Computationally Efficient Nonlinear Bell Inequalities for General Quantum Networks
The correlations in quantum networks have attracted strong interest with new types of violations of the locality. The standard Bell inequalities cannot characterize these multipartite correlations, which are generated by multiple sources. The main problem is that no computationally efficient method is available for constructing useful Bell inequalities for any quantum network. In this work, we show a significant improvement by presenting new, explicit Bell-type inequalities for general networks including cyclic networks. These nonlinear inequalities are related to the matching problem of an equivalent unweighted bipartite graph which allows constructing a polynomial-time algorithm. For any bipartite entangled pure states and Greenberger-Horne-Zeilinger (GHZ) states as quantum resources, we prove the generic non-multilocality of quantum networks with multiple independent observers using the new Bell inequalities. The violations are maximal with respect to the presented Tsirelson’s bound for the maximally entangled Einstein-Podolsky-Rosen (EPR) states and GHZ states. Moreover, these violations hold for Werner states or some general noisy states. Our results suggest that the new Bell inequalities can be used to characterize experimental quantum networks.
Bell’s well-known theorem  states that the predictions of quantum mechanics are inconsistent with classical causal relations that originate from a common local hidden variable (LHV). Specifically, the correlation between the outcomes of local measurements on a remotely shared entanglement cannot be described by a locally causal model. The study of quantum nonlocality has stimulated both remarkable developments in quantum theory [2-5] and potential applications [6-10].
Quantum nonlocality has been significantly generalized by considering complex causal structures beyond the standard LHV models [11-17]. These improvements aim to provide a rigorous theoretical framework of causal relations and structures [3,18-20] and are useful for deriving similar linear Bell inequalities [2-5,9,21]. In theory, these inequalities are derived from networks of a single source. Nonetheless, for general networks, there are various independent sources to distribute hidden states to space-like separated parties in terms of the generalized locally causal model (GLCM) [3,18-20]. As reasonable extensions of a single source, the correlations should be defined using multiple sources. Meanwhile, a useful Bell-type inequality enables the characterization of these correlations across the entire network. How to feature and verify the nonlocality of multipartite correlations not only are theoretically important to prove the supremacy , but also are experimentally challenging in the implementation of quantum networks [22,23] and quantum repeaters .
Unfortunately, the standard Bell inequalities derived from a single source are useless for characterizing the correlations of general quantum networks. Recently, for the simplest network of entanglement swapping with two shared entangled states, new nonlinear Bell inequalities have been proposed to verify the non-bilocality of tripartite correlations [12,25-27]. It is then extended for a general star-shaped network with multiple parties . For a small-sized general network, computational algebraic method  or the linear programming technique provides a reasonable route to explore polynomial Bell inequalities . Another method is to iteratively expand a given network to the desired network by adding independent sources . Despite these advances, no computationally efficient method is available to feature general quantum networks. Additionally, the nonlinear Bell inequalities imply that some projection subspaces of the multipartite correlation space are not convex [25,29-32], which reveal new features beyond the correlation polytopes bounded by linear Bell inequalities [1-5]. A natural problem is whether these characteristics are typical for quantum networks. One of our goals is to address this problem. Certain quantum networks have been experimentally realized using different physical systems to verify the nonlocality [33-36].
In this work, we propose simple and efficient nonlinear Bell inequalities to characterize the multipartite correlations of a general quantum network in terms of the GLCM [17-20]. Notably, our approach depends primarily on the maximal matching problem of the equivalent unweighted bipartite graph , which allows constructing new Bell inequalities within polynomial time complexity. We further prove that the multipartite correlations violate the presented nonlinear inequalities for all finite-size quantum networks with multiple observers who have not shared quantum resources. The violation holds for any bipartite entangled pure states and Greenberger-Horne-Zeilinger (GHZ) states as quantum resources, and are maximal with respect to the presented Tsirelson’s bound of Hermitian operators. The generic non-multilocality is different from the nonlocality of a single entanglement using the linear Bell inequality [38,39] or CHSH inequality [40,41]. Finally, we evaluate the critical visibilities of Werner states and general noisy states for which the non-multilocality is also true [30-32]. Remarkably, our result holds for lots of cyclic networks which have not been investigated [24,27-32]. The simplicity of the presented Bell inequalities makes them useful for experimental quantum networks.
Multilocality structure of a network. In what follows, we consider the simplest scenario of dichotomic inputs and outputs for all parties.
Inspired from Bell inequalities of two parties , the multilocality of correlations of a network follows from the GLCM [3,18-20]. Formally, all systems measured in the experiment are considered to be in the hidden states of , where is arbitrary and could exist prior to the measurement choices, and is the number of hidden states. The dichotomic output of any particular system can arbitrarily depend on the global state and the type of measurement but not on the measurements performed on systems (here, one bit denotes the type of measurement). Thus, the GLCM suggests a general representation of the probabilities of the measurement outcome or correlations as
where and , . Here, is the joint distribution of with the normalization condition ; is the conditional probability of outcome for the -th party with the knowledge of and ; and is the number of space-like separated parties.
Now, we consider a general network of finite size shown in Fig.1 in terms of the GLCM. Assume that there are sources , which are used to distribute the corresponding hidden states . Each party (or observer in quantum mechanics) receives some hidden states from the corresponding sources , where . By combining all dependent sources into one source, we suppose that all sources are independent. The joint probability distribution of hidden states has the form , where is the probability distribution of with the normalization condition , . Eq.(1) can be rewritten as
In the case of , Eq.(2) reduces to the locality assumption of one source and geometrically defines a correlation polytope, which contains all LHV distributions inside with the linear Bell inequalities as facets [1,18-20]. This fact is not true for . In particular, for the standard entanglement swapping , two hidden states enable a non-convex correlation polytope . For some special cases of , the correlation polytopes may be elucidated by exploring new Bell-type inequalities according to the acyclic graph approach [42,29], the linear programming technique [30,43,44] or the expansion method .
Explicit nonlinear Bell inequalities for general networks. Our method is based on geometric features of a general network. A network is called -independent if there are parties without sharing independent sources. Geometrically, no incoming edges of the independent parties share a vertex in Fig.1. The independence of a network is equivalent to the following -locality condition in terms of the GLCM: there are subsets of hidden states such that
Denote integer sets , and . Let be the measurement operator of party . With given measurements of all parties , we define the quantity of multipartite correlations for the network in Fig.1 as
where and are defined in Eq.(2). Similarly, with the other measurements of all parties with , we define the quantity of multipartite correlations as
One of the main results is that the following nonlinear inequality holds (Appendix A1):
when the network satisfies the -independent condition or the equivalent -locality.
For the quantum network of Fig.1, consider the quantum mechanical correlations between space-like separated observers. Assume that there are Hermitian observables , , , with , where two observables () with outcomes are defined for the observer . The second results is the following Cirel’son bound  (also written Tsirelson bound , Appendix A2)
when the quantum network has independent observers without sharing quantum resources, where the quantities and are distinguished from those derived from the generalized locally causal model.
For a given network, there may be different subsets of hidden states that satisfy the -locality condition in Eq.(3). Thus, various inequalities may be derived from different and . The inequality in Eq.(6) reduces to linear Bell inequality  when . Generally, is determined by the network topology. Intuitively, a larger implies a tighter polytope for a given network because more multipartite correlations are involved in quantities and in Eq.(6). Thus, it is reasonable to find the maximum and the corresponding independent parties. Unfortunately, the maximum is equivalent to the integer optimization problem, which is generally NP-hard (Appendix B1). In spite of that, an analytical method exists for some networks with simple features (see Fig.3). This suggests a great improvement to special networks [12,25-28,32]. For a general network, it is possible to find a suboptimal as an alternative. Notably, the problem can be reduced to the maximal matching of an equivalent unweighted bipartite graph (Appendix B2), for which Hopcroft and Karp provided a polynomial algorithm to find the maximal matching [47-49]. Each maximal matching may imply a suboptimal that always admits a useful inequality if .
The operator inequality in Eq.(7) provides a tight bound for the correlations of quantum networks. Although the upper bound is theoretically different from that in Eq.(6) for classical network in terms of the GLCM, it is difficult to verify for any quantum networks. Our following applications are to partially address this problem.
Generic non-multilocality of quantum networks with independent observers. The prediction of the quantum theory is incompatible with the local realism model . This feature is generic for any entangled state of two spin- particles [38,39]. A similar result holds for any multipartite entangled states  by using the CHSH inequality . A natural question is whether the inconsistence is typical for quantum networks. We aim to answer the question for the networks consisting of bipartite entangled pure states (including Einstein-Podolsky-Rosen (EPR) states ) and GHZ states  using the inequality in Eq.(6). Consider a quantum network of finite size, where V denotes all particles, and E denotes all edges (two particles are connected by one edge if they are entangled. An equivalent quantum network in Fig.2 exists when there are independent observers (without sharing quantum resources). denotes other observers in except for . For each equivalent network, we prove that the multipartite quantum correlations of violate the multilocality inequality in Eq.(6) for bipartite entangled pure states and GHZ states. It is formally stated as follows:
Theorem A.-For any quantum network consisting of bipartite entangled pure states and GHZ states, assume that there are independent observers. Then the following results hold:
A set of observables exists for all observers such that the multipartite quantum correlations are inconsistent with generalized local realism;
A set of observables exists for all observers such that the violation of the presented Bell inequality in Eq.(6) is maximal if and only if the maximally entangled EPR states and GHZ states are consisted of quantum resources.
Different from previous Bell inequalities for the star-shaped network consisting of EPR states [12,25,28,32], Theorem A shows that the presented inequalities in Eq.(6) are useful for almost all acyclic or cyclic networks consisting of EPR states and GHZ states. Furthermore, if the Werner states are used as quantum resources, we can evaluate the critical viabilities as follows:
Theorem B.-Assume that a quantum network with independent observers consists of Werner states: , where and denote the numbers of the respective EPR states and GHZ states. Then the product of critical viabilities is given by
for which the multipartite correlations violate the inequality in Eq.(6), where are EPR states, are GHZ states of particles, is the square identity matrix, , and .
Those results hold for any integer satisfying . Thus, various violations exist for the same quantum network. The relationships of different violations may provide insights of the supremacy derived from quantum networks [8,10] and are valuable for further explorations.
The main idea of the proofs is to construct proper observables for all observers [39,40]. For EPR states, these observables are dependent on specific parameters (Appendix C1). In addition, all observables of the network in Fig.2 should be equivalently defined for all observers of the original network . Thus, and derived from the network in Fig.2 are essentially linear combinations of multi-partite correlations generated by all observers of . A similar procedure holds for any bipartite entangled pure states (Appendix C2) or hybrid systems combined with GHZ states (Appendix C3). In particular, with these observables, the maximal violations with respect to Tsirelson’s bound in Eq.(7) exist for the maximally entangled EPR states and GHZ states (Appendix C4). For the unknown EPR states and GHZ states, our proof enables us to probabilistically verify the violations (Appendix C5). The product of visibilities of noisy EPR states and GHZ states is easily obtained (Appendix D).
Chain-shaped network.-The tripartite correlations of the standard quantum entanglement swapping violate the inequality in Eq.(6) with and [12,25,32]. The long-distance entanglement distributing generates a chain-shaped network in Fig.3(a). Theorem A shows that the multipartite quantum correlations violate the -locality inequality in Eq.(6) for any EPR states as quantum resources, where denotes the number of observers without sharing quantum resources, and denotes the smallest integer no less than . From Tsirelson’s bound in Eq.(7), the maximal violation exists for the maximally entangled EPR states. This answers a conjecture  and may beyond the violation . For Werner states, the product of the critical visibilities is no less than the product of the visibility of each EPR state (Appendix E).
Hybrid star-shaped network.-Unlike the star-shaped network [12,25,32], the new network in Fig.3(b) consists of EPR states and four-partite GHZ states. Theorem A implies that the multipartite quantum correlations violate the -locality inequality in Eq.(6) with for all EPR states and four-partite GHZ states, where denotes the maximal integer no more than . This violation is maximal with respect to Tsirelson’s bound in Eq.(7) for the maximal entanglements. A similar result holds for the noisy channels of Werner states from Theorem B. Notably, Scarani and Gisin  showed some partially entangled GHZ states do not violate some linear Bell inequalities [53-55]. Nevertheless, all GHZ states of even particles violate another Bell inequality . Our example and Theorem A go beyond these results in the case of the multilocality using the inequality in Eq.(6).
Cyclic network.-Consider a cyclic network in Fig.3(c) with EPR states as quantum resources. Theorem A shows that the multipartite quantum correlations violate the -locality inequality in Eq.(6) for any EPR states when . It is also maximal from Tsirelson’s bound in Eq.(7) for the maximally entangled EPR states. A similar result holds for Werner states from Theorem B. This is the first example of the nontrivial cyclic network discussed so far, for others see Appendix F.
For general noisy resources beyond Werner states, we provide one sufficient condition of the violation of the multilocality (Appendix G). Our condition only may be easily verified in applications. Further investigations are valuable for the non-multilocality and the entanglement witness . When multiple outputs or high-dimensional resources are considered, the linear method  may be inefficient to characterize all multipartite quantum correlations . The general representations of quantities are related to the famous conjecture of the Hadamard matrix . This raises three interesting problems: (1) how to characterize the common features of these networks; (2) how to explore new Bell inequalities for these networks; (3) how to characterize cyclic quantum network [25-32] (Appendix H).
In addition to interesting applications such as randomness amplification, interactive proofs and quantum games [6-10], quantum networks allow multipartite tasks. One notable problem is to address the supremacy of quantum networks in the case of multipartite interactive proofs or computational complexities. Its improvement may provide further relevance of quantum networks and classical problems. Moreover, the generic non-multilocality of special networks with hybrid entanglements would be fruitful for investigation because of the impossibility of classifying multipartite entanglements.
In conclusion, we presented explicit nonlinear Bell-inequalities for general networks with independent sources. These inequalities are computationally efficient and are used to prove the generic non-multilocality of quantum networks with independent observers. This result holds for any bipartite entangled pure states and GHZ states as quantum resources. The violations are maximal with respect to Tsirelson’s bound for the maximally entangled EPR states and GHZ states. Furthermore, the visibilities are presented for Werner states or general noisy states. These results may stimulate investigators to employ the non-multilocality for quantum information processing or quantum Internet.
We thank the helpful discussions of Luming Duan, Yaoyun Shi, M. Orgun, Huiming Li, Xiubo Chen, Yixian Yang, Xiaojun Wang, Yuan Su. This work was supported by the National Natural Science Foundation of China (No.61303039), Sichuan Youth Science and Technique Foundation (No.2017JQ0048), Fundamental Research Funds for the Central Universities (No.2682014CX095), Chuying Fellowship, CSC Scholarship, and EU ICT COST CryptoAction (No.IC1306).
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Appendix A1: Proof of the inequality in Eq.(6)
In this section, we prove the inequality in Eq.(6) for a general network in terms of the generalized locally causal model. From the definition of -locality in Eq.(3), has the decomposition in Eq.(2). Let . Define the expectation of the output as
Denote the integer sets , and . With the inequalities for , from Eqs.(4), (5) and (A1) it follows that
By setting , Eq.(A2) yields to
Similarly, we obtain that
From the Mahler inequality , Eqs.(A3) and (A4) follow that
where the inequality in Eq.(A5) is from the inequalities , for ; and Eq.(A6) is from the normalization condition of the probability distribution of hidden states.
Appendix A2: Proof of the inequality in Eq.(7)
In this subsection, we prove Tsirelson’s bound in Eq.(7). For the network in Fig.1, assume that there are Hermitian dichotomic operators with , where with outcomes are defined on the joint system of the quantum network. Note that these operators satisfy the commute condition for . Thus, there exist operators (up to proper unitary equivalence) satisfying , where denotes the direct summation of two operators performed on different Hilbert spaces, and denotes the identity operator on the system which does not belong to observer . All the observables have the eigenvalues and are performed on local systems.
We firstly prove the following lemma (which may be mathematically presented in some papers because of its simplicity)
Lemma 1. For any and integer , we obtain that the following inequality
where the equality holds if and only if .
Proof. The proof is completed by induction. For , the inequality (A7) is equivalent to
which implies that . This is satisfied for any .
Now, assume that for any , the inequality (A7) holds all . For even , from the assumption we obtain that
where the equality in Eq.(A9) holds if and only if and ; the equality in Eq.(A10) holds if and only if , and . Hence, the equality in Eq.(A7) holds if and only if .
For odd , by introducing an ancillary variable , from the assumption we obtain that
where the equality in Eq.(A11) holds if and only if and ; the equality in Eq.(A12) holds if and only if , and . Now, by setting , we get the equality . Thus, the inequality in Eq.(A12) yields that
which follows the inequality in Eq.(A7). The equality in Eq.(A7) holds if and only if .
Now, we continue to prove the inequality in Eq.(7). For the simplicity of the statement, let . Denote the norm of Hermitian operator on Hilbert space as , . From Eqs.(4) and (5), the inequalities , and the linearity of the expectation operation , we obtain that
where and are given by , , and represent different binary values, .
Moreover, using the commute conditions , Eq.(A13) yields to
from the equalities and the inequalities , where , , and denotes the identity operator.
In what follows, denote using the operators , which are performed on different subsystems. Note that because of . The inequality in Eq.(A14) is equivalent to
By setting with , we obtain that , . The inequality in Eq.(A15) follows that
where the inequality in Eq.(A16) is from the presented Lemma, and the equality in Eq.(A16) is from the equalities and . So, .
Appendix B: The number of independent parties in networks
Appendix B1: The maximum
In this subsection, we present the hardness of finding the maximum for a general network. Informally, we obtain that
Statement. The problem of finding the maximum of a general network is NP-Hard.
The following procedure starts from a new equivalent bipartite graph (in which the parties has not decomposed) of a given network in Fig.1. S denotes the set of independent sources . R denotes the set of parties . E denotes the set of all edges which schematically represent the relations of sources and parties. Denote as the number of sources which connect to party , . The maximum problem can be mathematically formulated as an integer program:
, , , ; .
Here, is the characteristic function of party , i.e., if party is included in the maximal set of independent parties for evaluating ; otherwise. The first condition is used to ensure each source distribute a hidden state to one party. The second condition is used to ensure that party is included in the set of independent parties, i.e., the number of nonzero should equal that of the edges connected to , . This problem is generally NP-hard  (there exist integer program problems which are NP-hard). Of course, there exists P-hard subsets of integer programs for special networks (see Fig.3).
Appendix B2: Efficient computation of
Despite the NP-hard problem for a general network, there exist computationally efficient algorithms to find suboptimal or possible values of . Let be the number of independent sources which distribute the hidden states to party . By schematically decomposing each party into different parties , we obtain an equivalent bipartite graph in Fig.S1, where , . All independent sources of the network in Fig.1 are regarded as the upper vertices in the set S while all decomposed parties are represented by the lower vertices in the set R with . Each edge represents the fact that a source distributes one hidden state to a decomposed party.
For bipartite graph , a matching is a subset of the edges satisfying that no two edges share a vertex . For any hidden states of the network in Fig.1 satisfying Eq.(3), it is easy to prove that all edges between the sources and corresponding parties are consisted of a matching set of the unweighted bipartite graph G in Fig.S1. This is from the independence assumption of the corresponding parties. Conversely, given a matching of the bipartite graph in Fig.S1, two steps are used to find the integer of the independent parties of the network in Fig.1 as follows:
Find all original parties who have at least one decomposed party in the vertex set of . Denote A as the desired set of all these parties;
Check the completeness of each party in A, where the completeness means all the decomposed parties of one original party in A are in the vertex set of .
Note that equals the number of original parties in Fig.1 satisfying the completeness. Hence, for each matching of the unweighted bipartite graph G, there may exist an integer and the corresponding independent parties in A; Otherwise, another matching should be used. In theory, one needs to find all matchings to obtain the largest for the tight form of Eq.(6).
Here, we present a simple example in Fig.S2. There are five parties , who receive the hidden states from 7 independent sources , , in Fig.S2(a). We firstly decompose into , , , , , , respectively. And then, we obtain an equivalent unweighted bipartite graph in Fig.S2(b). All the red edges are consisted of one of the maximal matchings of the graph. By checking the completeness of three parties , we obtain from this matching. Moreover, by checking all the maximal matchings, we obtain that the maximum equals 2.
Another accessible method is to get a small for a very large network as follows, where S is the set of vertexes (parties) and E is the set of edges, and each pair of vertexes (parties) is connected with an edge if they share an entanglement.
Randomly choose one vertex ;
Randomly choose one vertex , where is the reduced network by deleting all vertexes which have edges connected to ;
Continue the procedure until there is no remained vertexes.
Note that the deleting operation of one vertex is completed if there is one edge which has connected to the chosen vertex. So, this algorithm is efficient because we only need to deleting and random choosing operations which are polynomial in the total number of vertexes (not the total number of edges). In particular, one may choose one vertex with the minimal degree (the number of edges connected to it) in each step. Of course, this algorithm cannot ensure the maximal . Fortunately, our results holds for any . So, from the suboptimal or the minimum , we can obtain useful Bell inequalities.
Appendix C: Proof of Theorem A
In this section, we prove Theorem A for a general network in Fig.1 with arbitrary bipartite entangled pure states and GHZ states as quantum resources. In the following experiment of verifying the no-multilocality, after all parties except for (or the combined party ) perform some measurements depending on their input bits on their particles and obtain output bits , all parties performs some measurements depending on their input bits on their particles and obtain output bits , where .
The proof is completed by following the procedure from special quantum resources to general quantum resources. This is easy to follow the main idea.
Appendix C1: EPR states as quantum resources
In this subsection, we assume that the quantum resources consist of Einstein-Podolsky-Rosen (EPR) states :
where are EPR states with real coefficients satisfying the normalization condition