# Information-processing in theories constrained by no superluminal causation vs no superluminal signalling

V. Vilasini, Roger Colbeck

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V. Vilasini, Roger Colbeck. Information-processing in theories constrained by no superluminal causation vs no superluminal signalling. 2025. hal-04885726

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# Information-processing in theories constrained by no superluminal causation vs no superluminal signalling

V. Vilasini¹,\* and Roger Colbeck²,†

¹Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland

²Department of Mathematics, University of York, Heslington, York YO10 5DD.

(Dated: 21st February, 2024)

Relativistic causality principles constrain information processing possibilities in space-time. No superluminal causation (NSC) and no superluminal signaling (NSS) are two such principles which, although related, are distinct. In this work we study the consequence of these principles by considering the tasks of generating non-classical correlations within two space-time configurations. Considering theories constrained by NSC, we show that the first task is impossible in any classical theory and the second is impossible in any (possibly non-classical) theory. However, we construct a protocol enabling non-classical correlations to be generated in both configurations in a theory restricted by the weaker NSS principle. To do so we exploit theories that allow an effect called jamming. In our realisation, non-communicating agents sharing classical resources and assisted by jamming, can generate PR-box correlations. Using this protocol the violation of NSC without violating NSS would be verifiable. We discuss the implications of these findings for the speed of generation of non-classical correlations. Our work offers insights into the differences in information processing power of theories constrained by NSC, NSS and other relativistic causality principles.

## I. INTRODUCTION

Relativistic causality principles shape the landscape of information processing in space and time. One such fundamental principle is no superluminal causation (NSC), which constrains information flow between different space-time events. The principle of no superluminal signalling (NSS) is a distinct, yet related causality principle, constraining the ability of agents at different space-time locations to communicate through interventions.

In the context of relativistic causality in tripartite Bell scenarios, so-called jamming non-local theories were proposed in [1]. Jamming theories enable a party (say Bob) to influence the correlations between the measurement outcomes of space-like separated parties (Alice and Charlie) without influencing the individual outcomes. It has been suggested [1, 2] that such theories do not allow for superluminal signalling in Minkowski space-time. The role of other relativistic causality principles in these theories, such as NSC have remained little understood.

In recent works [3, 4], we have developed a general framework in which distinct relativistic causality principles can be fully formalised within a broad class of (possibly non-classical) theories. Applying this to jamming theories, we showed in [5] that these theories necessarily violate the NSC principle. Moreover, we found that under certain circumstances, jamming theories may also violate NSS (contrary to previous claims), and characterised the necessary and sufficient conditions under which jamming would adhere to the NSS principle. These conditions enable an identification of a concrete set of scenarios that violate NSC but respect NSS.

Building on this foundation, we investigate the information processing consequences of the NSC and NSS principles in different theories, highlighting their distinctions. In particular, we consider the generation of non-classical correlations in two space-time configurations. In the first, the task is impossible to achieve in any *classical* theory constrained by NSC, and in the second it is impossible to achieve in *any* theory (possibly non-classical) constrained by NSC. Although impossible under the NSC principle, we explore the possibility of these tasks in a theory constrained by the weaker principle of NSS.

We then investigate these tasks in jamming theories, finding that both tasks can be perfectly achieved in such theories without violating NSS. Specifically, we provide a protocol by which space-like separated agents Alice and Charlie sharing classical resources can generate maximally non-classical PR box correlations, in theories that permit a third agent Bob to jam Alice and Charlie's correlations. Crucially this only requires Alice and Charlie to communicate to Bob (with no other communication involved) and surprisingly, remains possible in a space-time configuration where the measurement outcomes can be arbitrarily earlier in time with respect to the corresponding measurement settings, in some reference frame. In light of these results, we discuss the ability to operationally certify the violation of NSC in a theory, without violating NSS and also discuss the speed of generating non-classical correlations in different scenarios.

We begin by providing an overview of jamming and of our causality framework for formalising NSC and NSS principles, before presenting the tasks and results.

## II. JAMMING NON-LOCAL CORRELATIONS

Consider a tri-partite Bell scenario where Alice, Bob and Charlie measure some shared system $\Lambda$ using freely

<footnote>\* vilasini@phys.ethz.ch</footnote>

<footnote>† roger.colbeck@york.ac.uk</footnote>
chosen settings A, B and C to obtain outcomes X, Y and Z respectively, and suppose that the measurements of the three parties occur in a space-time configuration where they are pairwise space-like separated. The standard tri-partite non-signalling conditions on the correlations $P(XYZ|ABC)$ which are inspired by the requirement of forbidding superluminal signalling in this space-time configuration require that the outcomes of any set of parties are independent of the settings of the complementary set of parties, in particular $P(XZ|ABC) = P(XZ|AC)$ holds.

If we are guaranteed that the joint future of the space-time locations where X and Z are embedded is contained in the future of the space-time location where B is embedded, then [1] propose that $P(XZ|ABC) = P(XZ|AC)$ is no longer necessary for avoiding superluminal signalling since the correlations $P(XZ|ABC)$ can only be computed in the joint future of the space-time locations of X and Z which is already in the future of the location of B. This leads to the concept of jamming non-local correlations, which allows Bob's setting to be jointly but not individually correlated with Alice and Charlie's outcome, allowing Bob to jam the correlations between Alice and Charlie depending on his setting. For the present paper, we consider B jamming X and Z, in which case jamming correlations are characterised by the conditions

$$ 
\begin{align} 
P(XZ|B) &\neq P(XZ) \nonumber \\ 
P(X|B) &= P(X) \tag{1} \\ 
P(Z|B) &= P(Z). \nonumber 
\end{align} 
$$

### III. CAUSAL STRUCTURES AND THEIR COMPATIBILITY WITH SPACE-TIME

Following the framework of [3], we describe information-theoretic causal structures (henceforth simply referred to as causal structures) through an approach based on causal modelling and inference.<fnref n="1" /> Here, causal structures are modelled as directed graphs with directed edges $\rightsquigarrow$, and each node can either be an observed node (which corresponds to a classical variable such as a measurement setting or outcome) or an unobserved node (which can be a classical, quantum or a post-quantum system depending on the underlying theory).

In a classical causal model over a causal structure $\mathcal{G}$, all nodes correspond to classical variables and the model is obtained by specifying, for each node $N$, a function $f_N$ that maps the values of the random variables that are parents of $N$ to the possible values of $N$ or, if $N$ is parentless, a probability distribution $P(N)$. In quantum

causal models, the deterministic functions are replaced with quantum channels, and in post-quantum theories, they are replaced with maps that describe the informational channels of that theory. We refer to these (functions, quantum channels or maps) as the *causal mechanisms* of the theory.

Causal models defined on acyclic causal structures are known to satisfy the *d-separation property* [6, 7]. Given a causal structure $\mathcal{G}$ this property allows one to read off conditional independences of any observed probability distribution, $P_G(N_{obs})$, associated with the observed nodes of $\mathcal{G}$ from the graph (see Appendix A for more detail).

As well as accounting for correlations, the framework we have developed in [3] also allows for interventions, which are needed to distinguish correlation and causation. An intervention takes a set $X$ of observed nodes in a causal structure $\mathcal{G}$ and associates them with a new causal structure $G_{do}(X)$, obtained from $\mathcal{G}$ by removing all incoming edges to $X$. This captures that the intervention on $X$ is freely chosen, i.e., independent of prior causes of $X$. For two disjoint subsets $X$ and $Y$ of observed nodes, we say that $X$ affects $Y$ if there exists a value $x$ of $X$ such that

$$ P_{G_{do}(X)}(Y|X=x) \neq P_G(Y). \quad (2) $$

This encapsulates the idea that the probability of $Y$ can be changed by intervening on $X$, and captures the ability of agents to signal to each other in an information-processing protocol. $X$ affects $Y$ implies that $X$ is a cause of $Y$ but the converse is not true, i.e., it is possible to have *causation without signalling*.

Causal models can be embedded in space-time by assigning to each observed node $X$, a space-time location, which leads to a corresponding space-time random variable $\mathcal{X}$.<fnref n="2" /> We use $\mathcal{X} \prec \mathcal{Y}$ to denote that $\mathcal{Y}$ is in the future light-cone of $\mathcal{X}$, and $\bar{\mathcal{F}}(\mathcal{X})$ to denote the inclusive future i.e., points in the future light-cone of $\mathcal{X}$ and its own space-time location.

Ref. [3] provides a mathematical condition specifying when a causal model is *compatible* with a given space-time embedding. This captures the requirement that the embedded causal model does not lead to superluminal signalling. The formalism of [3] highlights that causation and signalling are not equivalent, and hence NSC and NSS are also not equivalent. [Formal definitions of NSC and NSS can be found in Appendix B.] We now explore the consequences of this inequivalence for information-theoretic tasks in theories constrained by either NSC or NSS.

<footnote n="1">¹ Causal modelling and causal inference was originally developed in classical statistics [6] and recently generalised to quantum and other non-classical theories (e.g., [7, 8]).</footnote>

<footnote n="2">² For this purpose it is sufficient to model the space-time as a partially ordered set $\mathcal{T}$, where the partial order $\prec$ is specified by the light-cone structure [3].</footnote>
#### IV. TASKS THAT ARE IMPOSSIBLE IN NSC THEORIES

We consider a primary task $T$ relating to the generation of non-classical correlations, and define two sub-tasks 1 and 2 by instantiating $T$ in two different classes of space-time configurations.

**Task (Task $T$).** The task involves two agents, Alice and Charlie. Each agent owns a device which takes a freely chosen variable as input, producing a corresponding output variable. The input variables of Alice and Charlie are $A$ and $C$, while their output variables are $X$ and $Z$ respectively. The two devices may act on a shared system $\Lambda$. The task is for Alice and Charlie to generate non-classical correlations i.e., a distribution $P(XZ|AC)$ that cannot be expressed as follows for any variable $V$

$$P(XZ|AC) = \sum_V P(V)P(X|AV)P(Z|CV). \quad (3)$$

Alice and Charlie may collaborate with a third agent Bob with whom they may share systems.

So far, there are no constraints on communication between Alice and Charlie and $T$ can be trivially achieved using classical communication.

**Task 1.** Here, Alice and Charlie are required to achieve task $T$ given that variables $A, C, X$ and $Z$ are embedded in an acyclic space-time $T$ such that all four variables are spacelike separated except that $A \prec X$ and $C \prec Z$ (Figure 1a).

**Task 2.** Here, Alice and Charlie are required to achieve task $T$ given that $A, C, X$ and $Z$ are embedded in an acyclic space-time $T$ such that $\overline{F}(X) \cap \overline{F}(Z) \subseteq \overline{F}(A) \cap \overline{F}(C)$ holds, but $A \not\prec X$ and $C \not\prec Z$ (Figure 1b).

In both cases above, the shared system $\Lambda$ must not be embedded in the future of the inputs $A$ or $C$. Then, the following theorem shows that there are limitations to achieving these tasks in theories restricted by the principle of no superluminal causation.

**Theorem 1.** *Task 1 is impossible to achieve in any classical theory without superluminal causation, while Task 2 is impossible to achieve in any theory (classical or otherwise) without superluminal causation. This holds even when allowing for arbitrary collaboration strategies involving additional agents.*

The statement for 1 is in line with what one would expect from Bell's theorem [9–11]. On the other hand, for Task 2, we show a stronger statement, that in an NSC theory where $A$ and $C$ are freely chosen, it is impossible to generate any correlations between $\{A, C\}$ and $\{X, Z\}$ in the space-time configuration of 2 (Figure 1b) i.e., $P(XZ|AC) = P(XZ)$ always holds in such a theory. This implies the above theorem, a proof of which can be found in Appendix C.

#### V. ACHIEVING THE TASKS IN NSS THEORIES

The following theorem shows that the above tasks which are impossible in NSC theories become possible in NSS theories.

**Theorem 2.** *Tasks 1 and 2 can both be achieved in a classical theory that allows for jamming, without violating NSS and using the assistance of one additional agent Bob. Alice and Charlie only send classical communication to Bob, and exchange no communication otherwise. In particular, the maximally non-classical, PR box correlations can be generated in this manner.*

We now describe a protocol that achieves the above.

**Protocol 1.** Consider an additional agent Bob (associated with a variable $B$) who can jam the correlations between Alice and Charlie's output variables $X$ and $Z$, through his choice of $B$. All variables are taken to be binary. Alice and Charlie communicate their classical inputs $A$ and $C$ to Bob, who sets $B = A \cdot C$. Then through jamming, Bob can ensure that $B$ determines the parity of $X$ and $Z$ without being correlated individually with $X$ or $Z$, i.e., $B = X \oplus Z$. The result is $X \oplus Z = A \cdot C$ which correspond to PR-box correlations $P(XZ|AC)$.

By modifying the associated causal model other non-classical correlations are also possible as explained in Appendix C.

The jamming required in this protocol can be achieved through a classical common cause $\Lambda$ influencing $X$ and $Z$ as well as a superluminal causal influence from $B$ to $X$ or $Z$ [5]. The common cause $\Lambda$ must remain inaccessible to avoid the possibility of superluminal signalling using this influence. In Appendix C, we show that there is a causal model for this protocol whose affects relations can be compatibly embedded in the space-time configurations of Figures 1a and 1b. This proves that the protocol satisfies NSS. Moreover, there is also a natural connection between our protocol with jamming and non-local hidden variable theories, see Appendix D for details.

#### VI. CERTIFYING VIOLATION OF NSC WITHOUT VIOLATION OF NSS

We have seen that NSS theories can achieve strictly more informational tasks than NSC theories (as NSC implies NSS, see Appendix B for details). We now ask whether this can be used to certify the presence of a superluminal causal influence in NSS theories that achieve the tasks, as this would operationally establish that the violation of NSC is the relevant resource involve.

We observe that in Task 1, there is no theory-independent way to verify the superluminal causal influence. This is because the task is possible without superluminal causal influence in quantum and post-quantum non-classical theories. In particular, in box-world (a
<figure><img src="image_3.png" /><figcaption>FIG. 1. Space-time embeddings for the variables *A*, *C*, *X* and *Z* associated with (a) Task 1 and (b) Task 2, the space-time embedded variables are denoted in caligraphic font. The definition of the tasks as well as the impossibility results of Theorem 1 allow all possible space-time embeddings of *B*. The strategy which achieves the task in jamming theories (Theorem 2) requires the additional variable *B* to be embedded as shown here, where it can be signalled to by *A* and *C* without violating NSS. The shared classical system $\Lambda$ (which must be inaccessible to the agents to avoid NSS) is embedded in the joint past of all variables but is not depicted in the figures.</figcaption></figure>

post-quantum non-classical theory), Alice and Charlie can share a bipartite (non-classical) resource of the theory, measure it using their settings *A* and *C* to obtain the outcomes *X* and *Z* respectively which would yield the non-classical PR box correlations $X \oplus Z = A.C$ without invoking any superluminal influences even when the measurements take place in space-like separation.

By contrast, in Task 2, we can certify the existence of the superluminal causal influence in a theory-independent manner. In any theory without superluminal causation, we have shown that $P(XZ|AC) = P(XZ)$ holds in Task 2. Therefore repeating this experiment several times and collecting statistics, if we find that with access to *A* and *C* we can predict any function of *X* and *Z* better than without *A* and *C*, then we certify the existence of a superluminal causal influence in a theory-independent manner. Our model with jamming described in Protocol 1 allows the parity function $f(X, Z) = X \oplus Z$ to be predicted perfectly given *A* and *C*, even when the parity cannot be predicted better than a random guess when *A* and *C* are not given. Notice that in the space-time embedding of Figure 1b, *A* and *C* can be jointly accessed outside the joint future of *X* and *Z*, and thus, in a theory that allows jamming, an agent with access to *A* and *C* can perfectly predict the parity of *X* and *Z* even outside the joint future of *X* and *Z*.

VII. SPEED OF CREATION OF NON-CLASSICAL CORRELATIONS

Consider again the task of generating non-classical correlations $P(XZ|AC)$ between Alice and Charlie. For simplicity, we discuss the case of (1 + 1)-dimensional Minkowski space-time, though the idea generalises. Suppose that Alice and Charlie remain stationary at the spatial locations $x_A$ and $x_C$ respectively (let $x_C > x_A$ wlog)

while carrying out their measurements. In an NSC theory, any information travelling between the locations $x_A$ and $x_C$ that would correlate the variables produced there can at best travel at the speed of light *c* (even if the communication of this information was mediated by a third party, Bob). Therefore the minimum time required to establish any non-classical correlations between Alice and Charlie if they do not have any pre-shared non-classical resources would be $t_{NSC} = \frac{x_C - x_A}{c}$.<fnref n="3" />

On the other hand, consider the causal model of Protocol 1 embedded in space-time as per Figure 1a where Alice's setting and outcome are associated with the same spatial location and similarly for Charlie, but the timing is such that the measurements are space-like separated. Our results imply that in a theory that allows jamming of space-like separated correlations, Alice and Charlie can establish non-classical correlations (such as the PR-box correlations) in half this time i.e., $t_{jam} = \frac{x_C - x_A}{2c} = \frac{t_{NSC}}{2}$. They can do so with the help of a third party Bob situated half way between Alice and Charlie, who can receive information from them (which takes $t_{jam}$ amount of time) and immediately jam their outputs to produce the required PR box correlations. Thus non-classical correlations can be created at double the speed in such theories.

If we allow the outputs *X* and *Z* to be produced at different spatial locations $x_X \le x_A$ and $x_Z \ge x_C$, then in theories without superluminal causation, more time would be required for information to travel from the input of one party to the output of the other and we would have $t_{NSC} \ge \frac{x_C - x_A}{c}$. However, a surprising situation arises in jamming theories. It is possible to establish non-classical correlations even with a negative $t_{jam}$ i.e.,

<footnote n="3">³ In the common rest frame of the parties, since we have taken them to be stationary at their respective locations.</footnote>
where the outputs are produced at a much earlier time
than the inputs (in some frame of reference). Referring
to Figure 1b, we can move the RVs X and Z down along
a light-like surface such that they end up arbitrarily far
apart in space and time but their joint future (blue re-
gion) remains unaltered and continues to be contained
in the future of the ORV B, enabling Bob to jam them.
Thus we can always find a reference frame in which t<sub>jam</sub>
can be arbitrarily negative.

Thus a theory constrained by no superluminal sig-
nalling can in general create non-classical correlations
in a shorter time than in a theory constrained by the
stronger principle of no superluminal causation. Jam-
ming theories are an example of the former but can lead
to different times for creating non-classical correlations,
depending on the space-time configuration under consid-
eration.

VIII. CONCLUSIONS AND OUTLOOK

Our work highlights fundamental and practical distinc-
tions between the relativistic causality principles of no
superluminal causation (NSC) and no superluminal sig-
naling (NSS). We demonstrated that theories constrained
by the weaker NSS principle can offer larger information
processing possibilities over those restricted by NSC, by
considering concrete tasks. Specifically, we link a pre-
viously introduced class of theories, jamming non-local
theories to these questions, showing the existence of sce-
narios with jamming that violate NSC but satisfy NSS,
which are capable of perfectly achieving the aforemen-
tioned tasks.

Recent work [5] shows that jamming can lead to a vi-
olation of NSS in certain scenarios (contrary to previous
claims [1, 2]). Our results highlight that even in scenar-
ios where jamming obeys NSS, it can lead to counter-
intuitive effects due to the (certifiable) violation of NSC.
The space-time configuration of our second task allows

measurement outcomes to occur at earlier times than the
corresponding settings, yet jamming would enable the
generation of PR box correlations between these setting-
outcome variables. This also provides a connection be-
tween jamming non-local correlations with other post-
quantum correlations such as PR boxes.

There is scope to further investigate the information
processing power and limitations of theories with these
principles, as well as related principles such as *no causal
loops* (NCL) which has been formalised in our frame-
work [3, 4]. For example, in the context of comparing
NSC and NSS, it would be interesting to understand the
implications of being able to generate PR box correla-
tions in the space-time configuration of Task 2 for tasks
such as cryptography or communication complexity. Fur-
thermore, Ref. [4] shows, through a constructive example
that NSS does not imply NCL even in acyclic space-times
such as Minkowski space-time. Formally comparing in-
formation processing in theories constrained by NCL to
those constrained by NSS would allow us to identify
whether one can surpass the other.

The relationship between relativistic causality princi-
ples has bearing on our understanding of fundamental
physics. Considering the differences in information pro-
cessing power that arise from such principles facilitates a
better understanding of why the world works in the way
it does and may point to new axioms for physical laws.

ACKNOWLEDGMENTS

Part of this research was carried out when VV was
a PhD candidate supported by a Scholarship from the
Department of Mathematics at the University of York.
VV also acknowledges support from an ETH Post-
doctoral Fellowship, the ETH Zurich Quantum Cen-
ter, the Swiss National Science Foundation via project
No. 200021_188541 and the QuantERA programme via
project No. 20QT21_187724.

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## Appendix A: Further details of the causal modelling framework

In the main text, we have provided a rather non-technical overview of the causality framework of [3, 4] which yields a formalism for backing our main results about information processing and relativistic causality principles. Here, we present further technical details of this framework which are relevant for the arguments of this paper. A first important concept is that of d-separation, which is defined below.

**Definition 1 (Blocked paths).** Let $\mathcal{G}$ be a directed graph in which $X$ and $Y \neq X$ are nodes and let $Z$ be a set of nodes not containing $X$ or $Y$. A path from $X$ to $Y$ (not necessarily directed) is said to be blocked by $Z$ if it contains either $A \rightsquigarrow W \rightsquigarrow B$ with $W \in Z$, $A \rightsquigarrow W \rightsquigarrow B$ (a fork) with $W \in Z$ or $A \rightsquigarrow W \leftarrow B$ (a collider) such that neither $W$ nor any descendant of $W$ belongs to $Z$, where $A$ and $B$ are arbitrary nodes in the said path between $X$ and $Y$.

**Definition 2 (d-separation).** Let $\mathcal{G}$ be a directed graph in which $X$, $Y$ and $Z$ are disjoint sets of nodes. $X$ and $Y$ are d-separated by $Z$ in $\mathcal{G}$, denoted as $(X \perp^d Y|Z)_{\mathcal{G}}$ if every path from a variable in $X$ to a variable in $Y$ is blocked by $Z$, otherwise, $X$ is said to be d-connected with $Y$ given $Z$.

D-separation is itself purely a property of the graph. What we call the d-separation property, then links d-separation in the causal structure $\mathcal{G}$ to conditional in-dependences over the set $S_{obs}$ of observed nodes of $\mathcal{G}$.

**Definition 3 (d-separation property).** Let $\mathcal{G}$ be a causal structure associated with the set $S_{obs}$ of observed nodes, a distribution $P_{\mathcal{G}}(S_{obs})$ is said to satisfy the d-separation property with respect to $\mathcal{G}$ if and only if the following holds, where $X$, $Y$ and $Z$ are three disjoint subsets of $S_{obs}$ with $X$ and $Y$ being non-empty.

$$ (X \perp^d Y|Z)_{\mathcal{G}} \downarrow \qquad (A1) \\ P_{\mathcal{G}}(XY|Z) = P_{\mathcal{G}}(X|Z)P_{\mathcal{G}}(Y|Z). $$

Then a causal model is specified by a directed graph $\mathcal{G}$ associated with a set of $S_{obs}$ of observed nodes, together with a distribution observed $P_{\mathcal{G}}(S_{obs})$ that satisfies the d-separation property with respect to $\mathcal{G}$. This provides a minimal and general definition of a causal model.

In the main text, we have described how a classical, quantum or post-quantum causal model over a causal structure $\mathcal{G}$ can be obtained by specifying the causal mechanisms (the informational channels of the theory that we associate with the graph $\mathcal{G}$). The causal mechanisms allow us to compute the observed distribution $P_{\mathcal{G}}(S_{obs})$. For instance, in quantum theory, the causal mechanisms correspond to quantum channels, and the observed distribution can be computed using the Born

rule. It has been shown that for classical and non-classical causal models defined on directed acyclic graphs, the d-separation property holds for the resulting observed distribution [7, 8]. More generally, our framework of [3, 4] does not make assumptions about the causal mechanisms, but rather only assumes the d-separation property at the level of the observed distribution. This property can also be satisfied in several classical and non-classical models defined on cyclic graphs, making the framework more generally applicable to cyclic, non-classical causal structures. However for the present paper, only acyclic graphs will be relevant.

In the main text we have reviewed the concept of affects relations which capture the ability of agents to signal between sets of observed nodes through active interventions. Our formalism also introduces the concept of higher-order affects relations to model more general interventions which captures the ability of agents to signal from $X$ to $Y$ when given information about interventions performed on a third set $Z$ of observed nodes. These are defined as follows.

**Definition 4 (Higher-order affects relation).** Consider a causal model associated with a causal graph $\mathcal{G}$ over a set $N_{obs}$ of observed nodes and an observed distribution $P_{\mathcal{G}}$. Let $X$, $Y$, and $Z$ be three pairwise disjoint subsets of $N_{obs}$, with $X$ and $Y$ non-empty. We say that $X$ affects $Y$ given $do(Z)$ if there exists values $x$ of $X$, $z$ of $Z$ such that

$$ P_{\mathcal{G}_{do}(XZ)}(Y|X=x, Z=z) \neq P_{\mathcal{G}_{do}(Z)}(Y|Z=z), \quad (A2) $$

which we denote in short as $P_{\mathcal{G}_{do}(XZ)}(Y|X, Z) \neq P_{\mathcal{G}_{do}(Z)}(Y|Z)$. When $Z = \emptyset$, we refer to this as a zeroth-order affects relation.

In general two affects relations $X_1$ affects $Y$ and $\{X_1, X_2\}$ affects $Y$ can carry the same information, this happens for instance when $X_2$ is irrelevant to the scenario (e.g., it may have no causal connections with $X_1$ or $Y$). In such a case, the affects relation emanating from the smaller set is the relevant one, both for causal inference (as it tells us that $X_1$ is a cause of $Y$) and for relativistic causality principles (as requiring $Y$ to be embedded in the future lightcone of $X_1$ and $X_2$ in light of the second affects relation may not be necessary for avoiding superluminal signalling, if $X_2$ does not add any information to the signal). Then the following definition captures whether an affects is reducible, which is important for ensuring that our compatibility condition is both a necessary and a sufficient condition for having no superluminal signalling when a causal model is embedded in space-time.

**Definition 5 (Reducible and irreducible affects relations).** If for all proper subsets $s_X$ of $X$, $s_X$ affects $Y$ given $do(Z \tilde{s}_X)$, the affects relation $X$ affects $Y$ given $do(Z)$ is said to be irreducible. Otherwise, it is said to be reducible.
It is shown in [3] that for every reducible higher-order affects relation X affects Y given do(Z), there exists a proper subset $\tilde{s}_X$ of X such that $\tilde{s}_X$ affects Y given do(Z) i.e., X affects Y given do(Z) can be reduced to the affects relation $\tilde{s}_X$ affects Y given do(Z).

## Appendix B: Details of how we define relativistic causality principles

The two relativistic causality principles considered in this paper are: no superluminal causation (NSC) and no superluminal signalling (NSS). More precisely, NSC (/NSS) imposes that when we embed a informational protocol (which can be equivalently described as a causal model) in space-time by promoting the relevant variables of the protocols to space-time variables, then there is no causal influence (/signalling) from one space-time event to another event outside its future lightcone. In our framework, space-time is modelled as a partially ordered set $\mathcal{T}$ where $P \prec Q$, $P \succ Q$ and $P \not\prec \not\succ Q$ for $P, Q \in \mathcal{T}$ represent P being in the future light-cone of, past light-cone of, and space-like separated to Q respectively. An embedding $\mathcal{E}$ of a causal model with observed nodes $S_{\text{obs}}$ (or a set $\mathcal{A}$ of affects relations over a set $S_{\text{obs}}$ of variables) simply assigns a location in $\mathcal{T}$ to each random variable $X \in S_{\text{obs}}$, thereby generating an ordered random variable (ORVs) $\mathcal{X} := (X, O(X))$ where $O(X) \in \mathcal{T}$. ORVs are simply an abstraction of space-time random variables, to general partially ordered sets. Moreover, the inclusive future of an ORV is then given as

$$ \bar{\mathcal{F}}(\mathcal{X}) := \{P \in \mathcal{T} | P \succeq O(X)\}. \quad (B1) $$

Within this general formalisms, the no superluminal causation (NSC) and no superluminal signalling (NSS) principles can be defined as follows.

**Definition 6 (NSC theories).** Given a causal model over a causal structure $\mathcal{G}$ and an embedding $\mathcal{E}$ of the causal model in a space-time $\mathcal{T}$, we say that the embedded model satisfies NSC if and only if the following holds: whenever $X \rightsquigarrow Y$ for any two nodes X and Y in $\mathcal{G}$, then $\mathcal{X} \prec \mathcal{Y}$ in the space-time embedding. An NSC theory only allows for causal models of this type in any given space-time embedding.

The NSS principle is captured by the following compatibility condition between (higher-order) affects relations of the model and the space-time embedding, noting that higher-order affects relations capture the general way of signalling in such scenarios.

**Definition 7 (Compatibility of a set of affects relations with an embedding in a partial order).** Let $\mathcal{S}$ be a set of ORVs formed by embedding a set of RVs $\mathcal{S}$ in a space-time $\mathcal{T}$ with embedding $\mathcal{E}$. Then a set of affects relations $\mathcal{A}$ is said to be compatible with the embedding $\mathcal{E}$ if whenever X affects Y given do(Z) belongs to $\mathcal{A}$, and

<figure><img src="image_4.png" /><figcaption>FIG. 2. The causal structure corresponding to the causal model and protocol described in Protocol 1. The causal model is compatible with both the space-time embeddings of Figure 1.</figcaption></figure>

is irreducible with respect to the affects relations in $\mathcal{A}$, then $\bar{\mathcal{F}}(\mathcal{Y}) \cap \bar{\mathcal{F}}(\mathcal{Z}) \subseteq \bar{\mathcal{F}}(\mathcal{X})$ with respect to $\mathcal{E}$.

Since the affects relation X affects Y given do(Z) allows an agent with access to X to signal to an agent with access to Y and Z, the above compatibility condition captures the idea that in order to avoid superluminal signalling, we must require the affects relation to be embedded in space-time such that $\bar{\mathcal{F}}(\mathcal{Y}) \cap \bar{\mathcal{F}}(\mathcal{Z})$ (which is where Y and Z can be jointly accessed) is fully contained in $\bar{\mathcal{F}}(\mathcal{X})$ (which is where X can be accessed).

**Definition 8 (NSS theories).** Given a causal model over a causal structure $\mathcal{G}$ and an embedding $\mathcal{E}$ of the causal model in a space-time $\mathcal{T}$, we say that the embedded model satisfies NSS if and only if the set $\mathcal{A}$ of all affects relations of the causal models is compatible with the embedding $\mathcal{E}$ in the space-time $\mathcal{T}$. An NSS theory only allows for causal models of this type in any given space-time embedding.

Notice that since signalling between two variables, captured by an affects relation X affects Y, implies that X causes Y, it follows that no superluminal causation (NSC) implies no superluminal (NSS) in our formulation of these principles.

## Appendix C: Proofs of all results

**Theorem 1.** *Task 1 is impossible to achieve in any classical theory without superluminal causation, while Task 2 is impossible to achieve in any theory (classical or otherwise) without superluminal causation. This holds even when allowing for arbitrary collaboration strategies involving additional agents.*

*Proof.* We split the proof into two parts, covering the statement for Tasks 1 and 2 separately.

**Impossibility for Task 1:** Consider the distribution $P(XZ|AC)$ which is relevant for the task. Since we are
working in a classical theory, all the systems in the task can be taken to be classical random variables. In particular, let $\Lambda$ denote the system shared by Alice and Charlie. Then we can write $P(XZ|AC) = \sum_{\Lambda} P(XZ\Lambda|AC)$ and express this as follows using the rules of conditional probability.

$$P(XZ|AC) = \sum_{\Lambda} P(X|\Lambda ZAC)P(Z|\Lambda AC)P(\Lambda|AC). \qquad (C1)$$

The free choice of $A$ and $C$ means that these variables have no causes that are relevant to the experiment, which corresponds to regarding them as parentless nodes (i.e., having no incoming arrows) in the causal structure associated with the experiment. Thus $d$-connection between some variable and $A$ (or $C$) implies that the latter is a cause of the former. Using NSC in the space-time configuration for 1 along with free choice, one can see that the following $d$-separation relations must be respected in the causal structure, which yield corresponding conditional independences in the probability distribution, by the $d$-separation property (Definition 3).

1. Since $\Lambda$ is not embedded in the space-time future of $A$ or $C$, and $A$ and $C$ have no incoming arrows in the causal model, using NSC it follows that all paths between $\Lambda$ and $AC$ must contain a collider. Hence, $(\Lambda \perp^d AC)$, which implies the conditional independence $P(\Lambda|AC) = P(\Lambda)$.

2. Since $\mathcal{X}$ and $\mathcal{Z}$ are spacelike separated, NSC means there are no directed paths between $X$ and $Z$ in the causal model. There could be a collider where the central variable $W$ is embedded in the future of both $\mathcal{X}$ and $\mathcal{Z}$, or a fork where the central variable is embedded in the past of $\mathcal{X}$ and $\mathcal{Z}$. Candidates for the central variable in a fork are $\Lambda$ or a variable under the control of another agent lying in the past light-cone of both $\mathcal{X}$ and $\mathcal{Z}$. Without loss of generality we can bundle such a variable into $\Lambda$. It follows that $(X \perp^d Z|\Lambda)$, and also $(X \perp^d Z|\Lambda AC)$. The latter implies $P(X|\Lambda ZAC) = P(X|\Lambda AC)$.

3. Since $\mathcal{X}$ and $\mathcal{C}$ are spacelike separated, NSC means there are no directed paths between $X$ and $C$ in the causal model. Since $C$ has no incoming arrows, all paths between $X$ and $C$ must contain a collider with central variable in the future of $C$. It follows that $(X \perp^d C|\Lambda \Lambda)$, and consequently $P(X|\Lambda AC) = P(X|\Lambda A)$. Symmetrically, we also have $(Z \perp^d A|C \Lambda)$ and hence $P(Z|\Lambda AC) = P(Z|\Lambda C)$.

Plugging all of these conditional independences in Equation (C1), we obtain

$$P(XZ|AC) = \sum_{\Lambda} P(X|\Lambda A)P(Z|\Lambda C)P(\Lambda), \qquad (C2)$$

which is precisely the condition for the correlation $P(XZ|AC)$ being classical (Bell's local causality).

To extend the argument to allow for arbitrary collaborative strategies, let $S$ denote the set of all other systems which may be under the control of other agents who may assist Alice and Charlie in the task. Then we immediately notice that in all the above arguments, the $d$-separation criteria used in the proof follow from NSC and free choice of $A$ and $C$ even considering causal influences mediated by additional variables in $S$. This proves the statement for Task 1.

**Impossibility for Task 2:** Since $\mathcal{X}$, $\mathcal{Z}$, $A$ and $C$ are all spacelike separated, NSC means that there can be no directed paths between any of the 4 variables in the causal model. Furthermore, since $A$ and $C$ are parentless, any path between $AC$ and $XZ$ must contain a collider whose central variable is in the future of at least one of $A$ or $C$. It follows that $(XZ \perp^d AC)$, which implies $P(XZ|AC) = P(XZ)$, and hence that no non-classical correlations are possible. $\square$

**Theorem 2.** Tasks 1 and 2 can both be achieved in a classical theory that allows for jamming, without violating NSS and using the assistance of one additional agent Bob. Alice and Charlie only send classical communication to Bob, and exchange no communication otherwise. In particular, the maximally non-classical, PR box correlations can be generated in this manner.

*Proof.* We begin by detailing a causal model for Protocol 1 that achieves Tasks 1 and 2 under the conditions of the theorem. We show the compatibility of the causal model with the space-time configurations of Tasks 1 and 2, which formally establishes that NSS is indeed satisfied.

**Causal model.** Consider the causal structure of Figure 2, where $\Lambda$ is unobserved and all other nodes are observed. Suppose that all nodes correspond to classical binary variables and consider the classical causal model where $\Lambda$ is uniformly distributed, $X = \Lambda$, $B = A.C$ and $Z = \Lambda \oplus B$. This is precisely the jamming causal model presented in [4] (which is defined on $X$, $Z$, $B$ and $\Lambda$) with the addition of the parentless nodes $A$ and $C$ which causally influence $B$. This leads to the PR box correlations $X \oplus Z = A.C$, as in Protocol 1. Moreover, this model involves jamming correlations between $B$, $X$ and $Z$ since the conditions in (1) are satisfied: $B = X \oplus Z$ ensures the first condition and $\Lambda$ being uniform ensures the last two.

We can modify the above causal model by including a “noise” variable. For instance, defining $\Lambda = (E, F)$ where $E$ is uniformly distributed, we can have $X = E$, $B = A.C$ and $Z = E \oplus F \oplus B$ and, by varying the distribution over $F$, we can generate other (possibly quantum) correlations as well. The given model is asymmetric as there is only an influence of $B$ on $Z$ and not on $X$, but it can be symmetrised, for instance by including influences from $B$ to $X$ and $Z$ and considering the following functional dependences where we include an additional symmetrising variable in $\Lambda$: $\Lambda := (E, F, G)$ with $E$ and $G$ uniform, $B = A.C$, $X = E \oplus (G \oplus 1).(F \oplus B)$, $Z = E \oplus G.(F \oplus B)$.
**Compatibility with the space-time configurations.** The only remaining part of the proof is to show that the causal model describing this protocol is compatible with both the space-time embeddings of Figures 1a and 1b, which would show that the protocol in both these space-time configurations can be described by an NSS theory (cf. Definition 8) and hence that an NSS theory which is fully classical at an underlying level can perfectly achieve both tasks.

In the causal model of Protocol 1, the only observed variables are $X, Z, A$ and $C$. In this causal model, we have the affects relations $A$ affects $B$ (since $A$ is parentless, this affects relation is equivalent to $P(B|A) \neq P(B)$ which holds because $B = A.C$ and $C$ is not fixed to 0) and $C$ affects $B$ (by the same argument). In the embedding, we have $\mathcal{B} \succ_A \mathcal{A}$ and $\mathcal{B} \succ_C \mathcal{C}$ which are equivalent to $\bar{\mathcal{F}}(\mathcal{B}) \subseteq \bar{\mathcal{F}}(\mathcal{A})$ and $\bar{\mathcal{F}}(\mathcal{B}) \subseteq \bar{\mathcal{F}}(\mathcal{C})$, and hence these affects relations are compatible with the space-time embeddings of Figures 1a and 1b both which satisfy this property (cf. Definition 7).

Moreover, we can verify that we have $B$ affects $\{X, Z\}$ (since $P_G(XZ)$ is uniform but $P_{G_{do}(B)}(XZ|B)$ is deterministic due to $B = X \oplus Z$). Similarly we also have $\{A, C\}$ affects $\{X, Z\}$. Both these affects relations are compatible with the space-time embeddings of Figures 1a and 1b since both satisfy have $\bar{\mathcal{F}}(\mathcal{X}) \cap \bar{\mathcal{F}}(\mathcal{Z}) \subseteq \bar{\mathcal{F}}(\mathcal{B})$ and $\bar{\mathcal{F}}(\mathcal{X}) \cap \bar{\mathcal{F}}(\mathcal{Z}) \subseteq \bar{\mathcal{F}}(\mathcal{A}) \cap \bar{\mathcal{F}}(\mathcal{C})$.

Further, $B$ does not affect $X$ (since $P_G(X)$ and $P_{G_{do}(B)}(X|B)$ are both uniform when $\Lambda$ is uniform) and similarly, $B$ does not affect $Z$. No further compatibility conditions are imposed by the affects relations of this causal model, for instance, the higher-order affects relation $A$ affects $B$ given $do(X)$ also holds but the compatibility condition for this relation, namely $\bar{\mathcal{F}}(\mathcal{B}) \cap \bar{\mathcal{F}}(\mathcal{X}) \subseteq \bar{\mathcal{F}}(\mathcal{A})$ is already implied by the compatibility condition for $A$ affects $B$ which is $\bar{\mathcal{F}}(\mathcal{B}) \subseteq \bar{\mathcal{F}}(\mathcal{A})$, that has already been accounted for. This completes the proof. $\square$

## Appendix D: Relation to non-local hidden variable models

In Theorem 2 we have shown that Tasks 1 and 2 can be achieved in jamming theories which violate NSC but not NSS. The protocol achieving the tasks highlights a link between the resources of jamming and the PR box, which have been proposed in different classes of post-quantum theories. Here, we discuss how the two tasks can also be achieved in other NSS theories without jamming. For example, we can slightly modify the causal model of our jamming protocol, by removing Bob's node $B$ in its causal structure (Figure 2) and having $A$ and $C$ directly influence $X$. Then instead of the model $B = A.C$ and $X = \Lambda \oplus B$, we can directly set $X = \Lambda \oplus A.C$ to achieve the same correlations $P(XZ|AC)$ (in this case, PR box correlations).

In the space-time configurations for Tasks 1 and 2, the influence from $\mathcal{C}$ to $\mathcal{X}$ will be superluminal. As before, this model remains compatible with the space-time i.e., does not lead to superluminal signalling in both the space-time configurations, as long as $\Lambda$ remains unobserved. In Task 1, this would correspond to an explanation of non-classical correlations in a Bell experiment in terms of a non-local hidden variable theory [10]. Moreover, this model can also be adapted analogously as we did for jamming, to generate other non-classical correlations, by including a noise variable in $\Lambda$.

In the context of our discussion on the speed of creation of non-classical boxes, non-local hidden variable theories in-principle allow for instantaneous creation of non-classical correlations with $t_{\text{NHV}} = 0$, since the causal influence from $\mathcal{C}$ to $\mathcal{X}$ can be instantaneous. Furthermore, analogous to the case of jamming, the above model can also achieve arbitrarily negative times for creating non-classical correlations in the space-time configuration of Task 2 (Figure 1b).