Entropic nonsignaling correlations
We introduce the concept of entropic nonsignaling correlations, i.e., entropies arising from probabilistic theories that are compatible with the fact that we cannot transmit information instantaneously. We characterize and show the relevance of these entropic correlations in a variety of different scenarios, ranging from typical Bell experiments to more refined descriptions such as bilocality and information causality. In particular, we apply the framework to derive the first entropic inequality testing genuine tripartite nonlocality in quantum systems of arbitrary dimension and also prove the first known monogamy relation for entropic Bell inequalities. Further, within the context of complex Bell networks, we show that entropic nonlocal correlations can be activated.
Quantum nonlocality—the fact that correlations obtained in quantum experiments performed by distant parties are incompatible with local hidden variable (LHV) models Bell (1964)—brings to light an intriguing aspect of quantum mechanics (QM) and relativistic causality Popescu (2014). QM is in accordance with the nonsignaling (NS) principle, that is, local manipulations by an experimenter cannot influence the measurement outcomes of other distant experimenters. However, as demonstrated by Popescu and Rohrlich Popescu and Rohrlich (1994), special relativity alone cannot single out quantum mechanical correlations as there are theories, beyond QM, also in agreement with NS. This result not only has triggered the search for physically well-motivated principles for quantum mechanics Navascués and Wunderlich (2010); Navascués et al. (2015); Pawłowski et al. (2009); Fritz et al. (2013); Cabello (2013); Sainz et al. (2014) but also has led to new insights about its limitations for information processing van Dam (1999); Brassard et al. (2006); Chiribella et al. (2010); Almeida et al. (2010); Chaves et al. (2015a).
Given the intrinsic statistical nature of QM, probabilities give a natural framework for nonlocality. Indeed, Bell inequalities and NS relations are nothing other than constraints on probabilities arising in a given theory, local and NS, respectively Brunner et al. (2014). Nevertheless, different approaches are possible Abramsky and Brandenburger (2011); Braunstein and Caves (1988); Chaves and Fritz (2012). In particular, in the information-theoretic approach to nonlocality Braunstein and Caves (1988); Cerf and Adami (1997); Chaves and Fritz (2012); Fritz and Chaves (2013); Chaves et al. (2014a); Kurzyński and Kaszlikowski (2014); Raeisi et al. (2015) the basic objects are the Shannon entropies Yeung (2008) of the observed data.
The information-theoretic approach provides a novel and useful alternative for both conceptual and technical reasons. First, entropy is a key concept in both classical and quantum information theory, thus developing a framework that focuses on entropies rather than probabilities leads to new insights and applications Barnum et al. (2010); Dahlsten et al. (2012); Janzing et al. (2013); Chaves et al. (2014b); Henson et al. (2014); Poh et al. (2015); Chaves et al. (2015b); Janzing et al. (2015). For instance, the celebrated principle of information causality Pawłowski et al. (2009) is nothing other than an entropic inequality bounding the correlations that can be achieved by imposing a certain causal structure to quantum mechanics Chaves et al. (2015a). Second, entropies allow for a much simpler and compact characterization of classical and quantum correlations in a variety of scenarios Fritz (2012); Chaves et al. (2014a); Henson et al. (2014); Chaves et al. (2015a). In spite of that, as opposed to the usual probabilistic description, little is known about entropic Bell inequalities beyond very simple cases and remarkably nothing is known about the structure imposed by the nonsignaling principle on the entropies of measurement outcomes.
In this letter, we aim to further develop the information-theoretic approach to nonlocality and, in particular, to define the concept of entropic nonsignaling correlations, i.e., the entropies compatible with the nonsignaling principle. We characterize NS entropic correlations in a variety scenarios: from usual bipartite and tripartite, to genuine multipartite nonlocality Svetlichny (1987); Gallego et al. (2012); Bancal et al. (2013, 2013), bilocality Branciard et al. (2010), and information causality Pawłowski et al. (2009). Our framework can also be employed to derive monogamy relations Masanes et al. (2006); Pawlowski and Brukner (2009) between entropic Bell inequalities. Furthermore, our methods highlight the use of entropic NS correlations as a novel tool to derive Bell inequalities in scenarios otherwise intractable.
Marginal scenarios, local and NS correlations.— In a quantum experiment, only some of the relevant observables are jointly measurable; hence, we face fundamental restrictions on the empirically accessible joint probability distributions. This fact is encoded in the notion of a marginal scenario. Given random variables , a marginal scenario is defined as , , such that for each a joint probability distribution is accessible Chaves and Fritz (2012); Fritz and Chaves (2013). Clearly, it is sufficient to consider maximal subsets.
A typical example is a Bell experiment: two separated parties, Alice and Bob, at each run of the experiment can perform one of different measurements, labeled as and , respectively, on their shares of a joint system. Their marginal scenario is, then, , corresponding to the probability distributions fno (a)— where labels the outcome when measurement has been performed (similarly for )—estimated from the statistical data. As shown by Fine Fine (1982), a LHV model for the data can be equivalently defined as a joint probability distribution . Hence, a set of marginals is called local if it is consistent with a single joint probability distribution for all measurements. This, in turn, implies the existence of a joint entropy of all possible measurements and all its marginals Braunstein and Caves (1988), where stands for the Shannon entropy. They can be represented as a -dimensional vector . Notice that is defined to be , but it is convenient to include it to have a more compact representation of the constraints satisfied by the entropy vector (cf. Appendix I.1).
The difference between the probabilistic and entropic description solely relies on how we quantify correlations. A marginal probability distribution is local if we can construct a well-defined joint probability distribution . Similarly, marginal entropies are local if a joint entropy and all its marginals , arising from a (nonunique) joint probability distribution, can be defined. The existence of a well-defined joint description imposes strict constraints—the famous Bell inequalities—on the empirically observable marginal correlations Fine (1982); Pitowsky (1989, 1991). Similarly, marginal entropic correlations admitting an extension to also obey strict constraints. The closure set of well-defined entropy vectors defines a convex cone , that is, if and are in so are , with , and , with fno (b). An explicit characterization of is yet to be found, however, an outer approximation, characterized by finitely many linear inequalities Yeung (2008) or, equivalently, in terms of finitely many extremal rays (vectors defined up to a positive factor Aliprantis and Tourky (2007)), is known: the Shannon cone . Such inequalities basically amount to the positivity of the conditional entropy, i.e. , and the positivity of the conditional mutual information, i.e., , for disjoint subsets of variables (see Appendix I.1 for further details). Thus, in full analogy with the probabilistic case Budroni and Cabello (2012), entropic Bell inequalities can be understood as the constraints arising from the projection of onto observable coordinates, that is, the projection of into defining the Bell entropic cone .
On the other hand, NS probabilities are defined as those where the outcomes of a part do not depend on the measurements performed by another distant part, i.e., such that (similarly for and for any number of parties). NS correlations are then defined by the above linear constraints (NS conditions) together with the nonnegativity condition ; i.e., they are classical probability distributions whenever restricted to , with consistent marginals. Geometrically, they can be seen as the intersection of the simplex polytopes Boyd and Vandenberghe (2009) defining each of the probabilities and thus overlapping over the marginals and . We can then naturally define NS entropic cone, for a marginal scenario , as the intersection , where is the entropy cone associated with (see Fig. 1). For instance, in the bipartite scenario, the NS cone is given by the intersection of cones corresponding to the subsets of variables appearing in the marginal scenario and respecting the basic constraints given by , and . This intersection can be understood as follows: since each contains a restricted set of variables, we embed each in a bigger space where the variables not in are unconstrained.
In the following, we apply the above framework to analyze from an entropic perspective a broad range of scenarios. Notice that for variables, the entropy cone corresponds to the Shannon cone, i.e., Yeung (2008); hence, all results for the bipartite and tripartite cases lead to the exact description of the NS cones fno (c). Further discussions and technical details can be found in SM (), including the derivation of all Bell inequalities that, nicely, can be proven by simple sums of Shannon inequalities.
Bipartite and tripartite scenarios.— We start with the simplest Bell scenario as above, for . In contrast to the probabilistic case, entropic correlations are concisely defined for an arbitrary number of measurement outcomes, highlighting another advantage of the entropic approach. For dichotomic observables (), the only nontrivial probabilistic Bell inequality is the Clauser-Horne-Shimony-Holt (CHSH) inequality Clauser et al. (1969)
is valid for any number of outcomes, where represents the mutual information.
Both inequalities are maximally violated by an extremal point or ray characterizing the NS correlations. Eq. (1) is maximally violated by the Popescu-Rohrlich (PR)-box . However, is entropically equivalent to the classical correlation and thus cannot violate (2). On the other hand, Eq. (2) is maximally violated by , with marginals and the number of outcomes. Thus, this correlation can be interpreted as the entropic counterpart of a PR-box. For , these entropies are obtained as an equal mixture of and . The mixing with is exactly the method proposed in Chaves (2013) to turn entropic inequalities into necessary and sufficient conditions for nonlocality detection. It is thus appealing that the NS entropic cone naturally retrieves this sort of correlations.
Another important result of our approach is the derivation of the first entropic monogamy relation for Bell inequalities. The monogamy of Bell inequalities violations is a general feature of NS theories Masanes et al. (2006), and it can be understood by the following example. For a tripartite distribution with binary inputs and outputs, whenever the marginal distribution violates the CHSH inequality necessarily must be local. Similarly, from the definition of NS entropic cone, we can prove that
meaning that both entropic Bell inequalities, between Alice-Bob and Alice-Charlie, cannot be violated at the same time, with the notable difference that this monogamy inequality is valid for any number of outcomes.
The similarities between the probabilistic and entropic approaches, which may suggest a deeper geometric connection Chaves et al. (2014a), already disappear in the tripartite scenario. For the case of three parties and two settings, the probabilistic NS correlations for dichotomic measurements consist of different classes of extremal points, with of them nonlocal Pironio et al. (2011). In turn, the entropic NS cone is characterized by different classes of extremal rays, of which correspond to nonlocal correlations fno (d). As it turns out, already at the tripartite case we obtain a much more complex structure than the one we could naively presume from the probabilistic description.
Genuine tripartite entropic nonlocality.— In analogy to entanglement Horodecki et al. (2009), when moving beyond the bipartite case, different classes of nonlocality arise. With three parties, one can introduce the notion of genuine tripartite nonlocality, that is, a stronger form of nonlocality that cannot be reproduced even if any two of the parties are allowed to share some nonlocal resources Svetlichny (1987); Gallego et al. (2012); Bancal et al. (2013). We focus our attention to nonsignaling resources (e.g., a PR-box) and two possible measurements per party, extensions to more measurements and parties are straightforward. For a given bipartition, say , a hybrid local-nonsignaling (LNS) model is equivalent to the existence of probability distributions , with consistent marginal , i.e., Alice has local correlations and Bob and Charlie share nonsignaling correlations. Genuine tripartite nonlocal (GTNL) correlations correspond to marginals that cannot be explained as a convex combination of models of the type , , and .
Analogously to the NS case, an entropic model corresponds to the joint entropies , , and all its marginals, and similarly for , and . We can then define the LNS entropic correlations via the cone , constructed as the convex hull (i.e., set of convex combinations) of the entropic cones for each of the models , , and . In turn, GTNL entropic correlations are those lying outside .
From the different classes of extremal nonlocal rays defining the tripartite scenario, correspond to GTNL correlations. One of these rays correspond to the distribution , that is, the mixing of Barrett et al. (2005a) with classical correlations . The GTNL character of this correlation can be witnessed by the violation of the following entropic inequality valid for any LNS correlation with arbitrary number of outcomes:
Furthermore, inequality (Entropic nonsignaling correlations) can also be used to witness the GTNL in quantum states, for instance using -dimensional Greenberger-Horne-Zeilinger (GHZ) states and projective measurements Collins et al. (2002). Results are plotted in Fig. 2 up to .
Activating entropic nonlocality in networks.— The tripartite scenario permits also another possibility: that the correlations between the parties are mediated by independent sources. The paradigmatic example is the entanglement swapping experiment Zukowski et al. (1993). Two independent pairs of entangled particles are distributed among three spatially separated parties: Bob receives one particle of each pair, and Alice and Charlie the remaining two. By jointly measuring his particles, Bob can generate (upon conditioning on outcomes) entanglement and nonlocal correlations between the two remaining particles, even though the latter have never interacted. A probabilistic and local realistic description of this experiment involves two independent hidden variables, the so called bilocality assumption Branciard et al. (2010); Tavakoli et al. (2014); Chaves et al. (2015c); Chaves (2016); Rosset et al. (2016), implying the independence relation , i.e., no correlations between Alice and Charlie. The local and NS correlations in the bilocality scenario are defined by infinitely many extremal points and is extremely challenging to characterize Chaves (2016).
The advantages of the entropic description are here apparent: independence constraints are encoded in simple linear relations, e.g., . Geometrically, a set of extra linear constraints , as the one above, corresponds to the intersection of the (polyhedral) convex cone (e.g., or ) with a linear subspace, which is still a (polyhedral) convex cone fno (e).
For the case of two settings per party, we have fully characterized the set of NS bilocal correlations: we found different classes of extremal rays, of which are nonlocal. Out of these, are genuinely nonbilocal, i.e., the correlations admit a LHV model but not a bilocal LHV model. A particularly interesting extremal correlation is the following: and . It can be understood as the case where Bob and Charlie always measure the same observable (no measurement choice) while Alice still can perform two different measurements. Clearly, since only one of the parties has measurement choices, all correlations arising in this scenario are compatible with a LHV model. However, this correlation is not bilocal, as it can be witnessed by the violation of the entropic inequality valid for any bilocal decomposition:
The entropic correlation above arises from a probability distribution , obtained when Alice and Bob share a PR-box while Bob and Charlie share a classical correlated distribution . To that aim, Bob assigns to the output of his share of the PR-box that takes as input the bit that is classically correlated with the output of Charlie. This result illustrates two novel aspects of the bilocality scenario. First, we see that the nonlocality of the PR-box, which in CHSH scenario is entropically equivalent to a classical correlation, can be activated by employing it in a network. Even more remarkable is the fact that the emergence of nonlocal correlations only requires one out of the three parties to have access to measurement choices. This is similar to what has been observed in Branciard et al. (2012); Fritz (2012), where it has been argued that since the role of Charlie can be interpreted as defining measurement choices for Bob, this scenario can be mapped to the CHSH one. In our case, however, we do not need to hinge on this mapping, since we violate a new sort of entropic Bell inequality. Thus, as opposed to Refs. Branciard et al. (2012); Fritz (2012), our result does not rely on Bell’s theorem.
Another genuinely nonbilocal extremal entropic ray is associated with the probability . Its nonbilocality can be witnessed via the violation of
specifically, with value . As opposed to other known inequalities Branciard et al. (2010, 2012); Chaves (2016); Rosset et al. (2016), Eq. (Entropic nonsignaling correlations) includes marginal terms, and it is valid for an arbitrary number of outcomes.
Information causality.— Information causality (IC) Pawłowski et al. (2009) is a principle introduced to explain the limitation of quantum correlations, i.e., Tsirelson bound Cirel’son (1980). It can be understood as a game: Alice receives two independent random bits and and the task of Bob is to guess, at each run of the experiment, the value of one of them, having as resources some preshared correlations with Alice and some classical communication ( bits) sent by her. For shared quantum correlations, the following inequality holds Pawłowski et al. (2009):
where denotes Bob’s guess of .
To characterize the set of NS entropic correlations associated to IC scenario [i.e, including post-quantum correlations violating (7)], first notice that the mutual information between Alice’s inputs and Bob’s guesses should be limited, according to the assumed causal structure, by the amount of communication, that is, , otherwise they could also communicate superluminally Popescu (2014). Here, similarly to what has been done in Pawłowski et al. (2009), we consider the marginals . The NS cone is thus given by the intersection of the Shannon cone defined by with the constraints () arising from the causal structure of the game Chaves et al. (2015a). We found to be characterized by extremal rays, of which respect Eq. (7). The extremal ray violating Eq. (7) is given by . It is achieved when the parties share a PR-box and apply the protocol used in Pawłowski et al. (2009). It is once more appealing that the NS cone approach naturally retrieves an entropic correlation of special importance.
Discussion.— Nonlocality stands nowadays as one of the cornerstones in our understanding of quantum theory. In turn, entropy is a key concept in the foundations and applications of quantum information science. It is thus surprising that still so little is known about their relations and in particular what nonsignaling—another guiding principle permeating all physics—has to say about the entropies that can be generated by the outcomes of physical measurements. Here, we introduced the notion of entropic nonsignaling correlations characterizing the entropies compatible with the fact that we cannot transmit information instantaneously. To illustrate its relevance and novelty, we have applied it to understand a broad range of different phenomena from an entropic perspective: from monogamy relations and nonlocality activation in networks, to genuine multipartite nonlocality.
Nonsignaling also lies at the heart of the device-independent approach to quantum information, which has lately attracted growing attention Ekert (1991); Barrett et al. (2005b); Colbeck (2009); Pironio et al. (2010); Colbeck and Renner (2012); Gallego et al. (2010); Chaves et al. (2015b), and we believe our results provide a new tool also for practical applications. In addition, the entropic approach provides the natural ground to treat generalized Bell scenarios Tavakoli et al. (2014); Chaves et al. (2015c); Chaves (2016); Rosset et al. (2016) and understand novel forms of nonlocal correlations emerging from it. Future lines of research also include monogamy relations Pawlowski and Brukner (2009), the role of non-Shannon type inequalities Yeung (2008) in multipartite scenarios and possible applications in nonlocal games Brukner et al. (2004).
Finally, as demonstrated by information causality Pawłowski et al. (2009); Chaves et al. (2015a), many of our current guiding principles are stated in terms of entropy. Our current framework can help to devise new entropic principles, in particular for the multipartite case Gallego et al. (2011).
Acknowledgements.The authors thank Nikolai Miklin for discussions. R.C. acknowledges financial support from the Excellence Initiative of the German Federal and State Governments (Grants ZUK 43 & 81), the FQXi Fund, the US Army Research Office under Contracts W911NF-14-1-0098 and W911NF-14-1-0133 (Quantum Characterization, Verification, and Validation), the DFG (GRO 4334 & SPP 1798). C.B. acknowledges financial support from the EU (Marie Curie CIG 293993/ENFOQI), the FQXi Fund (Silicon Valley Community Foundation), and the DFG.
i.1 The Shannon and Bell entropic cones
Given a collection of discrete random variables , we denote by the set of indices and its power set. For every , let be the vector and the associated Shannon entropy, given by . We can define the vector . Not every vector will correspond to an entropy vector, as, e.g., entropies are nonnegative. The entropy cone is defined as the closure of the region
is known to be a convex cone but a tight and explicit description is still to be found Yeung (2008). However, an outer approximation to the entropic cone is known, the so-called Shannon cone.
The Shannon cone is a polyhedral closed convex cone, i.e., a subset of defined by a finite set of linear inequalities, known as basic Shannon-type inequalities, plus a normalization constraint. The first type of inequalities are given by monotonicity conditions, for example, , stating that the uncertainty about a set of variables should always be larger than or equal to the uncertainty about any subset of it. The second type is the strong subadditivity condition given by which is equivalent to the positivity of the conditional mutual information. The normalization constraint imposes that .
Given variables, all the associated Shannon type inequalities (and thus the Shannon cone) are characterized by the following elemental (non-redundant) inequalities Yeung (2008)
for all , and . Thus, the Shannon cone associated with variables is described by inequalities plus one normalization constraint.
Given a marginal scenario , we are interested in the projection of in the subspace of representing only observable terms, that is, the Bell cone associated with the marginal scenario in question. Since we are given a description of the Shannon cone in terms of linear inequalities, to perform this projection we need to eliminate from this system of inequalities all terms corresponding to non-observables terms. In practice, this is achieved via a Fourier-Motzkin (FM) elimination Williams (1986). The final set of inequalities obtained via the FM elimination (and after eliminating over redundant inequalities) gives all the facets of the associated Bell cone, the non-trivial of which are exactly the entropic Bell inequalities Braunstein and Caves (1988); Chaves and Fritz (2012); Fritz and Chaves (2013); Chaves et al. (2014a). By non-trivial, we denote those inequalities that are not simply basic Shannon type inequalities like (8).
To exemplify the general framework it is useful to analyze the particular case of the CHSH scenario discussed in the main text. In this case, the existence of an underlying LHV model implies the existence of (and all its marginals) respecting the elementary inequalities in (8). Proceeding with the FM elimination and keeping only the observable terms , and (with ) we observe that the only non-trivial inequality (up to permutations) is given by
This is exactly the inequality originally derived in Braunstein and Caves (1988). Replacing the bipartite entropies by mutual informations in (9), that is, using , we obtain the form (2) discussed in the main text and that can be understood as the entropic analogue of the CHSH inequality. However, opposed to the probabilistic version of the CHSH inequality, its entropic version can be used as a nonlocality witness for an arbitrary finite number of measurement outcomes.
i.2 A tool for the derivation of entropic Bell inequalities
The formalism outlined above provides a general framework for the derivation of entropic Bell inequalities in basically any scenario Chaves and Fritz (2012); Fritz and Chaves (2013); Chaves et al. (2014a). In short, we have to perform the projection –via a Fourier-Motzkin (FM) elimination – of the Shannon cone to the subspace of observable coordinates defining the Bell cone . The problem with this approach is that it turns out to be computationally extremely demanding. The FM algorithm eliminate a variable from a system of inequalities by summing, after proper normalization, inequalities where appears with coefficient and with coefficient , and keeping the rest. Eliminating over variables in a system of inequalities can lead to a number of inequalities, that is, double exponential. Since the number of initial inequalities describing is itself exponential Yeung (2008), this leads to a triple-exponential complexity algorithm, which is limited, in practice, to very simple cases. In fact, no systematic characterization of entropic inequalities is known beyond particular instances of the bipartite case, although particular multipartite inequalities have been derived Chaves et al. (2014a); Raeisi et al. (2015). In particular, no entropic Bell inequality witnessing genuine multipartite nonlocality Svetlichny (1987) was known to this date.
Nevertheless, checking whether a given inequality is valid for a scenario of interest is computationally much simpler. First, notice that any valid entropic inequality must follow from a FM elimination over the system of inequalities defining the scenario at question, for example, entropic Bell inequalities follow from the basic Shannon type inequalities (8) plus possible additional linear constraints. Given the entropy vector , any linear inequality can be written as the inner product , where is a vector to the inequality. Similarly, a system of inequalities, e.g., Eq. (8), can be represented as a matrix , such that . To check the validity of an inequality with respect to the system , one simply needs to solve the following (efficient) linear program Yeung (2008):
If the minimum of is larger or equal to zero, then the inequality is valid.
Moreover, one can extend this result to inequalities for the projected cone, even without performing the corresponding FM elimination on the system . More precisely, given and , where is the projection on the first coordinates, we prove that the inequality , with for is valid for the polyhedral cone if and only if it is valid for the polyhedral cone . If there exists such that , then and . Vice versa, given such that , there exists such that and .
Similarly, it is again a linear program (in fact, a feasibility problem) to check if a given observed entropic correlation is local or nonlocal. It is simply given by
where in this case can be any linear objective function. We see that in this case, we not only impose the constraints but also impose that some of the coordinates of the vector (those given by ) should correspond to the observable quantities . In case the linear program above has no solution (non-feasible), the correlations at question are thus nonlocal.
These linear programs together with the characterization of the entropic NS cone provide novel tools in the derivation of entropic Bell inequalities, as explained below.
A nonlocal extremal correlation given by will imply a non-feasible LP of the form (11). However, not all the constraints encoded in the matrix will be necessary for witnessing this non-feasibility: many of the inequalities in can be eliminated until we reach a (not unique) minimum set of inequalities that will still lead to a non-feasible LP. Given this minimum set we can then perform a FM elimination that thus will lead to an inequality that is violated by the given correlation . Notice that in general the obtained inequalities are not necessarily going to be facets of the marginal cone of interest. However, we can obtain tight inequalities by adding noise to the extremal correlations (for instance, white noise). By doing that we guarantee that we are deriving inequalities probing the nonlocal character of the given correlation until the noise is strong enough to make it enter in the marginal cone of interest (and thus become local). Notice that a similar procedure can be applied to obtain the minimum set of inequalities required to prove the validity of a given inequality bounding the marginal cone of interest.
i.3 Entropic NS cones in a variety of scenarios
For the entropic NS cone description, we use the elemental Shannon type inequalities (8) for each subset of mutually compatible variables defined by a given marginal scenario. In the sections below, we describe in details the entropic NS cone in the bipartite and tripartite scenario, and the hybrid local-nonsignaling models. Notice that for , the Shannon cone correspond with the true entropy cone, so our bounds are tight. For each of the scenarios we consider, we have catalogued all the inequivalent classes of extremal rays and characterized whether they correspond to local or nonlocal correlations.
In a bipartite Bell scenario, two parties Alice and Bob can measure possible different observables. Thus, the bipartite entropic NS cone is defined by the elemental inequalities for each subset of mutually compatible observables , that is, it is simply described in terms of the elemental inequalities , and for .
As discussed in the main text, for the case (corresponding to the CHSH scenario) is characterized by different classes of extremal NS rays, of which correspond to local correlations and only correspond to nonlocal correlations. In Table (1) we list all the different classes (a class is defined by the inequalities that are equivalent up to the permutation of parties and/or observables).
The class of extremal rays corresponds to the case where one of the variables has maximal entropy, e.g, (where is the number of outcomes), while all other variables have null entropy (that is, they represent probability distributions with deterministic outcomes). Class represents perfect (anti)correlations between one observable of Alice and one of Bob, e.g., while all other entropies are null. Class represents perfect (anti)correlations between the two observables of Alice with one observable of Bob, e.g, , while the other observable of Bob has null entropy. Class represents perfect (anti)correlations between all the observables of Alice and Bob, that is, . All these rays clearly correspond to local correlations.
In turn, the class corresponds to entropic nonlocal correlations. For , it can be understood as the entropic version of the paradigmatic PR-box Popescu and Rohrlich (1994)
First, notice that is entropically equivalent to the purely classical correlations Chaves (2013)
That is, both (13) and (14) correspond to the class of extremal rays. The reason for this equivalence is due to the fact that entropies are unable to distinguish between correlations and anti-correlations Chaves (2013). Interestingly, the class correspond to a probability distribution obtained by mixing with equal weights the distributions (13) and (14), that is, corresponding to three perfect (anti)correlated pairs of variables and one uncorrelated pair . The class violates the entropic version of the CHSH inequality (9) up to the algebraic maximum, achieving .
We have performed the same analysis for the case and obtained different classes of extremal rays, of which correspond local and to nonlocal correlations. The different classes of extremal rays are listed in Table (2). The interpretation of the local rays is identical to the one we have detailed above to the case of measurement settings. Regarding the nonlocal rays, we focus attention to the classes an . Class basically correspond to the entropic version of the PR-box discussed above, since . In turn, class can be understood as the NS correlation that maximally violates the entropic version of the Collins-Gisin inequality Collins and Gisin (2004) given by Chaves et al. (2014a)
In a tripartite Bell scenario, three parties Alice, Bob and Charlie perform different measurements in their shares of a joint state. Thus, the tripartite entropic NS cone is defined by the elemental inequalities for each subset of mutually compatible observables (with ).
We have obtained the full characterization of in terms of extremal rays for . There are different inequivalent classes of extremal rays, of which are local and are nonlocal. Furthermore, as discussed in details below, in the tripartite case we can introduce the notion of genuine tripartite nonlocal (GTNL) correlations. From the different classes of extremal nonlocal rays, of them are GTNL.
There are thus rays corresponding to nonlocal correlations but displaying no GTNL. We focus our attention to a particular class of these rays, given by , and and . It can be obtained by the mixing of the nonlocal correlation
with the classical correlation
The nonlocal character of this distribution can be witnessed via the following tripartite entropic inequality (obtained via the approach described in Sec. I.2)
since the correlations above imply that thus violating the inequality. To prove this inequality analytically it is sufficient to consider the chain rule for entropies, implying that
Using the basic inequalities saying that and , we can turn the chain rule decomposition above in the inequality
that can be rewrite exactly as if we use that .
i.3.3 Proving the monogamy inequality for the entropic CHSH
In order to prove that the monogamy inequality (3) of the main text holds for nonsignaling correlations, we must show that if follows from the elemental inequalities for each subset of mutually compatible observables . It is sufficient to add the the following elemental inequalities
that can be rewritten exactly as
i.3.4 Genuine tripartite nonlocality
Similarly to what happens to entanglement Horodecki et al. (2009), when moving from the bipartite case, different classes of nonlocality can be categorized. In the tripartite scenario one can introduce the notion of genuine tripartite nonlocality, that is, a stronger form of nonlocality that cannot be reproduced by LHV models even if two of the parties are allowed to share some nonlocal resources. As discussed in the main text, hybrid local-nonsignaling (LNS) models are those that can be decomposed as
where and where (similarly to the other permutations) represent some nonlocal resource. The different nonlocal resources we allow the parties to share will lead to distinct notions of genuine multipartite nonlocality. See Gallego et al. (2012); Bancal et al. (2013) for a discussion of the different (non-signalling or signalling) nonlocal resources that can be used in the definition of genuine multipartite nonlocality. Here we will focus our attention to nonsignaling nonlocal resources, as for example, nonlocal quantum correlations or PR-boxes Popescu and Rohrlich (1994). In this case, (I.3.4) is equivalent to the existence of probability distributions , and such that the marginals coincide .
To simplify the discussion let us consider that each of the parties can perform two possible measurements. The first term in the decomposition (I.3.4), that is, a model with decomposition given by
is equivalent, in the entropic description, to the existence of and all its marginals. For each value of we have therefore a collection of four variables respecting the Shannon type inequalities (8).
Similarly, the two other terms in (I.3.4), namely
imply the existence of and respectively. Thus, following the general prescription, the entropic description of hybrid local-nonsignaling (LNS) correlations corresponds to the intersection of the Shannon cones defined for each of subsets of variables