Genuinely Multipartite Noncausality
Abstract
The study of correlations with no definite causal order has revealed a rich structure emerging when more than two parties are involved. This motivates the consideration of multipartite “noncausal” correlations that cannot be realised even if noncausal resources are made available to a smaller number of parties. Here we formalise this notion: genuinely partite noncausal correlations are those that cannot be produced by grouping parties into two or more subsets, where a causal order between the subsets exists. We prove that such correlations can be characterised as lying outside a polytope, whose vertices correspond to deterministic strategies and whose facets define what we call “2causal” inequalities. We show that genuinely multipartite noncausal correlations arise within the process matrix formalism, where quantum mechanics holds locally but no global causal structure is assumed, although for some inequalities no violation was found. We further introduce two refined definitions that allow one to quantify, in different ways, to what extent noncausal correlations correspond to a genuinely multipartite resource.
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1 Introduction
Understanding the correlations between events, or between the parties that observe them, is a central objective in science. In order to provide an explanation for a given correlation, one typically refers to the notion of causality and embeds events (or parties) into a causal structure, that defines a causal order between them Reichenbach (1956); Pearl (2009). Correlations that can be explained in such a way, i.e., that can be established according to a definite causal order, are said to be causal Brukner (2014).
The study of causal correlations has gained a lot of interest recently as a result of the realisation that more general frameworks can actually be considered, where the causal assumptions are weakened and in which noncausal correlations can be obtained Oreshkov et al. (2012). Investigations of causal versus noncausal correlations first focused on the simplest bipartite case Oreshkov et al. (2012); Branciard et al. (2016), and were soon extended to multipartite scenarios, where a much richer situation is found Baumeler and Wolf (2014); Baumeler et al. (2014); Oreshkov and Giarmatzi (2016); Abbott et al. (2016)—this opens, for instance, the possibility for causal correlations to be established following a dynamical causal order, where the causal order between events may depend on events occurring beforehand Hardy (2005). When analysing noncausal correlations in a multipartite setting, however, a natural question arises: is the noncausality of these correlations a truly multipartite phenomenon, or can it be reduced to a simpler one, that involves fewer parties? The goal of this paper is precisely to address this question, and provide criteria to justify whether one really deals with genuinely multipartite noncausality or not.
To make things more precise, let us start with the case of two parties, and . Each party receives an input , , and returns an output , , respectively. The correlations shared by and are described by the conditional probability distribution . If the two parties’ events (returning an output upon receiving an input) are embedded into a fixed causal structure, then one could have that causally precedes —a situation that we shall denote by , and where ’s output may depend on ’s input but not vice versa: —or that causally precedes —, where . (It can also be that the correlation is not due to a direct causal relation between and , but to some latent common cause; such a situation is however still compatible with an explanation in terms of or , and is therefore encompassed in the previous two cases.) A causal correlation is defined as one that is compatible with either or , or with a convex mixture thereof, which would describe a situation where the party that comes first is selected probabilistically in each run of the experiment Oreshkov et al. (2012); Brukner (2014).
Adding a third party with input and output , and taking into account the possibility of a dynamical causal order, a tripartite causal correlation is defined as one that is compatible with one party acting first—which one it is may again be chosen probabilistically—and such that whatever happens with that first party, the reduced bipartite correlation shared by the other two parties, conditioned on the input and output of the first party, is causal (see Definition 1 below for a more formal definition, and its recursive generalisation to parties) Oreshkov and Giarmatzi (2016); Abbott et al. (2016). In contrast, a noncausal tripartite correlation cannot for instance be decomposed as
(1) 
with bipartite correlations that are causal for each . Nevertheless, such a decomposition may still be possible for a tripartite noncausal correlation if one does not demand that (all) the bipartite correlations are causal. Without this constraint, the correlation (1) is thus compatible with the “coarsegrained” causal order , if and are grouped together to define a new “effective party” and act “as one”. This illustrates that although a multipartite correlation may be noncausal, there might still exist some definite causal order between certain subsets of parties; the intuition that motivates our work is that such a correlation would therefore not display genuinely multipartite noncausality.
This paper is organised as follows. In Sec. 2, we introduce the notion of genuinely partite noncausal correlations in opposition to what we call 2causal correlations, which can be established whenever two separate groups of parties can be causally ordered; we furthermore show how such correlations can be characterised via socalled 2causal inequalities. In Sec. 3, as an illustration we analyse in detail the simplest nontrivial tripartite scenario where these concepts make sense; we present explicit 2causal inequalities for that scenario, investigate their violations in the process matrix framework of Ref. Oreshkov et al. (2012), and generalise some of them to partite inequalities. In Sec. 4, we propose two possible generalisations of the notion of 2causal correlations, which we call causal and sizecausal correlations, respectively. This allows one to refine the analysis, and provides two different hierarchies of criteria that quantify the extent to which the noncausality of a correlation is a genuinely multipartite phenomenon.
2 Genuinely partite noncausal correlations
The general multipartite scenario that we consider in this paper, and the notations we use, are the same as in Ref. Abbott et al. (2016). 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 vectors of inputs and outputs are denoted by and . The correlations between the parties are given by the conditional probability distribution . For some (nonempty) subset of , we denote by and the vectors of inputs and outputs of the parties in ; with this notation, and (or simply and for a singleton ) denote the vectors of inputs and outputs of all parties that are not in . For simplicity we will identify the parties’ names with their labels, so that , and similarly for any subset .
2.1 Definitions
The assumption that the parties in such a scenario are embedded into a welldefined causal structure restricts the correlations that they can establish. In Refs. Oreshkov and Giarmatzi (2016); Abbott et al. (2016), the most general correlations that are compatible with a definite causal order between the parties were studied and characterised. Such correlations include those compatible with causal orders that are probabilistic or dynamical—that is, the operations of parties in the past can determine the causal order of parties in the future. These socalled causal correlations—which, for clarity, we shall often call fully causal here—can be defined iteratively in the following way:
Definition 1 ((Fully) causal correlations).

For , any valid probability distribution is (fully) causal;

For , an partite correlation is (fully) causal if and only if it can be decomposed in the form
(2) with for each , , where (for each ) is a singleparty probability distribution and (for each ) is a (fully) causal partite correlation.
As the tripartite example in the introduction shows, there can be situations in which no overall causal order exists, but where there still is a (“coarsegrained”) causal order between certain subsets of parties, obtained by grouping certain parties together. The correlations that can be established in such situations are more general than causal correlations, but nevertheless restricted due to the existence of this partial causal ordering. If we want to identify the idea of noncausality as a genuinely partite phenomenon, we should, however, exclude such correlations, and characterise correlations for which no subset of parties can have a definite causal relation to any other subset. This idea was already suggested in Ref. Abbott et al. (2016); here we define the concept precisely.
Note that if several different nonempty subsets do have definite causal relations to each other, then clearly there will be two subsets having a definite causal relation between them—one can consider the subset that comes first and group the remaining subsets together into the complementary subset, which then comes second. We shall for now consider partitions of into just two (nonempty) subsets and , and we thus introduce the following definition:
Definition 2 (causal correlations).
An partite correlation (for ) is said to be causal if and only if it can be decomposed in the form
(3) 
where the sum runs over all nonempty strict subsets of , with for each , , and where (for each ) is a valid probability distribution for the parties in and (for each ) is a valid probability distribution for the remaining parties.
For , the above definition reduces to the standard definition of bipartite causal correlations Oreshkov et al. (2012), which is equivalent to Definition 1 above. In the general multipartite case, it can be understood in the following way: each individual summand for each bipartition describes correlations compatible with all the parties in acting before all the parties in , since the choice of inputs for the parties in does not affect the outputs for the parties in . The convex combination in Eq. (3) then takes into account the possibility that the subset acting first can be chosen randomly.^{1}^{1}1One can easily see that it is indeed sufficient to consider just one term per bipartition in the sum (3). That is, for some given , some correlations and , and some weights with , the convex mixture is also of the same form (with and ). This already implies, in particular, that 2causal correlations form a convex set.
For correlations that are not 2causal, we introduce the following terminology:
Definition 3 (Genuinely partite noncausal correlations).
An partite correlation that is not 2causal is said to be genuinely partite noncausal.
Thus, genuinely partite noncausal correlations are those for which it is impossible to find any definite causal relation between any two (complementary) subsets of parties, even when taking into consideration the possibility that the subset acting first may be chosen probabilistically.
2.2 Characterisation of the set of 2causal correlations as a convex polytope
As shown in Ref. Branciard et al. (2016) for the bipartite case and in Refs. Oreshkov and Giarmatzi (2016); Abbott et al. (2016) for the general partite case, any fully causal correlation can be written as a convex combination of deterministic fully causal correlations. As the number of such deterministic fully causal correlations is finite (for finite alphabets of inputs and outputs), they correspond to the extremal points of a convex polytope—the (fully) causal polytope. The facets of this polytope are given by linear inequalities, which define socalled (fully) causal inequalities.
As it turns out, the set of causal correlations can be characterised as a convex polytope in the same way:
Theorem 4.
The set of 2causal correlations forms a convex polytope, whose (finitely many) extremal points correspond to deterministic 2causal correlations.
Proof.
For a given nonempty strict subset of , defines an “effectively bipartite” correlation, that is, a bipartite correlation between an effective party with input and output and an effective party with input and output , which are formed by grouping together all parties in the respective subsets. That effectively bipartite correlation is compatible with the causal order^{2}^{2}2The notation (or simply for singletons ), already used in the introduction, formally means that the correlation under consideration satisfies . It will also be extended to more subsets, with meaning that for all . . As mentioned above, the set of such correlations forms a convex polytope whose extremal points are deterministic, effectively bipartite causal correlations Branciard et al. (2016)—which, according to Definition 2, define deterministic 2causal partite correlations.
Eq. (2) then implies that the set of 2causal correlations is the convex hull of all such polytopes for each nonempty strict subset of ; it is thus itself a convex polytope, whose extremal points are indeed deterministic 2causal correlations. ∎
As any fully causal correlation is 2causal, but not vice versa, the fully causal polytope is a strict subset of what we shall call the 2causal polytope (see Fig. 1). Every vertex of the 2causal polytope corresponds to a deterministic function that assigns a list of outputs to the list of inputs , such that the corresponding probability distribution is 2causal, and thus satisfies Eq. (3). Since can only take values or , there is only one term in the sum in Eq. (3), and it can be written such that there is a single (nonempty) strict subset that acts first. That is, is such that the outputs of the parties in are determined exclusively by their inputs , while the outputs of the remaining parties are determined by all inputs . The facets of the 2causal polytope are linear inequalities that are satisfied by all 2causal correlations; we shall call these 2causal inequalities (see Fig. 1).
3 Analysis of the tripartite “lazy scenario”
In this section we analyse in detail, as an illustration, the polytope of 2causal correlations for the simplest nontrivial scenario with more than two parties. In Ref. Abbott et al. (2016) it was shown that this scenario is the socalled tripartite “lazy scenario”, in which each party receives a binary input , has a single constant output for one of the inputs, and a binary output for the other. By convention we consider that for each , on input the output is always , while for we take . The set of fully causal correlations was completely characterised for this scenario in Ref. Abbott et al. (2016), which will furthermore permit us to compare the noncausal and genuinely tripartite noncausal correlations in this concrete example.
As is standard (and as we did in the introduction), we will denote here the three parties by , , , their inputs , , , and their outputs , and . Furthermore, we will denote the complete tripartite probability distribution by [i.e., ] and the marginal distributions for the indicated parties by , , etc. [e.g., ].
3.1 Characterisation of the polytope of 2causal correlations
3.1.1 Complete characterisation
We characterise the polytope of 2causal correlations in much the same way as the polytope of fully causal correlations was characterised in Ref. Abbott et al. (2016), where we refer the reader for a more indepth presentation. Specifically, the vertices of the polytope are found by enumerating all deterministic 2causal probability distributions , i.e., those which admit a decomposition of the form (3) with (because they are deterministic) a single term in the sum (corresponding to a single group of parties acting first). One finds that there are such distributions, and thus vertices.
In order to determine the facets of the polytope, which in turn correspond to tight 2causal inequalities, a parametrisation of the 19dimensional polytope must be fixed and the convex hull problem solved. We use the same parametrisation as in Ref. Abbott et al. (2016), and again use cdd Fukuda (2012) to compute the facets of the polytope. We find that the polytope has facets, each corresponding to a 2causal inequality, the violation of which would certify genuinely tripartite noncausality. Many inequalities, however, can be obtained from others by either relabelling outputs or permuting parties, and as a result it is natural to group the inequalities into equivalence classes, or “families”, of inequalities. Taking this into account, we find that there are 476 families of facetinducing 2causal inequalities, 3 of which are trivial, as they simply correspond to positivity constraints on the probabilities (and are thus satisfied by any valid probability distribution). While the 2causal inequalities all detect genuinely partite noncausality, it is interesting to note that all except 22 of them can be saturated by fully causal correlations (and all but 37 even by correlations compatible with a fixed causal order).
We provide the complete list of these inequalities, organised by their symmetries and the types of distribution required to saturate them, in the Supplementary Material SM (), and will analyse in more detail a few particularly interesting examples in what follows. First, however, it is interesting to note that only 2 of the 473 nontrivial facets are also facets of the (fully) causal polytope for this scenario (one of which is Eq. (8) analysed below), and hence the vast majority of facetinducing inequalities of the causal polytope do not single out genuinely tripartite noncausal correlations. Moreover, none of the 2causal inequalities we obtain here differ from facetinducing fully causal inequalities only in their bound, and, except for the aforementioned cases, our 2causal inequalities thus represent novel inequalities.
3.1.2 Three interesting inequalities
Of the nontrivial 2causal inequalities, those that display certain symmetries between the parties are particularly interesting since they tend to have comparatively simple forms and often permit natural interpretations (e.g., as causal games Oreshkov et al. (2012); Branciard et al. (2016)).
For example, three nontrivial families of 2causal inequalities have forms (i.e., certain versions of the inequality within the corresponding equivalence class) that are completely symmetric under permutations of the parties. One of these is the inequality
(4) 
which can be naturally expressed as a causal game. Indeed, it can be rewritten as
(5) 
where if , if (i.e., , where denotes addition modulo 2), and similarly for and , and where it is implicitly assumed that all inputs occur with the same probability. This can be interpreted as a game in which the goal is to collaborate such that the product of the nontrivial outputs (i.e., those corresponding to an input 1) is equal to the product of the inputs, and where the former product is taken to be 1 if all inputs are 0 and there are therefore no nontrivial outputs (in which case the game will always be lost). The probability of success for this game can be no greater than if the parties share a 2causal correlation. This bound can easily be saturated by a deterministic, even fully causal, distribution: if every party always outputs 0 then the parties will win the game in all cases, except when the inputs are all 0 or all 1.
Another partypermutationsymmetric 2causal inequality is the following:
(6) 
whose interpretation can be made clearer by rewriting it as
(7) 
The lefthand side of this inequality is simply the sum of three terms corresponding to conditional “lazy guess your neighbour’s input” (LGYNI) inequalities Abbott et al. (2016), one for each pair of parties (conditioned on the remaining party having input ), while the negative bound on the righthand side accounts for the fact that any pair of parties that are grouped together in a bipartition may maximally violate the LGYNI inequality between them (and thus reach the minimum algebraic bound ). This inequality can be interpreted as a “scored game” (as opposed to a “winorlose game”) in which each pair of parties scores one point if they win their respective bipartite LGYNI game and the third party’s input is 0, and where the goal of the game is to maximise the total score, given by the sum of all three pairs’ individual scores. The best average score (when the inputs are uniformly distributed) for a 2causal correlation is , corresponding to the 2causal bounds of in Eq. (3.1.2) and in Eq. (3.1.2).^{3}^{3}3The bound of these inequalities, and the best average score of the corresponding game, can be reached by a 2causal strategy in which one party, say , has a fixed causal order with respect to the other two parties grouped together, who share a correlation maximally violating the corresponding LGYNI inequality. For example, the distribution , where is the Kronecker delta function, is compatible with the order (or with ) and saturates Eqs. (3.1.2) and (3.1.2). It is also clear from the form of Eq. (3.1.2) that for fully causal correlations the lefthand side is lowerbounded by . This inequality is thus amongst the 22 facetinducing 2causal inequalities that cannot be saturated by fully causal distributions.
In addition to the inequalities that are symmetric under any permutation of the parties, there are four further nontrivial families containing 2causal inequalities which are symmetric under cyclic exchanges of parties. One interesting such example is the following:
(8) 
This inequality can again be interpreted as a causal game in the form (where we again implicitly assume a uniform distribution of inputs for all parties)
(9) 
where the goal of the game is for each party, whenever they receive the input 1 and their righthand neighbour has the input 0, to output the input of their lefthand neighbour (with being considered, in a circular manner, to be to the left of ).^{4}^{4}4The bound of on the probability of success can, for instance, be reached by the fully causal (and hence 2causal) distribution , compatible with the order , which wins the game in all cases except when . This inequality is of additional interest as it is one of the two nontrivial inequalities which is also a facet of the standard causal polytope for this scenario. (The second such inequality, which lacks the symmetry of this one, is presented in the Supplementary Material SM ().)
3.2 Violations of 2causal inequalities by process matrix correlations
One of the major sources of interest in causal inequalities has been the potential to violate them in more general frameworks, in which causal restrictions are weakened. There has been a particular interest in one such model, the process matrix formalism, in which quantum mechanics is considered to hold locally for each party, but no global causal order between the parties is assumed Oreshkov et al. (2012). In this framework, the (possibly noncausal) interactions between the parties are described by a process matrix , which, along with a description of the operations performed by the parties, allows the correlations to be calculated.
It is wellknown that process matrix correlations can violate causal inequalities Abbott et al. (2016); Baumeler and Wolf (2014); Baumeler et al. (2014); Branciard et al. (2016); Oreshkov et al. (2012), although the physical realisability of such processes remains an open question Araújo et al. (2017); Feix et al. (2016). In Ref. Abbott et al. (2016) it was shown that all the nontrivial fully causal inequalities for the tripartite lazy scenario can be violated by process matrices. However, for most inequalities violation was found to be possible using process matrices that are compatible with acting last, which means the correlations they produced were necessarily 2causal. It is therefore interesting to see whether process matrices are capable of violating 2causal inequalities in general, and thus of exhibiting genuinely partite noncausality. We will not present the process matrix formalism here, and instead simply summarise our findings; we refer the reader to Refs. Araújo et al. (2015); Oreshkov et al. (2012) for further details on the technical formalism.
Following the same approach as in Refs. Branciard et al. (2016); Abbott et al. (2016) we looked for violations of the 2causal inequalities. Specifically, we focused on twodimensional (qubit) systems and applied the same “seesaw” algorithm to iteratively perform semidefinite convex optimisation over the process matrix and the instruments defining the operations of the parties.
As a result, we were able to find process matrices violating all but 2 of the 473 nontrivial families of tight 2causal inequalities (including Eqs. (3.1.2) and (8) above) using qubits, and in all cases where a violation was found, the best violation was given by the same instruments that provided similar results in Ref. Abbott et al. (2016). We similarly found that 284 families of these 2causal inequalities (including Eq. (8)) could be violated by completely classical process matrices,^{5}^{5}5Incidentally, exactly the same number of families of fully causal inequalities were found to be violable with classical process matrices in Ref. Abbott et al. (2016). It remains unclear whether this is merely a coincidence or the result of a deeper connection. a phenomenon that is not present in the bipartite scenario where classical processes are necessarily causal Oreshkov et al. (2012).
While the violation of 2causal inequalities is again rather ubiquitous, the existence of two inequalities for which we found no violation is curious. One of these inequalities is precisely Eq. (3.1.2), and its decomposition in Eq. (3.1.2) into three LGYNI inequalities helps provide an explanation. In particular, the seemingly best possible violation of a (conditional) LGYNI inequality using qubits is approximately Abbott et al. (2016); Branciard et al. (2016), whereas it is clear that a process matrix violating Eq. (3.1.2) must necessarily violate a conditional LGYNI inequality between one pair of parties by at least . Moreover, in Ref. Branciard et al. (2016) it was reported that no better violation was found using three or fourdimensional systems, indicating that Eq. (3.1.2) can similarly not be violated by such systems. It nonetheless remains unproven whether such a violation is indeed impossible, and the convex optimisation problem for three parties quickly becomes intractable for higher dimensional systems, making further numerical investigation difficult. The second inequality for which no violation was found can similarly be expressed as a sum of three different forms (i.e., relabellings) of a conditional LGYNI inequality, and a similar argument thus explains why no violation was found. Recall that, as they can be expressed as a sum of three conditional LGYNI inequalities with a negative 2causal bound, these two 2causal inequalities cannot be saturated by fully causal distributions; it is interesting that the remaining inequalities that require noncausal but 2causal distributions to saturate can nonetheless be violated by process matrix correlations.
3.3 Generalised 2causal inequalities for parties
Although it quickly becomes intractable to completely characterise the 2causal polytope for more complicated scenarios with more parties, inputs and/or outputs, as is also the case for fully causal correlations, it is nonetheless possible to generalise some of the 2causal inequalities into inequalities that are valid for any number of parties .
The inequality (3.1.2), for example, can naturally be generalised to give a 2causal inequality valid for all .^{6}^{6}6We continue to focus on the lazy scenario defined earlier for concreteness, but we note that the proofs of the generalised inequalities (3.3) and (12) in fact hold in any nontrivial scenario, of which the lazy one is the simplest example. The bounds for the corresponding causal games and whether or not the inequalities define facets will, however, generally depend on the scenario considered. Specifically, one obtains
(10) 
where and , which can be written analogously to Eq. (5) as a game (again implicitly defined with uniform inputs) of the form
(11) 
We leave the proof of this inequality and its 2causal bound to Appendix A. It is interesting to ask if this inequality is tight (i.e., facet inducing) for all . For it reduces to the LGYNI inequality which is indeed tight, and for it was also found to be a facet. By explicitly enumerating the vertices of the 2causal polytope for (of which there are ) we were able to verify that is indeed also a facet, and we conjecture that this is true for all . Note that, as for the tripartite case it is trivial to saturate the inequality for all by considering the (fully causal) strategy where each party always outputs 0.
It is also possible to generalise inequality (3.1.2) to parties—which will prove more interesting later—by considering a scored game in which every pair of parties gets one point if they win their respective bipartite LGYNI game and all other parties’ inputs are 0, and the goal of the game is to maximise the total score of all pairs. If two parties belong to the same subset in a bipartition, then they can win their respective LGYNI game perfectly, whereas they are limited by the causal bound if they belong to two different groups. The 2causal bound on the inequality is thus given by the maximum number of pairs of parties that belong to a common subset over all bipartitions, times the maximal violation of the bipartite LGYNI inequality. Specifically, we obtain the 2causal inequality
(12) 
where is a binomial coefficient and
(13) 
Each term defines a bipartite conditional LGYNI inequality with the causal bound , and the minimum algebraic bound (i.e. the maximal violation) . The minimum algebraic bound of is thus . The validity of inequality (12) for 2causal correlations (which corresponds to a maximal average score of —compared to the maximal algebraic value of —for the corresponding game with uniform inputs) is again formally proved in Appendix A.
We note that in contrast to Eq. (3.3), is not a facet of the 4partite 2causal polytope, and thus the inequality is not tight in general. Inequality (12) can nonetheless be saturated by 2causal correlations for any . For example, consider and take the distribution
(14) 
with if contains exactly two inputs 1, and otherwise. is clearly 2causal since it is compatible with the causal order (indeed, also with ). One can then easily verify that saturates (12), since all pairs of parties in can win their respective conditional LGYNI game perfectly, and therefore contribute with a term of to the sum in Eq. (12).
4 Refining the definition of genuinely multipartite noncausal correlations
So far we only discussed correlations that can or cannot arise given a definite causal order between two subsets of parties. It makes sense to consider more refined definitions that discriminate, among noncausal correlations, to what extent and in which way they represent a genuinely multipartite resource. The idea will again be to see if a given correlation can be established by letting certain groups of parties act “as one”, while retaining a definite causal order between different groups. The number and size of the groups for which this is possible can be used to give two distinct characterisations of how genuinely multipartite the observed noncausality is.
4.1 causal correlations
We first want to characterise the correlations that can be realised when a definite causal order exists between certain groups of parties, while no constraint is imposed on the correlations within each group.
Let us consider for this purpose a partition of —i.e., a set of nonempty disjoint subsets of , such that , see Fig. 2. Note that if contains at least two subsets, then for a given subset , also represents a partition of . Let us then introduce the following definition:
Definition 5 (causal correlations).
For a given partition of , an partite correlation is said to be causal if and only if is causal when considered as an effective partite correlation, where each subset in defines an effective party.
More precisely, analogously to Definition 1:

For , any partite correlation is causal;

For , an partite correlation is causal if and only if it can be decomposed in the form
(15) with for each , , where (for each ) is a valid probability distribution for the parties in and (for each ) is a causal correlation for the remaining parties.
In the extreme case of a singleset partition (), any correlation is by definition trivially causal; at the other extreme, for a partition of into singletons (), the definition of causal correlations above is equivalent to that of fully causal correlations, Definition 1 Oreshkov and Giarmatzi (2016); Abbott et al. (2016). Between these two extreme cases, a causal correlation identifies the situation where, with some probability, all parties within one group act before all other parties; conditioned on their inputs and outputs, another group acts second (before all remaining parties) with some probability; and so on. We emphasise that no constraint is imposed on the correlations that can be generated within each group, since we allow them to share the most general resource conceivable—in particular, there might be no definite causal order between the parties inside a group.
Since the definition of causal correlations above matches that of causal correlations for the effective parties defined by , all basic properties of causal correlations (see Ref. Abbott et al. (2016)) generalise straightforwardly to causal correlations. Note in particular that the definition captures the idea of dynamical causal order, where the causal order between certain subsets of parties in may depend on the inputs and outputs of other subsets of parties that acted before them. The following result also follows directly from what is known about causal correlations Oreshkov and Giarmatzi (2016); Abbott et al. (2016):
Theorem 6.
For any given , the set of causal correlations forms a convex polytope, whose (finitely many) extremal points correspond to deterministic causal correlations.
We shall call this polytope the causal polytope; its facets define causal inequalities. Theorem 6 implies that any causal correlation can be obtained as a probabilistic mixture of deterministic causal correlations. It is useful to note that, similarly to Ref. Abbott et al. (2016), deterministic causal correlations can be interpreted in the following way: a set of parties acts with certainty before all others, with their outputs being a deterministic function of all inputs in that set but independent of the inputs of any other parties, . The inputs of the first set also determine which set comes second, , where , whose outputs can depend on all inputs of the first and second sets; and so on, until all the sets in the partition are ordered. As one can see, each possible vector of inputs thus determines (in a not necessarily unique way) a given causal order for the sets of parties in .
4.2 Noninclusion relations for causal polytopes
As suggested earlier, our goal is to quantify the extent to which a noncausal resource is genuinely multipartite in terms of the number or size of the subsets one needs to consider in a partition to make a given correlation causal. A natural property to demand of such a quantification is that it defines nested sets of correlations: if a correlation is genuinely multipartite noncausal “to a certain degree”, it should also be contained in the sets of “less genuinely multipartite noncausal” correlations (and, eventually, the set of simply noncausal correlations). It is therefore useful, before providing the relevant definitions in the next subsections, to gather a better understanding of the inclusion relations between causal polytopes.
One might intuitively think that there should indeed be nontrivial inclusion relations among those polytopes. For example, one might think that a causal correlation should also be causal if is a “coarsegraining” of (i.e., is obtained from by grouping some of its groups to define fewer but larger subsets)—or, more generally, when contains fewer subsets than , i.e. . This, however, is not true. For example, in the tripartite case, a fully causal correlation (i.e., a causal one for ) compatible with the fixed order , where comes between and , may not be causal for , since one cannot order with respect to when those are taken together. In fact, no nontrivial inclusion exists among causal polytopes, as established by the following theorem, proved in Appendix B.
Theorem 7.
Consider an partite scenario where each party has at least two possible inputs and at least two possible outputs for one value of the inputs. Given two distinct nontrivial^{7}^{7}7If one of the two partitions is trivial, say , then the causal polytope is of course contained in the trivial causal one (which contains all valid probability distributions). Note that for there is only one nontrivial partition; the theorem is thus only relevant for scenarios with . partitions and of with , the causal polytope is not contained in the causal one, nor vice versa.
One may also ask whether, for a given causal correlation , there always exists a partition with such that is also causal (recall that the case is trivial). The answer is negative when mixtures of different causal orders are involved: e.g., in the tripartite case with , a fully causal correlation of the form , where each correlation in the sum is compatible with the corresponding causal order, may not be causal for any of the form , as there is always a term in above for which comes between and . For an explicit example one can take the correlation above to be a mixture of 6 correlations introduced in Appendix B.^{8}^{8}8To see that thus defined is indeed not causal for any such bipartition, first note that, by symmetry, it suffices to show it is not causal for . One can readily show that all such causal inequalities must obey the LGYNItype inequality (which, moreover, is a facet of the causal polytope). It is easily verified that violates this inequality with the lefthand side obtaining the value .
The above results tell us that causal polytopes do not really define useful classes to directly quantify how genuinely multipartite the noncausality of a correlation is. One may wonder whether considering convex hulls of causal polytopes allows one to avoid these issues. For example, is it the case that any causal correlation is contained in the convex hull of all causal correlations for all partitions with a fixed value of ?^{9}^{9}9Note that a convex combination of causal correlations for various partitions with a fixed number of subsets is not necessarily causal for any single partition with the same value of . For this is trivial, and this remains true for : any causal correlation can be decomposed as a convex combination of causal correlations for various partitions with . Eq. (15) is indeed such a decomposition, with the partitions . This is also true, for any value of , for causal correlations that are compatible with a fixed causal order between the subsets in (or convex mixtures thereof): indeed, such a correlation is also causal for any coarsegrained partition of where consecutive subsets (as per the causal order in question, or per each causal order in a convex mixture) of are grouped together. However, this is not true in general for when dynamical causal orders are involved. It is indeed possible to find a 4partite, fully causal correlation that cannot be expressed as a convex combination of causal correlations with all ; an explicit counterexample is presented in Appendix C.
From these observations we conclude that, although grouping parties into subsets seems to be a stronger constraint than grouping parties into some subsets, the fact that a correlation is causal for some (or more generally, that it is a convex combination of various causal correlations with all ) does not guarantee that it is also causal for some —unless (or , trivially)—nor that it can be decomposed as a convex combination of causal correlations with all . In particular, fully causal correlations may not be causal for any with , or convex combinations of such causal correlations. This remark motivates the definitions in the next subsection.
4.3 causal correlations
4.3.1 Definition and characterisation
With the previous discussion in mind, we propose the following definition, as a first refinement between the definitions of fully causal and 2causal correlations.
Definition 8 (causal correlations).
An partite correlation is said to be causal (for ) if and only if it is a convex combination of causal correlations, for various partitions of into subsets.
More explicitly: is causal if and only if it can be decomposed as
(16) 
where the sum is over all partitions of into subsets or more, with for each , , and where each is a causal correlation.
For , any correlation is trivially 1causal, since for any correlation is causal. For , the definition of causal correlations above is equivalent to that of fully causal correlations, Definition 1 Oreshkov and Giarmatzi (2016); Abbott et al. (2016).
For , the above definition is equivalent to that of 2causal correlations as introduced through Definition 2. To see this, recall first (from the discussion in the previous subsection), that any causal correlation with can be written as a convex combination of some causal correlations, for various bipartitions with . It follows that, for , it would be equivalent to have the condition instead of in Definition 8 of causal correlations. Definition 2 is then recovered when writing the bipartitions in the decomposition as , using Eq. (15) from the definition of causal correlations, and rearranging the terms in the decomposition. Hence, Definition 2 is in fact equivalent to saying that 2causal correlations are those that can be written as a convex mixture of causal correlations, for different partitions of into subsets, thus justifying further our definition of genuinely partite noncausal correlations as those that cannot be written as such a convex mixture (or equivalently, those that are not causal for any ). Note that since we used the constraint rather than in Eq. (16),^{10}^{10}10Replacing the condition by in Definition 8 for arbitrary , we could define “causal correlations”, which would be distinct from causal correlations for . We would also have that “causal correlations” form a convex polytope; however, the various “causal polytopes” would not necessarily be included in one another for distinct values of , as discussed in the previous subsection. our definition establishes a hierarchy of correlations as desired, with causal causal if .
With the above definition of causal correlations, we have the following:
Theorem 9.
For any given value of (with ), the set of causal correlations forms a convex polytope, whose (finitely many) extremal points correspond to deterministic causal correlations, for all possible partitions with —that is, deterministic causal correlations.
Proof.
According to Eq. (16), the set of causal correlations is the convex hull of the polytopes of causal correlations with . Since there is a finite number of such polytopes, the set of causal correlations is itself a convex polytope; its extremal points are those of the various causal polytopes with , namely deterministic causal correlations (see Theorem 6). ∎
We thus obtain a family of convex polytopes—which we shall call causal polytopes—included in one another, see Fig. 3. The facets of these polytopes are causal inequalities, which define a hierarchy of criteria: e.g., if all causal inequalities are satisfied, then so are all causal inequalities if —or equivalently: if some causal inequality is violated, then some causal inequality must also be violated if . Given a correlation , one can in principle test to which set it belongs. The largest for which is causal can be used as a measure of how genuinely multipartite its noncausality is: it means that can be obtained as a convex combination of causal correlations with all , but not with all —indeed, if then violates some causal inequality for any (with ). If that is , is a genuinely partite noncausal correlation; if it is , then is fully causal, hence it displays no noncausality (genuinely multipartite or not).
4.3.2 A family of causal inequalities
The general partite 2causal inequality (12) can easily be modified to give an causal inequality that is valid—although not tight in general, as observed before—for all and (with ), simply by changing the bound. Indeed, this bound is derived from the largest possible number of pairs of parties that can be in a single subset of a given partition, and this can easily be recalculated for subset partitions rather than bipartitions. We thus obtain that
(17) 
for any causal correlation. This updated bound is proved in Appendix A.
As for Eq. (12) it is easy to see that, for any , there are causal correlations saturating the inequality (17). Consider, for instance, the partition of with and for , and take the distribution
(18) 
where we use the same function as in Eq. (14). Analogous reasoning shows that this correlation indeed reaches the bound (17).
Since this (reachable) lower bound is different for each possible value of , this implies, in particular, that (for the partite lazy scenario) all the inclusions causal causal 3causal 2causal in the hierarchy of causal polytopes are strict. In fact, redas for inequalities (3.3) and (12) (see Footnote 6), the proof of Eq. (17) holds in any nontrivial scenario (with arbitrarily many inputs and outputs), of which the lazy scenario is the simplest example for all . Moreover, one can saturate it in such scenarios by trivially extending the causal correlation (18) (e.g., by producing a constant output on all other inputs) and thus these inclusions are strict in general.
4.4 Sizecausal correlations
In the previous subsection we used the number of subsets needed in a partition to quantify how genuinely multipartite the noncausality of a correlation is. Here we present an alternative quantification, based on the size of the biggest subset in a partition, rather than the number of subsets.
Intuitively, the bigger the subsets in a partition needed to reproduce a correlation, the more genuinely multipartite noncausal the corresponding causal correlations are. However, the discussion of Sec. 4.2 implies that, as was the case with causal correlations, it is not sufficient to simply ask whether a given correlation is causal for some partition with subsets of a particular size. We therefore focus on classes of correlations that can be written as mixtures of causal ones whose largest subset is not larger than some number . For convenience, we introduce the notation
(19) 
We then take the following definition:
Definition 10 (Sizecausal correlations).
An partite correlation is said to be sizecausal (for ) if and only if it is a convex combination of causal correlations, for various partitions whose subsets are no larger than .
More explicitly: is sizecausal if and only if it can be decomposed as
(20) 
where the sum is over all partitions of with no subset of size larger than , with for each , , and where each is a causal correlation.
Any partite correlation is trivially sizecausal, while sizecausal correlations coincide with fully causal correlations. Furthermore, noting that if and only if , we see that the set of sizecausal correlations coincides with that of causal correlations. Hence, the definition of sizecausal correlations is another possible generalisation of that of 2causal ones. From this new perspective, causal correlations can be seen as those that can be realised using (probabilistic mixtures of) noncausal resources available to groups of parties of size or less. This further strengthens the definition of causal correlations as the largest set of correlations that do not possess genuinely partite noncausality.
Without repeating in full detail, it is clear that sizecausal correlations define a structure similar to that of causal correlations: for each , sizecausal correlations define sizecausal polytopes whose vertices are deterministic sizecausal correlations and whose facets define sizecausal inequalities. For , all sizecausal correlations are also sizecausal, so that the various sizecausal polytopes are included in one another. The lowest for which a correlation is sizecausal also provides a measure of how genuinely multipartite the corresponding noncausal resource is, distinct to that defined by causal correlations.
It is also possible here to generalise inequality (12) to sizecausal correlations by changing the bound. As proven in Appendix A, we thus obtain the sizecausal inequality
(21) 
(where denotes the largest integer smaller than or equal to ). Although, once again, this inequality is not tight in the sense that it does not define a facet of the sizecausal polytope, its lower bound can be saturated by a sizecausal correlation for each value of , for instance by considering the partition of into groups of parties, and (if is not a multiple of ) a last group with the remaining parties, and by taking the deterministic correlation
(22) 
(with again the same function as in Eq. (14)). Since the (reachable) lower bounds in Eq. (