frac{\sqrt{2}}{2}, $$
$$\cos(\alpha_{1}+\beta_{1}-\gamma_{0})=\frac{\sqrt{2}}{2}, \cos(\alpha_{1}+\beta_{0}-\gamma_{0})=\frac{\sqrt{2}}{2}, $$
$$\cos(\alpha_{0}+\beta_{1}-\gamma_{0})=\frac{\sqrt{2}}{2}, \cos(\alpha_{0}+\beta_{0}-\gamma_{0})=-\frac{\sqrt{2}}{2}. $$

Thus,

$$|\langle S_{3}\rangle_{\rho}|=\frac{1}{4}\times 8\times\frac{\sqrt{2}}{2} =\sqrt{2}>1, $$

a contradiction. This shows that ρ is not {1;2;3}-local.

Similarly, by taking ± 1-valued observables

$$A^{0}_{a_{1}}=\sigma_{x}\otimes\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}},\ A^{1}_{a_{1}}=-\sigma_{y}\otimes\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}},\ A^{0}_{a_{2}}=\sigma_{x},\ A^{1}_{a_{2}}=\sigma_{y}$$

and using (3.12), we get

$$\begin{array}{@{}rcl@{}} \langle S_{2}\rangle_{\rho} &=&\frac{1}{2}\left\langle \sigma_{x}\otimes\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}} \otimes\sigma_{x}+\sigma_{x}\otimes\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}} \otimes\sigma_{y}\right.\\ &&\left.-\sigma_{y}\otimes\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes \sigma_{x}+\sigma_{y}\otimes\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes \sigma_{y}\right\rangle_{\rho}\\ &=&\frac{1}{2}\times(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} -\frac{1}{\sqrt{2}})\\ &=&-\sqrt{2}. \end{array} $$

This shows that |〈S2〉 ρ | > 1 and then ρ is not {1,2;3}-local (Remark 3.3).

Finally, by taking

$$A^{0}_{a_{1}}=-\sigma_{y},\ A^{1}_{a_{1}}=\sigma_{x},\ A^{0}_{a_{2}}=\frac{-\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes\sigma_{y},\ A^{1}_{a_{2}}=\frac{-\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes\sigma_{x}$$

and using (3.12), we get

$$\begin{array}{@{}rcl@{}} \langle S_{2}\rangle_{\rho} &=&\frac{1}{2}\left\langle -\sigma_{y}\otimes\frac{-\sigma_{x}+\sigma_{y}}{\sqrt{2}} \otimes\sigma_{y}-\sigma_{y}\otimes\frac{-\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes \sigma_{x}\right.\\ &&\left.+\sigma_{x}\otimes\frac{-\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes \sigma_{y}-\sigma_{x}\otimes\frac{-\sigma_{x}+\sigma_{y}}{\sqrt{2}}\otimes\sigma_{x} \right\rangle_{\rho}\\ &=&\frac{1}{2}\times(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} -\frac{1}{\sqrt{2}})\\ &=&-\sqrt{2}. \end{array} $$

This shows that |〈S2〉 ρ | > 1 and therefore ρ is not {1;2,3}-local (Remark 3.3). □

As a conclusion, ρ is Λ-nonlocal for every Λ. The proof is completed.

4 A Class of 2-separable Nonlocal States

According to Refs. [23, 24], a 2-partition A1|A2 of the index set {1,2,…,n} consists of a pairwise disjoint subsets

$$A_{1}=\{m_{1},m_{2},\ldots,m_{j}\}, \ A_{2}=\{m_{j + 1},\ldots,m_{n}\},$$

such that A1 ∪ A2 = {1,2,…,n}. An n-partite pure state \(|\psi \rangle \in \mathcal {H}_{1}\otimes \mathcal {H}_{2}\otimes \ldots \otimes \mathcal {H}_{n}\) was called 2-separable in [23, 24] means that there is a 2-partition A1|A2, and two states \(|\psi _{1}\rangle _{A_{1}}\) of \(\mathcal {H}_{m_{1}}\otimes \mathcal {H}_{m_{2}}\otimes \ldots \otimes \mathcal {H}_{m_{j}}\) and \(|\psi _{2}\rangle _{A_{2}}\) of \(\mathcal {H}_{m_{j + 1}}\otimes \mathcal {H}_{m_{j + 2}}\otimes \ldots \otimes \mathcal {H}_{m_{n}}\) such that \(|\psi \rangle =|\psi _{1}\rangle _{A_{1}}\otimes |\psi _{2}\rangle _{A_{2}}\). An n-partite mixed state ρ was called 2-separable [23, 24] if it can be written as a convex combination of 2-separable pure states: \(\rho ={\sum }_{m = 1}^{d}p_{m}|\varphi _{m}\rangle \langle \varphi _{m}|, \)where |φ m 〉(m = 1,2,…,d) are 2-separable, but may be with respect to different 2-partitions.

It is well-known that a separable bipartite state must be Bell-local. For multipartite case, k-separability [23, 24] does not imply locality, because that there exists a class of 2-separable states which are nonlocal, see Example 4.1.

Example 4.1

Consider the following pure states of \(\mathbb {C}^{2}\otimes \mathbb {C}^{2}\otimes \mathbb {C}^{2}\):

$$|\psi_{1}\rangle=\frac{1}{\sqrt{2}}|0\rangle_{1}\otimes(|00\rangle+|11\rangle)_{23},$$
$$|\psi_{2}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{12}\otimes|0\rangle_{3},$$

being 2-separable with respect to 2-partitions {1}|{2,3} and {1,2}|{3}, respectively. Thus, the mixed state

$$\rho=p|\psi_{1}\rangle\langle\psi_{1}|+(1-p)|\psi_{2}\rangle\langle\psi_{2}|$$

is 2-separable for all 0 ≤ p ≤ 1 according to the definition in [23, 24]. Next, we show that ρ is neither {1,2;3}-local when \(\sqrt {2}-1<p\leq 1\) nor {1;2,3}-local when \(0\leq p<2-\sqrt {2}\). In other words, ρ has no any locality.

In fact, we compte that

$$\begin{array}{@{}rcl@{}} \rho &=&\frac{p}{2}|000\rangle\langle000|+\frac{p}{2}|000\rangle\langle011| +\frac{p}{2}|011\rangle\langle000|+\frac{p}{2}|011\rangle\langle011|\\ &&+\frac{1-p}{2}|000\rangle\langle000|+\frac{1-p}{2}|000\rangle\langle110| +\frac{1-p}{2}|110\rangle\langle000|\\&&+\frac{1-p}{2}|110\rangle\langle110|. \end{array} $$

Let P,Q be as in (3.1). Then

$$\begin{array}{@{}rcl@{}} \langle \sigma_{z}\otimes\sigma_{z}\otimes P\rangle_{\rho} &=&\text{tr}(\sigma_{z}\otimes\sigma_{z}\otimes P)\rho\\ &=&\frac{p}{2}\text{tr}(P|0\rangle\langle0|)-\frac{p}{2}\text{tr}(P|1\rangle\langle1|)\\ &&+\frac{1-p}{2}\text{tr}(P|0\rangle\langle0|)\\&&+\frac{1-p}{2}\text{tr}(P|0\rangle\langle0|)\\ &=&\frac{2-p}{2}\langle 0|P|0\rangle-\frac{p}{2}\langle 1|P|1\rangle, \end{array} $$
$$\begin{array}{@{}rcl@{}} \langle \sigma_{z}\otimes\sigma_{x}\otimes P\rangle_{\rho} &=&\text{tr}(\sigma_{z}\otimes\sigma_{x}\otimes P)\rho\\ &=&\frac{p}{2}\text{tr}(P|0\rangle\langle1|)+\frac{p}{2}\text{tr}(P|1\rangle\langle0|)\\ &=&\frac{p}{2}\langle 1|P|0\rangle+\frac{p}{2}\langle 0|P|1\rangle. \end{array} $$

Similarly,

$$\begin{array}{@{}rcl@{}} \langle \sigma_{z}\otimes \sigma_{z}\otimes Q\rangle_{\rho}&=&\frac{2-p}{2}\langle 0|Q|0\rangle -\frac{p}{2}\langle 1|Q|1\rangle,\\ \langle \sigma_{z}\otimes\sigma_{x}\otimes Q\rangle_{\rho}&=&\frac{p}{2}\langle 1|Q|0\rangle+\frac{p}{2}\langle 0|Q_|1\rangle. \end{array} $$

So,

$$\begin{array}{@{}rcl@{}} {\Delta}&:=&|\langle \sigma_{z}\otimes\sigma_{z}\otimes P\rangle_{\rho}+\langle \sigma_{z}\otimes \sigma_{z}\otimes Q\rangle_{\rho}+\langle \sigma_{z}\otimes\sigma_{x}\otimes P\rangle_{\rho} -\langle \sigma_{z}\otimes\sigma_{x}\otimes Q\rangle_{\rho}|\\ &=&\left|\frac{2-p}{2}\langle 0|P|0\rangle-\frac{p}{2}\langle 1|P|1\rangle +\frac{2-p}{2}\langle 0|Q|0\rangle -\frac{p}{2}\langle 1|Q|1\rangle\right.\\ &&\left.+\frac{p}{2}\langle 1|P|0\rangle+\frac{p}{2}\langle 0|P|1\rangle -\frac{p}{2}\langle 1|Q|0\rangle -\frac{p}{2}\langle 0|Q|1\vphantom{\frac{2-p}{2}}\rangle\right|\\ &=&\left|\frac{2-p}{2}\langle 0|P+Q|0\rangle -\frac{p}{2}\langle 1|P+Q|1\rangle +\frac{p}{2}\langle 1|P-Q|0\rangle +\frac{p}{2}\langle 0|P-Q|1\rangle\right|. \end{array} $$

Especially, when

$$P=\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}},\ Q=\frac{\sigma_{z}-\sigma_{x}} {\sqrt{2}},$$

have

$$P+Q=\sqrt{2}\sigma_{z}=\left( \begin{array}{cc} \sqrt{2} & 0 \\ 0 & -\sqrt{2} \end{array} \right) ,\ P-Q=\sqrt{2}\sigma_{x}=\left( \begin{array}{cc} 0 & \sqrt{2} \\ \sqrt{2} & 0 \end{array} \right) .$$

Thus

$${\Delta}=\frac{\sqrt{2}}{2}(2-p)+\frac{\sqrt{2}}{2}p+\frac{\sqrt{2}}{2}p +\frac{\sqrt{2}}{2}p=\sqrt{2}+\sqrt{2}p.$$

Clearly,

$${\Delta}>2\Leftrightarrow p>\sqrt{2}-1.$$

Therefore, we have if \(1\ge p>\sqrt {2}-1\), then ρ is {1,2;3}-nonlocal.

Similarly, we can prove that ρ is {1;2,3}-nonlocal when \(0\leq p<2-\sqrt {2}\).

5 Conclusions

In this paper, we have introduced Λ-locality and Λ-nonlocality of n-partite states, discussed some related properties and established some related nonlocality inequalities for {1,2;3}-local, {1;2,3}-local, and Λ-local states, respectively. The violation of one of these inequalities exhibits Λ-nonlocality. As application, we have checked genuinely nonlocality of a tripartite state by a violation of nonlocality inequality. Finally, we have given a class of 2-separable nonlocal states, which shows that a 2-separable tripartite state is not necessarily local. When n = 2,k = 2,A1 = {1},A2 = {2}, Λ-locality is equivalent to Bell-locality. This implies that Λ-locality generalizes the Bell-locality of bipartite states and Λ-nonlocality inequalities are generalization of the usual Bell-inequalities.