Abstract
The possibility of Bell inequality violations in quantum theory had a profound impact on our understanding of the correlations that can be shared by distant parties. Generalizing the concept of Bell nonlocality to networks leads to novel forms of correlations, the characterization of which is, however, challenging. Here, we investigate constraints on correlations in networks under the natural assumptions of nosignaling and independence of the sources. We consider the triangle network with binary outputs, and derive strong constraints on correlations even though the parties receive no input, i.e., each party performs a fixed measurement. We show that some of these constraints are tight, by constructing explicit local models (i.e. where sources distribute classical variables) that can saturate them. However, we also observe that other constraints can apparently not be saturated by local models, which opens the possibility of having nonlocal (but nonsignaling) correlations in the triangle network with binary outputs.
Introduction
The nosignaling principle states that instantaneous communication at a distance is impossible. This imposes constraints on the possible correlations between distant observers. Consider the socalled Bell scenario^{1}, where each party performs different local measurements on a shared physical resource distributed by a single common source. In this case, the nosignaling principle implies that the choice of measurement (the input) of one party cannot influence the measurement statistics observed by the other parties (their outputs). In other words, the marginal probability distribution of each party (or subset of parties) must be independent of the input of any other party. These are the wellknown nosignaling conditions, which represent the weakest conditions that correlations must satisfy in any reasonable physical theory^{2}, in the sense of being compatible with relativity. More generally, the nosignaling principle ensures that the information cannot be transmitted without any physical carrier. This provides a useful framework to investigate quantum correlations (which obviously satisfy the nosignaling conditions, but do not saturate them in general^{2}) within a larger set of physical theories satisfying nosignaling; see e.g., refs. ^{2,3,4,5,6,7,8,9}.
Recently, the concept of Bell nonlocality has been generalized to networks, where separated sources distribute physical resources to subsets of distant parties (Fig. 1). Assuming the sources to be independent from each other^{10,11}, arguably a natural assumption in this context, leads to many novel effects. Notably, it becomes now possible to demonstrate quantum nonlocality without the use of measurement inputs^{11,12,13,14,15}, but only by considering the output statistics of fixed measurements. Just recently, a first example of such nonlocality genuine to networks was proposed^{15,16}. This radically departs from the standard setting of Bell nonlocality, and opens many novel questions. Characterizing correlations in networks (local or quantum) is however still very challenging at the moment, despite recent progress^{17,18,19,20,21,22,23,24,25,26,27,28}.
Moving beyond quantum correlations, this naturally raises the question of finding the limits of possible correlations in networks, assuming only nosignaling and independence (NSI) of the sources^{22,29,30,31,32,33}. Here, we investigate this question and derive limits on correlations, which we refer to as NSI constraints. While our approach can in principle be applied to any network, we focus here on the wellknown triangle network with binary outputs and no inputs, for which we obtain strong, and even tight NSI constraints. Specifically, we show that, despite the absence of an input, some statistics imply the possibility for one party to signal to others by locally changing (or not changing) the structure of the network. Formally, this amounts to considering a specific class of socalled network inflations, as introduced in ref. ^{22}, which we show can lead to general and strong NSI constraints. Moreover, we prove that some of our NSI constraints are in fact tight, by showing that they can be saturated by correlations from explicit trilocal models, in which the sources distribute classical variables. Interestingly, however, it appears that not all of our NSI constraints can be saturated by trilocal models, which opens the possibility of having nonlocal (but nevertheless nonsignaling) correlations in the triangle network with binary outputs. Finally, we conclude with a list of open questions.
Results
NSI constraints
The triangle network (sketched in Fig. 1a) features three observers: Alice, Bob, and Charlie. Every pair of observers is connected by a (bipartite) source, providing a shared physical system. Importantly, the three sources are assumed to be independent from each other. Hence, the three observers share no common (i.e., tripartite) piece of information. Based on the received physical resources, each observer provides an output (a, b, and c, respectively). Note that the observers receive no input in this setting, contrary to standard Bell nonlocality tests. The statistics of the experiment are thus given by the joint probability distribution p(a, b, c). We focus on the case of binary outputs: a, b, c ∈ {+1, −1}. It is then convenient to express the joint distribution as follows:
where E_{A}, E_{B}, and E_{C} are the singleparty marginals, E_{AB}, E_{BC}, and E_{AC} are the twoparty marginals, and E_{ABC} is the threebody correlator. Note that the positivity of p(a, b, c) implies constraints on marginals, in particular p(+ + +) + p(− − −) ≥ 0 implies
In the following, we will derive nontrivial constraints bounding and relating the singleparty and twoparty marginals of p(a, b, c) under the assumption of NSI. While it seems a priori astonishing that the nosignaling principle can impose constraints in a Bell scenario, featuring no inputs for the parties, we will see that this is nevertheless the case in the triangle network.
The main idea is the following. Although one party (say Alice) receives no input, she could still potentially signal to Bob and Charlie by locally modifying the structure of the network. To see this, consider the hexagon network depicted in Fig. 1b, and focus on parties Bob and Charlie. From their point of view, the two networks (triangle and hexagon) should be indistinguishable. This is because all the modification required to bring the triangle network to the hexagon (e.g., by having Alice adding extra parties and sources) occurs on Alice’s side, and can therefore be spacelike separated from Bob and Charlie. If Alice, by deciding which network to use, could remotely influence the statistics of Bob and Charlie, this would clearly lead to signaling. Hence, we conclude that the local statistics of Bob and Charlie (i.e., the singleparty marginals E_{B} and E_{C}, as well as the twoparty marginals E_{BC}) must be the same in the triangle and in the hexagon. To see that this condition really captures the possibility to signal, we could imagine a thought experiment in which we would give an input to Alice, which determines whether she modifies her network structure or not. If she does so and this has an incidence on the E_{BC} marginal, then Bob and Charlie can learn about Alice’s input, hence breaking the usual notion nosignaling condition. Note that the input considered here is however purely fictional, Alice’s input is not present in the actual experiment.
From the above reasoning, we conclude that the joint output probability distribution for the hexagon, i.e., \(p(a,b,c,a^{\prime} ,b^{\prime} ,c^{\prime} )\), must satisfy several constraints. In particular, one should have that
where all sums go over all outputs \(a,b,c,a^{\prime} ,b^{\prime} ,c^{\prime}\). From the independence of the sources, we obtain additional constraints, namely
Clearly, we also get similar constraints when considering signaling between any other party (Bob or Charlie) to the remaining two.
Altogether, we see that NSI imposes many constraints on \(p(a,b,c,a^{\prime} ,b^{\prime} ,c^{\prime} )\). Obviously, we also require that
Now reversing the argument, we see that the nonnegativity of \(p(a,b,c,a^{\prime} ,b^{\prime} ,c^{\prime} )\) imposes nontrivial constraints relating the single and twoparty marginals of the triangle distribution p(a, b, c). To illustrate this, let us proceed with an example in a slightly simplified scenario, assuming all singleparty marginals to be uniformly random, i.e., E_{A} = E_{B} = E_{C} = 0. In this case, we obtain
Importantly, notice that the above expression contains a number of variables (of the form F_{X}) that are uncharacterized; these represent Xparty correlators in the hexagon network, see Supplementary Note 1 for more details. Hence, we obtain a set of inequalities imposing constraints on our variables of interest (i.e., E_{AB}, E_{BC}, and E_{AC}), but containing also additional variables that we would like to discard. This can be done systematically via the algorithm of Fourier–Motzkin elimination^{34}. Note that here we need to treat the squared terms, such as \({E}_{{\rm{AB}}}^{2}\), as new variables, independent from E_{AB}, so that we get a system of linear inequalities. Solving the latter, and taking into account positivity constraints as in Eq. (2), we obtain a complete characterization of the set of twobody marginals (i.e., E_{AB}, E_{BC}, and E_{AC}) that are compatible with NSI in the triangle network (for a hexagon inflation and uniform singleparty marginals), in terms of a single inequality
and its symmetries (under relabeling of the parties and of the outputs). This implies a more symmetric, but slightly weaker inequality:
Note that when E_{AB} = E_{BC} = E_{AC} ≡ E_{2}, we get simply \({E}_{2}\le \sqrt{2}1\approx 0.41\).
Next, we consider the symmetric case (i.e., E_{A} = E_{B} = E_{C} ≡ E_{1} and E_{AB} = E_{BC} = E_{AC} ≡ E_{2}) and obtain nontrivial NSI constraints on the possible values of E_{1} and E_{2} (Fig. 2). In particular, correlations compatible with NSI must satisfy the following inequality
Let us move now to the most general case, with arbitrary values for single and twoparty marginals. For a given set of values E_{A}, E_{B}, E_{C}, E_{AB}, E_{BC}, and E_{AC}, it is possible here to determine via a linear program whether this set is compatible with NSI or not (Supplementary Note 1). More generally, obtaining a characterization of the NSI constraints in terms of explicit inequalities (as above) is challenging, due mainly to the number of parameters and nonlinear constraints. We nevertheless obtain that the following inequality represents an NSI constraint
A proof of this general inequality is given in Supplementary Note 1. Note that this inequality reduces to Eq. (14) when E_{A} = E_{B} = E_{C} = 0, as well as to Eq. (15) for the symmetric case.
It is worthwhile discussing the connection between our approach and the inflation technique presented in refs. ^{22,25}. There, the main focus is on using inflated networks for deriving constraints on correlations achievable, with classical resources. In that case, information can be readily copied, so that sources can send the same information to several parties. Ultimately, this allows for a full characterization of correlations achievable with classical resources^{22}. Copying information is however not possible in our case, as nosignaling resources cannot be perfectly cloned in general^{6}. Hence only inflated networks with bipartite sources can be considered in our case, such as the hexagon. A discussion of these ideas can be found in Section V.D of ref. ^{22}, where the idea of using inflation to limit nosignaling correlations in networks is mentioned. Here, we derive explicitly bounds that all correlations satisfying the NSI constraints, whether quantum of postquantum, have to satisfy, and identify the physical principle behind them.
Finally, the choice of the hexagon inflation deserves a few words. As seen from Fig. 1b, it is judicious to consider inflated networks forming a ring, with a number of parties that is a multiple of three. Intuitively, this should enforce the strongest constraints on the correlations of the inflated network; in particular, all single and twobody marginals are fixed by the correlations of the triangle. This would not be the case when considering inflations to ring networks, with a number of parties that is not divisible by three.
Tightness
A natural question is whether the constraints we derived above, that are necessary to satisfy NSI, are also sufficient. There is a priori no reason why this should be the case. Of course, starting from the triangle network, there are many (in fact infinitely many) possible extended networks that can be considered, and nosignaling must be enforced in all cases. For instance, instead of extending the network to a hexagon (as in Fig. 1), Alice could consider an extension to a ring network featuring 9, 12, or more parties. Clearly, such extensions could lead to stronger constraints than those derived here for the hexagon network.
Nevertheless, we show that some of the constraints we obtain above are in fact tight, i.e., necessary and sufficient for NSI. We prove this by presenting explicit correlations (constructed within a generalized probabilitic theory satisfying NSI) that saturate these constraints. In fact, we consider simply the case where all sources distribute classical variables to each party, which we refer to as trilocal models. The latter give rise to correlations of the form
where α, β, and γ represent the three local variables distributed by each source, with arbitrary probability densities μ(α), ν(β), and ω(γ). Also, p_{A}(a∣β, γ) represents an arbitrary response function for Alice, and similarly for p_{B}(b∣α, γ) and p_{C}(c∣α, β). Note that such trilocal models represents a natural extension of the concept of Bell locality to networks (see e.g., refs. ^{10,19}).
We first consider the case of symmetric distributions, i.e., characterized by the two parameters E_{1} and E_{2}, and seek to determine the set of correlations that can be achieved with trilocal models. As shown in Fig. 2, it turns out that almost all NSI constraints can be saturated in this case, in particular the inequality (15). After performing a numerical search, we could construct explicitly some of these trilocal models, which involve up to ternary local variables (see Supplementary Note 2 for details). Moreover, we compare our NSI constraint (15) to the one derived in ref. ^{22} (see Eq. (34)), and find that the present one is stronger, and in fact tight (Fig. 2). Note also that a previous work derived an NSI constraint based on entropic quantities^{29}; such constraints are however known to be generally weak, as entropies are a coarsegraining of the statistics, which no longer distinguishes between correlations and anticorrelations.
As seen from Fig. 2, there is however a small region (in yellow) that is compatible with NSI (considering the hexagon inflation), but for which we could not construct a trilocal model. Whether this gap can be closed by considering more sophisticated local models (using variables of larger alphabet) or whether stronger nosignaling bounds can be obtained is an interesting open question. For the triangle network with binary outcomes, any trilocal distribution can be obtained by considering shared variables of dimension (at most) six, and deterministic response functions^{24}.
In fact, another (and arguably much more interesting) possibility would be that this gap cannot be closed, as it would feature correlations with binary outcomes satisfying NSI, but that are nevertheless nontrilocal. To further explore this question, let us now focus on the case where singleparty marginals vanish, i.e., E_{1} = 0. We investigate the relation between twoparty marginals E_{2} and the threeparty correlator E_{3} = E_{ABC}, comparing NSI constraints and trilocal models. Notice that the NSI constraints we obtain here do not involve E_{3} (as the latter cannot be recovered within the analysis of the hexagon). Hence NSI imposes only \({E}_{2}\le \sqrt{2}1\), while positivity of p(a, b, c) imposes other constraints. This is shown in Fig. 3, where we also seek to characterize the set of correlations achievable via trilocal models (proceeding as above). Interestingly, we find again a potential gap between trilocal correlations and NSI constraints. This should however be considered with care. First, the NSI constraints obtained from the hexagon may not be optimal (see Discussion section). Second, there could exist more sophisticated trilocal models (e.g., involving higherdimensional variables) that could lead to a stronger correlations (i.e., cover a larger region in Fig. 3). Note also that we investigated whether quantum distributions satisfying the independence assumption exist outside of the trilocal region, but we could not find any example (we performed a numerical search, considering entangled states of dimension up to 4 × 4).
Finally, note that we also performed a similar analysis for the case where singleparty marginals vanish, but twobody marginals are not assumed to be identical to each other. Here, we find that inequality (13) can be saturated in a few specific cases. However, there also exist correlations satisfying the NSI bounds that do not seem to admit a trilocal model; details in Supplementary Note 1.
Discussion
We discussed the constraints arising on correlations in networks, under the assumption of NSI of the sources. We focused our attention on the triangle network with binary outputs for which we derived strong constraints, including tight ones. Our work raises a number of open questions that we now discuss further.
A first question is whether the constraints we derive (necessary under NSI), could also be sufficient. We believe this not to be the case, as stronger NSI constraints could arise from inflations of the triangle to more complex networks (e.g., loop networks with an arbitrary number of parties). Note that there could also exist different forms of nosignaling constraints, that cannot be enforced via inflation. In this respect, we compare in Supplementary Note 1 our NSI constraints with the recent work of ref. ^{32} proposing a very different approach to this problem, using the Finner inequality. A notable difference is that the latter imposes constraints on tripartite correlations, which is not the case here.
Another important question is whether there could exist nonlocality in the simplest triangle network with binary outcomes. That is, can we find a p(a, b, c) that satisfies NSI, but that is nevertheless nontrilocal? While we identified certain potential candidate distributions for this, we could not prove any conclusive result at this point. We cannot exclude the possibilities that (i) these correlations are in fact not compatible with NSI (as there exist stronger NSI constraints) or (ii) these correlations can in fact be reproduced by a trilocal model. In order to address point (i), one could try to reproduce these correlations via an explicit NSI model, for instance considering that all sources emit nosignaling resources (such as nonlocal boxes^{2}) which could then be wired together by the parties. To address point (ii), one could show that these correlations violate a multilocality inequality for the triangle network. Of course finding such inequalities is notably challenging, see e.g., ref. ^{13}.
Furthermore, it would be interesting to derive NSI constraints for other types of networks. Indeed, the approach developed here can be straightforwardly used. Cases of high interest are general loop networks, as well as the triangle network with larger output alphabet (where examples of quantum nonlocality are proven to exist^{11,15}).
Finally, a more fundamental question is whether any correlation satisfying the complete NSI constraints can be realized within an explicit physical theory satisfying nosignaling (the latter are usually referred to as generalized probabilistic theories^{6}). While this is the case in the standard Bell scenario (where all parties share a common resource), it is not clear if that would also be the case in the network scenario.
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Acknowledgements
We thank Stefano Pironio, MarcOlivier Renou, Denis Rosset, and Elie Wolfe for discussions. We acknowledge financial support from the Swiss national science foundation (Starting grant DIAQ, NCCRQSIT, and NCCRSwissmap). E.Z.C. acknowledges support by the Swiss National Science Foundation via the Mobility Fellowship P2GEP2_188276.
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N.G., S.P., and N.B. came up with the idea of the method. N.G., J.D.B., Y.C., P.R., A.T., E.Z.C., S.P., and N.B. participated in deriving the results, and writing and editing the manuscript.
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Gisin, N., Bancal, JD., Cai, Y. et al. Constraints on nonlocality in networks from nosignaling and independence. Nat Commun 11, 2378 (2020). https://doi.org/10.1038/s41467020161374
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DOI: https://doi.org/10.1038/s41467020161374
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