Multipartite Causal Correlations: Polytopes and Inequalities

# Multipartite Causal Correlations: Polytopes and Inequalities

Alastair A. Abbott Institut Néel, CNRS and Université Grenoble Alpes, 38042 Grenoble Cedex 9, France    Christina Giarmatzi Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4072, Australia Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4072, Australia    Fabio Costa Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4072, Australia    Cyril Branciard Institut Néel, CNRS and Université Grenoble Alpes, 38042 Grenoble Cedex 9, France
August 23, 2019
###### Abstract

We consider the most general correlations that can be obtained by a group of parties whose causal relations are well-defined, although possibly probabilistic and dependent on past parties’ operations. We show that, for any fixed number of parties and inputs and outputs for each party, the set of such correlations forms a convex polytope, whose vertices correspond to deterministic strategies, and whose (nontrivial) facets define so-called causal inequalities. We completely characterize the simplest tripartite polytope in terms of its facet inequalities, propose generalizations of some inequalities to scenarios with more parties, and show that our tripartite inequalities can be violated within the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed.

## I Introduction

One of the most surprising features of quantum mechanics is that it generates correlations that cannot be obtained with classical systems. The most studied scenario involves parties that cannot communicate with each other. As first proved by Bell Bell (1964), two such parties sharing entangled quantum states can generate nonsignaling correlations that are not achievable with classical resources. The no-signaling constraint corresponds to a particular causal structure, where all correlations are due to a common cause. There is a growing interest in studying and characterizing more general causal structures Branciard et al. (2012); Fritz (2012); Chaves et al. (2014); Fritz (2016) and understanding when the corresponding correlations can be produced with classical or quantum systems Henson et al. (2014); Pienaar (2016).

But what are the most general correlations achievable in any causal structure? And can quantum mechanics generate even more general correlations not compatible with any definite causal structure? This question was considered in Oreshkov et al. (2012), where a framework was developed that assumes the validity of quantum mechanics in local laboratories, with no assumptions about the causal structure in which the laboratories are embedded. It was found that such a framework allows for correlations that can violate causal inequalities, constraints that are necessarily satisfied by correlations generated in any definite causal order. However, no clear physical interpretation was found for such ‘noncausal correlations’.

A physical process in which operations are performed ‘in a superposition’ of causal orders was first proposed in Chiribella et al. (2013). This process—the quantum switch—can in principle provide advantages for computation Araújo et al. (2014) and communication Feix et al. (2015); Guérin et al. (2016), and a first experimental proof of principle has been recently demonstrated Procopio et al. (2015). However, the quantum switch requires device-dependent tests to detect its ‘causal nonseparability’ Araújo et al. (2015) (i.e., the lack of definite causal order) and it cannot be used to violate any causal inequality Araújo et al. (2015); Oreshkov and Giarmatzi (2016). Interestingly, the quantum switch requires the coordinated action of three parties to detect causal nonseparability, while none of the known bipartite examples of causally nonseparable processes seem to have physical interpretations. This motivates a systematic study of multipartite scenarios: if there is a process in nature that can violate causal inequalities, it may indeed require more than two parties.

Here we consider general scenarios involving a finite but arbitrary number of parties with finite numbers of possible inputs and outputs, and prove that ‘causal correlations’—those that can be generated in a well-defined causal structure, be it fixed or dynamical, deterministic or probabilistic—form a convex polytope whose vertices correspond to deterministic strategies. That is, any causal correlation can be expressed as a probabilistic mixture of deterministic causal strategies. These are strategies where the output of each party is a deterministic function of its own input and of the inputs of parties in its past, and where the causal relations amongst a set of parties are functions of the inputs of the parties in the past of that set. We further completely characterize the simplest nontrivial polytope for three parties in terms of its facets, which define causal inequalities. We interpret some of these inequalities in terms of device-independent ‘causal games’ Oreshkov et al. (2012), for which the probability of success has a nontrivial upper bound whenever the parties are constrained by a definite causal order. We also generalize some of these inequalities to the -partite case. Finally, we show that all of these nontrivial tripartite inequalities can be violated within the process matrix formalism of Ref. Oreshkov et al. (2012).

## Ii Multipartite causal correlations

### ii.1 Scenarios and notation

In this paper we will consider situations where a finite number of parties each receive an input from some finite set (which can in principle be different for each party), and generate an output that also belongs to some finite set (and which may also differ for each input). The fixed number of parties, of possible inputs for each party, and of possible outputs for each input, define together a ‘scenario’; throughout the paper we will always (often implicitly) assume that such a scenario is fixed. We define the vectors of inputs and outputs and . The correlation established by the parties in a given scenario is then described by the conditional probability distribution .

For any (nonempty) subset of with elements, we shall denote by and the lists of inputs and outputs for the parties in . This will in particular allow us to consider marginal correlations and write, for instance (noting that , up to a reordering of the parties), . For ease of notation, a singleton will simply be written , and the vectors of inputs and outputs corresponding to the parties in (obtained by removing just one party, ) will simply be denoted by and .

### ii.2 Defining multipartite causal correlations

We wish to investigate here the correlations that can be established in a scenario where each party’s events—namely, the choice of an input and the generation of an output—happen within a well-defined causal structure, with well-defined causal relations between the parties.

The case of two parties is rather clear. The only possible causal relations are that causally precedes —a case that we denote by , and which implies that ’s marginal probability distribution or ‘response function’ should not depend on ’s input: —or vice versa, , where . The case where and are causally independent can be included in either or , as it is compatible with both. As originally considered in Ref. Oreshkov et al. (2012), one may also allow for situations where the causal order is not fixed, but chosen probabilistically. A bipartite correlation that is compatible with , or , or a probabilistic mixture of the two is said to be ‘causal’ Oreshkov et al. (2012); Araújo et al. (2015); Brukner (2014); Oreshkov and Giarmatzi (2016); Branciard et al. (2016).

Moving now to three or more parties, more complex possibilities arise. Indeed, the action of the first party could control the causal relations of the following parties, in perfect agreement with the idea of a well-defined causal structure. For instance, if a party is first, they could decide to set before if their input is , or before if it is (or they could choose the causal order between and as the result of a coin toss, where the coin’s bias depends on the input). We should thus allow for such dynamical causal orders Hardy (2005); Oreshkov and Giarmatzi (2016) (sometimes also referred to as adaptive causal orders Baumeler and Wolf (2014)) to establish the correlations we are interested in.

In any case, even allowing for dynamical causal orders, the compatibility with a definite causal structure will always require that one party acts first; which party this is can be chosen probabilistically, as in the bipartite case. The response function of that first party (say ) should then not depend on the other parties’ inputs: . The action of that party would then determine the causal structure of the following parties, so that the correlation shared by the latter, conditioned on the input and output of the former, should also be compatible with a definite causal structure. These observations lead us to introduce the following inductive definition for what we shall call ‘causal correlations’:

###### Definition 1 (Multipartite causal correlations).

• For , any valid probability distribution is causal;

• For , an -partite correlation is causal if and only if it can be decomposed in the form

 P(→a|→x)=∑k∈N qk Pk(ak|xk) Pk,xk,ak(→a∖k|→x∖k), (1)

with for each , , where (for each ) is a single-party (and hence causal) probability distribution and (for each ) is a causal -partite correlation.

It is easy to check that this general definition is compatible with that for the bipartite case recalled above. We note that the multipartite case was first investigated in Ref. Oreshkov and Giarmatzi (2016). It was shown there that the above definition characterizes precisely the correlations that are compatible with the intuition about causality that (paraphrasing Oreshkov and Giarmatzi (2016)) the choice of input for one party cannot affect the outputs of other parties that acted before it (or which are not causally related, being neither in the past nor in the future of it), nor the causal order between those previous parties and the party in question. This also implies that causal correlations are those for which a classical ‘hidden variable’ exists, whose value determines the causal order between all the parties, and signaling is only possible from parties in the causal past to those in the future according to the given causal order Oreshkov and Giarmatzi (2016). These arguments provide further justification to the above choice of definition for multipartite causal correlations.

### ii.3 Basic properties of causal correlations

Let us mention some basic properties of this definition of multipartite causal correlations. The proofs of the claims below are given in the Appendix.

#### ii.3.1 Convexity of causal correlations

From the previous discussion it should be clear that any probabilistic mixture of causal correlations (for a given scenario) must also be causal, so that the set of causal correlations is convex. Although this is not immediately evident from the definition of Eq. (1), it can indeed be shown to be the case.

#### ii.3.2 Ignoring certain parties

Any marginal correlation, for any subset of parties, of a causal correlation is causal.

More specifically, consider an -partite causal correlation and a nonempty subset . Then the -partite correlation

 P→xN∖K(→aK|→xK):=P(→aK|→x)=∑→aN∖KP(→a|→x) (2)

is causal for all .

The correlation above is still conditioned on the inputs of the parties whose outputs are discarded. One can remove this dependence by averaging it out (for a given input distribution): by the convexity of causal correlations, the resulting -partite correlation remains causal. Note, on the other hand, that the correlation , conditioned also on the outputs of the discarded parties, is not necessarily causal: post-selection indeed allows one to turn a causal correlation into a noncausal one.111To see this, consider for instance a bipartite ‘causal game’ as in Oreshkov et al. (2012); Branciard et al. (2016), and add a third party to whom all inputs and outputs of the first two parties are sent, and who outputs if and only if the winning conditions for the causal game are met. Postselecting on that output, the bipartite correlation shared by the first two parties clearly wins the game perfectly (which implies that it is noncausal), although it could be established within a well-defined causal structure.

#### ii.3.3 Combining causal correlations ‘one after the other’

Consider two (nonempty) sets of parties and , and two causal correlations: for the first set, and for the second, which may depend on the inputs and outputs of the first set of parties (the parties in the set are thus understood to ‘act before’ those of the set ). Then the -partite correlation obtained by combining those in the form

 P(→a|→x):=P(→aK|→xK) P→xK,→aK(→aN∖K|→xN∖K) (3)

is causal.

#### ii.3.4 An equivalent characterization of causal correlations

An equivalent characterization, for the case , is that an -partite correlation is causal if and only if it can be decomposed in the form (cf. also Ref. Oreshkov and Giarmatzi (2016))

 P(→a|→x)=∑∅KNqK PK(→aK|→xK) PK,→xK,→aK(→aN∖K|→xN∖K),

where the sum runs over all nonempty strict subsets of (with elements), with for each , , where (for each ) is a -partite causal correlation, and (for each ) is an -partite causal correlation.

This equivalent characterization implies in particular that a correlation of the form

 P(→a|→x)=∑k∈N qk Pk(→a∖k|→x∖k) Pk,→x∖k,→a∖k(ak|xk), (5)

with for each , , where (for each ) is a causal -partite correlation and (for each ) is a single-party probability distribution, is causal. (It is indeed obtained from (LABEL:def:causal_correlation_v2) by summing over the subsets , and relabelling certain subscripts to .) Compared to Eq. (1), each term in the above sum distinguishes a given party that ‘comes last’, rather than first. Note for instance that the correlations obtained from the so-called quantum switch Chiribella et al. (2013) are precisely of this form, and are hence causal; see Refs. Oreshkov and Giarmatzi (2016); Araújo et al. (2015).

It should be emphasized however that for the contrary is not true: not all causal correlations are of the form (5). Correlations with dynamical causal order, such as the one given below, provide counter-examples.

#### ii.3.5 Examples: fixed-order, mixtures of fixed orders, and dynamical-order causal correlations

Let us finish this section with some examples.

The simplest example of a causal correlation one can think of is one that is compatible with a fixed causal order between all the parties that is independent of any party’s input and output. For instance, a correlation compatible with the causal order can be written:

 P(→a|→x) = P(a1|x1) Px1,a1(a2|x2) Px1,x2,a1,a2(a3|x3) (6) ×⋯×P→x∖N,→a∖N(aN|xN),

which clearly satisfies the definition of a causal correlation given by Eq. (1).

Beyond this simplest case, by the convexity of the definition (see Sec. II.3.1 above), any probabilistic mixture of fixed-order causal correlations is causal. For example, if the correlation is compatible with the fixed order and is compatible with (where and are two permutations of ), then for any , is also causal. The interpretation is simply that, with probability , the correlation is compatible with the fixed causal order defined by , while with probability it is compatible with .

For , this mixture of fixed-order causal correlations is not yet the most general type of causal correlation. Indeed, as discussed above, the inputs and outputs of the party (or parties) acting first could influence the causal order between the subsequent parties: the causal order can be dynamical Oreshkov and Giarmatzi (2016). As a concrete example, consider for instance the tripartite scenario with binary inputs , for all parties, a single fixed output for (which we can therefore ignore) and binary outputs , for and , and the following (deterministic) correlation:

 P(a2,a3|x1,x2,x3) = δx1,0 δa2,0 δa3,x2 (7) + δx1,1 δa2,x3 δa3,0,

where is the Kronecker delta. This example can be understood causally as follows (recall the discussion of Subsection II.2): the party acts first; their input ( or ) then determines the causal order between the following two parties ( or , respectively), where the second party must always output (corresponding to or , resp.) and the last party must produce the input of the second party ( or , resp.) as output. This correlation can thus be established in a well-defined, although dynamical, causal order and is thus causal. One can check that it is indeed of the form (1) (with only one term in the sum, for ), but not of the form (5): there is indeed no party that always acts last (note that, since the correlation is deterministic, the sum in (5) would also need to have only one term, which would single out a fixed last party).

Finally, a generalization of the previous example is a situation in which the order between and is chosen probabilistically with a probability depending on the input of . An example of this type is

 P (a2,a3|x1,x2,x3)= (8) δx1,0(q0δa2,0 δa3,x2+(1−q0)δa2,x3 δa3,0) +δx1,1(q1δa2,0 δa3,x2+(1−q1)δa2,x3 δa3,0),

with . In this example, with probability if and with probability if . Once again, this correlation is of the form (1) and can be established in a well-defined causal order. However, for , it is not a probabilistic mixture of fixed-order causal correlations.222 To see this, assume that we can write , , where and are causal correlations with the fixed orders and , respectively. (Since has no output, we can assume they act first; see Sec. III.3.) Note that ’s marginal distribution satisfies , and thus also. But the causal order of requires , so that . Similarly, since we have , so that . Together, this implies . Analogous reasoning for when implies that and thus we must have if is a mixture of fixed-order causal correlations.

## Iii Characterization of causal correlations as a convex polytope

As noted earlier, any convex combination of causal correlations is causal, meaning that causal correlations (for a given scenario) form a convex set. It was already argued in Refs. Oreshkov and Giarmatzi (2016); Branciard et al. (2016), more precisely, that this set is a convex polytope, the so-called ‘causal polytope’. Here we will prove this more explicitly by showing how any causal correlation can be written as a convex combination of deterministic causal correlations. (This was already proved for the bipartite case in Ref. Branciard et al. (2016).) The polytope structure then follows from the fact that, for any given scenario, the number of such deterministic causal correlations is finite. The facets of the causal polytope can be expressed as linear inequalities that are satisfied by all causal correlations: when nontrivial, these correspond to (tight) ‘causal inequalities’ Oreshkov et al. (2012); Branciard et al. (2016).

### iii.1 Decomposing causal correlations into deterministic ones

Let us first introduce some more notation. A correlation is deterministic if the list of outputs, , is a deterministic function of the list of inputs, : . We shall then denote the corresponding probability distribution by , such that

 Pdet→α(→a|→x)=δ→a,→α(→x). (9)

We will now prove the following theorem:

###### Theorem 2.

Any -partite causal correlation can be written as a convex combination

 P(→a|→x)=∑→αq→α Pdet→α(→a|→x) (10)

with , , where the sum is over all functions that define a deterministic causal correlation .

The proof is by induction:

• For , it is a well-known fact that any correlation can be written as a convex combination of deterministic ones (see, e.g., Ref. Fine (1982)), and any single-party correlation is causal.

• For any given we shall prove the following implication: if it is true that all -partite causal correlations can be written as convex combinations of deterministic ones (the induction hypothesis), then the same is true for -partite causal correlations.

Consider an -partite causal correlation , decomposed in the form (1), with the correlations being -partite causal correlations (for all ). By the induction hypothesis, the latter can be decomposed as in Eq. (10):

 Pk,xk,ak(→a∖k|→x∖k)=∑→α∖kq→α∖k(k,xk,ak)Pdet→α∖k(→a∖k|→x∖k), (11)

where the weights depend in general on , and the sum is over all functions that define a deterministic causal correlation . This decomposition does not yet prove the theorem, because we need to express as a convex combination with weights that do not depend on the inputs and outputs. However, we can remove this dependency by appropriately rearranging the sum (11). To this end, we shall first prove the following lemma:

###### Lemma 3.

Consider a set of points () belonging to some linear space, and different points () in their convex hull, written as convex combinations of the extremal points in the following way:

 P(z)=M∑m=1qm(z) Qm, (12)

with weights that depend on (such that, for each , all and ).

Then, each point can also be written as

 P(z)=M∑m1=1⋯M∑mZ=1~qm1,…,mZ Qmz, (13)

where it is now the extremal points that depend on , while the new weights , are fixed.

###### Proof.

The new weights are defined as

 ~qm1,…,mZ:=Z∏z=1qmz(z). (14)

Then for a given ,

 ∑m1,…,mz−1,mz+1,…,mZ~qm1,…,mZ = qmz(z), (15)

and

 ∑m1,…,mZ~qm1,…,mZ Qmz = ∑mz∑m1,…,mz−1,mz+1,…,mZ~qm1,…,mZ Qmz (16) = ∑mzqmz(z) Qmz=P(z),

as required. ∎

Returning to the proof of Theorem 2, we rename the party-input-output variables as . We can now apply Lemma 3 to Eq. (11) and write

 Pk,xk,ak(→a∖k|→x∖k)=∑→α1∖k,…,→αZk∖k~q→α1∖k,…,→αZk∖k Pdet→αzk∖k(→a∖k|→x∖k), (17)

where the correlations are taken from the same set as the ’s above, and hence are deterministic and causal.

The single-party probability distributions in Eq. (1) can also be decomposed as a combination of deterministic correlations,

 Pk(ak|xk)=∑αkq′αk Pdetαk(ak|xk). (18)

Using Eqs. (17) and (18), we can now expand correlations of the form of (1) as

 (19)

This is indeed a convex combination of deterministic causal correlations, with weights independent of inputs and outputs, which thus completes the proof. ∎

### iii.2 Describing deterministic causal strategies

As mentioned above, a deterministic ‘strategy’ (or correlation) can be characterized by a deterministic function of the list of inputs , which determines the list of outputs .

Of course, not any such function will make the correlation causal. In order to be causal, must indeed have a decomposition of the form (1). Since can only take values 0 or 1, this implies in particular that the weights are also 0 or 1 and hence, there is only one term in the sum.

That is, the causal deterministic strategy can be understood as follows: it determines a party that acts first. The output of that party is then a deterministic function of its input (which is also specified by ). For each input of that party (and the corresponding deterministic output ), the remaining parties must then also share a deterministic and causal correlation—which in turn must be compatible with one specific party acting first. Hence, the input of the first party also determines the party that acts second (recall that causal correlations allow for dynamical causal orders, see the example in Sec. II.3.5); the response function of that party is then a deterministic function of the input of the first party and its own input. Continuing in this fashion, the party that acts third then depends on the inputs of and , and its output is a deterministic function of the inputs of those two parties and its own input; etc.

Thus, each given set of inputs can be viewed as being processed in a particular causal order333If some parties are causally independent then this order may not be unique. (with , etc.), so that the correlation can be written as

 Pdet→α(→a|→x) = P(ak1|xk1)P(ak→x2|xk1,xk→x2) (20) ×⋯×P(ak→xN−1|→x∖k→xN)P(ak→xN|→x).

It follows in particular that for each given , there exist nested subsets with , , such that for all , the marginal distribution

 Pdet→α(→aK→xK|→x)=Pdet→α(→aK→xK|→xK→xK) (21)

does not depend on the inputs of the last parties in the causal order realized on input .

### iii.3 Causal correlations in scenarios with trivial inputs or outputs

Consider a scenario in which one party has a fixed output for all its possible inputs (the output being fixed, we could just ignore it and equivalently say that has no output).444This scenario also arises when one averages over ’s outputs. It is well known that, for Bell-type local correlations, this scenario is equivalent to the similar one in which is simply ignored. More generally, if a single input has a fixed (or, equivalently, no) output, then the local polytope is simply equivalent to that obtained by discarding the input completely Pironio (2005). In contrast, for the case of causal inequalities it has already been noted that one can obtain interesting correlations with ‘nontrivial inputs with fixed outputs’ Branciard et al. (2016). What can we therefore say more generally about the causal polytope when has a fixed output for all its inputs?

In such a scenario we can write the -partite correlation as

 P(→a|→x)=P(→a∖k|→x∖k,xk)=Pxk(→a∖k|→x∖k). (22)

If is causal for all then is trivially of the form (1), and therefore causal. Conversely, if is causal then, by the remark discussed in Subsection II.3.2, is also causal for each .

Thus, the -partite correlation is causal if and only if all of the conditional -partite correlations obtained for each possible input of are causal. In order to test whether is causal it therefore suffices to test whether the -partite correlations are causal, and one can always assume that is located before all the other parties.555Hence, in an -partite scenario where one party has a trivial output, a noncausal correlation can only be obtained if some reduced -partite correlation is already noncausal. Note that in contrast, in the framework of process matrices Oreshkov et al. (2012) (see Section V), the property of causal nonseparability of a process—which is the ‘device-dependent’ analog of noncausality for correlations Araújo et al. (2015); Branciard et al. (2016)—can be witnessed in a scenario where some parties only have trivial outputs (e.g., where they simply implement unitary operations), while all reduced processes involving fewer parties are causally separable Araújo et al. (2015); Branciard (2016).

Another important scenario to understand is that in which a party has a single fixed input (or equivalently, as before, no input). In this case we have (from the definition of conditional probabilities)

 P(→a|→x)=P(→a∖k,ak|→x∖k)=P(→a∖k|→x∖k)P→x∖k,→a∖k(ak), (23)

with . If is causal then is clearly of the form (5), and thus causal. Conversely, referring again to the remark in Sec. II.3.2, if is causal then so is .

Thus, as is the case for locality, the causality of the -partite correlation is equivalent to the causality of the -partite correlation obtained by discarding the party with a fixed input. Causally, one may consider that always acts after the other parties. Note that this is also true, more generally, whenever a party cannot signal to any other party or set of parties—as is indeed the case when they have a fixed (or no) input.

## Iv Simplest tripartite inequalities

With the basic properties of the multipartite causal polytope laid out, we wish to study in detail the simplest scenario with more than two parties (i.e., which is not reducible to the bipartite scenario that was characterized in Ref. Branciard et al. (2016)). In contrast to the case for Bell inequalities, where the simplest such case is the ‘’ scenario with 3 parties all having binary inputs and outputs, the discussion in the previous section suggests that a simpler tripartite scenario exists for causal correlations. This is the scenario where each party has a binary input , a single constant output for one of the inputs, and a binary output for the other. Specifically, we consider that for each , the input has the constant output , while the input has two possible outputs, or .

As is standard, let us denote the three parties (i.e., ), their inputs (instead of ) and their outputs (instead of ). We will denote below by the complete tripartite probability distribution (i.e., ), and by , , etc. the marginal distributions for the parties indicated by the subscript (e.g., , etc.). Note that every marginal distribution retains a dependency on all three inputs.

### iv.1 Characterizing the causal polytope

The vertices of the causal polytope for this scenario can be found by enumerating all the deterministic probability distributions compatible with any of the 12 possible definite causal orders (for each of the 3 parties acting first, there are 4 possible causal orders for the remaining 2 parties: two fixed orders, and two dynamical ones, where the order depends on the input of the first party). One finds that there are 680 such strategies (and thus vertices), of which 488 are compatible with a fixed causal order, while the remaining 192 require a dynamical order to be realized.

The causal polytope is -dimensional, since this is the minimum number of parameters needed to completely specify any probability : for each set of inputs , if of them are non-zero then one needs values to specify the probabilities completely for these inputs (normalization determines the remaining value)—so that the dimension of the problem is indeed .

In order to determine the facets of this polytope, which correspond directly to tight causal inequalities, a parametrization of the polytope must be fixed and the convex hull problem solved Branciard et al. (2016). Several such parametrizations of are possible but we found that, because of the size of the polytope, the ability to solve the convex hull problem depended critically on the chosen parametrization. Using the software cdd Fukuda (2012) we were able to compute the facets of the polytope from its description in terms of its vertices with the following parametrization:

 →P=( PA(1|100),PB(1|010),PC(1|001), PAB(10|110),PAB(01|110),PAB(11|110), PBC(10|011),PBC(01|011),PBC(11|011), PAC(01|101),PAC(10|101),PAC(11|101), PABC(100|111),PABC(010|111), PABC(001|111),PABC(110|111), PABC(011|111),PABC(101|111), PABC(111|111)). (24)

In total, the polytope was found to have facets, each corresponding to a causal inequality. However, inequalities that can be obtained from one another, either by relabeling outputs or permuting parties, can be considered to be equivalent. Once such equivalences are taken into account one finds that there are 305 equivalence classes, or ‘families’, of inequalities. A complete list of these families can be found in the Supplemental Material SM , but in what follows we will focus on some specific interesting examples.

### iv.2 Three simple inequalities

As is standard for polytopes of correlations, several facets correspond to trivial inequalities of the form . Specifically, there are three inequivalent such families, corresponding to 1, 2 or 3 inputs being 1. One also recovers conditional versions of the nontrivial bipartite ‘lazy guess-your-neighbor’s-input’ (LGYNI) inequalities , where denotes addition modulo 2 (and where this notation implicitly assumes that the inputs and are uniformly distributed) Branciard et al. (2016). These can equivalently be written as

 PA(1|10z)+PB(1|01z)−PAB(11|11z) ≥ 0. (25)

There are two inequivalent families of such inequalities, for the two cases of or .

Amongst the remaining families of inequalities there are 5 which are completely symmetric under exchange of parties, and which are good candidates for simple inequalities that nontrivially involve all three parties. The following two are of particular interest due to their simple form, and the fact that they can be seen as natural generalizations of the LGYNI inequality (25):

 I1= PAB(11|110)+PBC(11|011) +PAC(11|101)−PABC(111|111) ≥ 0, (26)

and

 I2= PA(1|100)+PB(1|010) +PC(1|001)−PABC(111|111) ≥ 0. (27)

As is the case for the LYGNI inequality (25), these two inequalities can be expressed as ‘causal games’ Oreshkov et al. (2012); Branciard et al. (2016). They can indeed be written as

 P(xy(ab⊕z)=yz(bc⊕x) =xz(ac⊕y)=0) ≤ 7/8 (28)

and

 P(x(y⊕z⊕1)(a⊕yz)=y(x⊕z⊕1)(b⊕xz) =z(x⊕y⊕1)(c⊕xy)=0) ≤ 7/8, (29)

respectively, where it is implicitly assumed that all inputs occur with the same probability. More precisely, the first inequality can be interpreted as a game in which the goal is to collaborate so that, whenever two parties both receive the input 1, the product of their outputs should match the input of the other party (in all other cases any output wins the game). The second inequality can be interpreted as a similar game in which the goal is to ensure that, whenever a party receives the input 1 and the other two parties receive the same input, that party’s output should match the other parties’ inputs. In both cases, the probability of success can be no greater than if the three parties follow a causal strategy. It is simple to saturate this bound with a deterministic causal strategy: for example, if all parties always output , both games are won in all cases except that where all inputs are , giving indeed a success probability of .

Another, simple inequality of interest that is symmetric only under a cyclic permutation of parties, and can also be seen as a generalization of the LGYNI input inequality, is the following:

 I3= 2−PAB(01|110)−PBC(01|011) −PAC(10|101) ≥ 0. (30)

As for the previous two inequalities, this causal inequality can be interpreted as a causal game in the form (still implicitly assuming a uniform distribution of inputs for all parties):

 P(xy(z⊕1)((a⊕1)b⊕1)=yz(x⊕1)((b⊕1)c⊕1) =xz(y⊕1)((c⊕1)a⊕1)=0) ≤ 7/8, (31)

where the goal of the game is to ensure that whenever exactly two parties receive the input 1, each of them must guess the input of their left-hand neighbor (where is considered, in a circular manner, to be to the left of ). The bound on the probability of success can for instance be reached with the causal strategy, compatible with the order , where the parties output , and : this strategy indeed wins the game in all cases except when the inputs are .

### iv.3 Generalizing tripartite causal inequalities

For scenarios more complicated than the ‘simplest’ tripartite one considered above, the convex hull problem—and thus the characterization of the causal polytope—very quickly becomes intractable. For example, the polytope for the ‘complete binary’ tripartite case where binary outputs are allowed for both inputs, has vertices and is 56-dimensional. Beyond this, it moreover becomes difficult to even enumerate the different vertices of the causal polytope.

Although we were hence unable to enumerate the causal inequalities for this complete binary tripartite scenario, by enumerating the vertices of the polytope we were able to verify that the three causal inequalities discussed above are in fact facets in this scenario as well. To see this, one can enumerate all the deterministic strategies and thus vertices of the polytope, and use the fact that an inequality is a facet of the polytope if and only if a) it is satisfied by every vertex of the polytope, and b) there are affinely independent vertices saturating the inequality, where is the dimension of the polytope Pironio (2005).

Another facet of this complete binary tripartite polytope has independently been found by Araújo and Feix Araújo and Feix (2016) (reproduced here with their permission) and can be written as

 I4= 2−PABC(000|000)−PABC(011|110) −PABC(101|011)−PABC(110|101) ≥ 0. (32)

This inequality can be interpreted as a causal game that generalizes that of Eq. (31):

 P((x⊕y⊕z⊕1)( (b⊕x⊕1)(c⊕y⊕1) ×(a⊕z⊕1)⊕1)=0)≤ 3/4, (33)

where the goal of the game is, whenever an even number of parties receive the input 1, for every party to guess the input of their left-hand neighbor.666The bound on the probability of success can for instance be reached with the causal strategy, compatible with the order , where the parties output , and : this strategy indeed wins the game in all cases except when the inputs are or . This game is equivalent to a form of the original multipartite, cyclic, guess your neighbor’s input (GYNI) game Almeida et al. (2010) in which each party must always guess their left-hand neighbor’s input, but a non-uniform distribution of inputs is considered (namely, only the four input combinations appearing in Eq. (32) are allowed). It is interesting to note that the inequality corresponding to the variant of the GYNI game with a uniform distribution of all inputs is not a facet of the polytope.

Thus, this game is a form of the original multipartite, cyclic, guess-your-neighbor’s-input (GYNI) game introduced in Ref. Almeida et al. (2010), where different input distributions where considered. It is interesting to note that the inequality corresponding to the GYNI game with a uniform distribution over all inputs is not a facet of the polytope.

As we noted earlier, the inequalities discussed in the previous subsection can be seen as natural possible generalizations of the LGYNI bipartite inequalities from Ref. Branciard et al. (2016). This suggests that similar generalizations to -partite scenarios might provide tight causal inequalities for arbitrarily many parties. In particular, the natural generalizations of the first two inequalities, Eqs. (26) and (27), to -parties would be:777To prove that Eq. (34) indeed defines a valid causal inequality for all , it is sufficient to prove that it holds for all deterministic causal correlations. From the remark at the end of Subsection III.2 we know that, given a deterministic causal correlation , the input fixes a particular causal order, and in particular a ‘last’ party , such that . This implies that