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# A causal modelling analysis of Bell scenarios in space-time: implications of
jamming non-local correlations for relativistic causality principles

V. Vilasini<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH
Zurich, 8093 Zürich, Switzerland  Roger Colbeck<EMAIL_ADDRESS>Department of Mathematics, University of York, Heslington, York YO10 5DD.

###### Abstract

Bell scenarios involve space-like separated measurements made by multiple
parties. The standard no-signalling constraints ensure that such parties
cannot signal superluminally by choosing their measurement settings. In
tripartite Bell scenarios, relaxed non-signalling constraints have been
proposed, which permit a larger class of post-quantum theories. This includes
so-called jamming non-local theories, the idea being that no superluminal
signalling would continue to hold. To analyse whether this is the case and,
more generally, the role of non-signalling constraints in preserving
relativistic causality principles, we apply a framework that we have recently
developed for defining information-theoretic causal models in non-classical
theories and their compatibility with relativistic causality in a space-time.
We show that any theory (classical or non-classical) that generates jamming
correlations in a Bell scenario between space-like separated parties must
necessarily do so through causal fine-tuning and by means of superluminal
causal influences. Moreover, within our framework, we show that jamming
theories can lead to superluminal signalling (contrary to previous claims)
unless it is ensured that certain systems are fundamentally inaccessible to
agents. This highlights limitations for the physicality of jamming theories.
Finally, we analyse relativistic causality in Bell scenarios showing that no-
signalling constraints on correlations are generally insufficient for ruling
out superluminal signalling when general interventions are allowed. In this
way, we identify necessary and sufficient conditions for ruling out
superluminal signalling in Bell scenarios, and demonstrate through examples
that the no-signalling constraints on correlations are neither necessary nor
sufficient for ruling out causal loops. These results solidify our
understanding of relativistic causality principles in information processing
tasks in space-time, involving classical, quantum or post-quantum resources.

###### Contents

  1. I Introduction
    1. I.1 Outline of paper and main contributions
  2. II The standard non-signalling conditions in Bell-type scenarios
  3. III Jamming non-local correlations and claims about relativistic causality
    1. III.1 Issues with these claims
  4. IV A framework for relating information-processing and relativistic causality in general theories
    1. IV.1 Causal models, affects relations and causal loops
    2. IV.2 Space-time embeddings and compatibility
  5. V Formal definitions of Bell scenarios and corresponding space-time embeddings
  6. VI Causal modelling analysis of jamming and implications for relativistic causality
    1. VI.1 Jamming requires fine-tuning and superluminal causal influence
    2. VI.2 Jamming can lead to superluminal signalling
  7. VII Relativistic causality in Bell scenarios: accounting for arbitrary interventions
    1. VII.1 Non-signalling constraints and higher-order affects relations
    2. VII.2 Conditions for preserving relativistic causality
      1. VII.2.1 Conditions for ruling out superluminal signalling
      2. VII.2.2 Conditions for ruling out causal loops
  8. VIII Conclusions and outlook
  9. A Further technical details relating to our framework
    1. A.1 D-separation and causal models
    2. A.2 Higher-order affects relations and causal inference
    3. A.3 Compatibility with space-time
  10. B Proofs of all results
    1. B.1 Proofs of Lemmas VI.1, VI.2 and Theorem VI.1
    2. B.2 Proofs of Theorems VII.1 and VII.2
    3. B.3 Proofs of Theorems VII.3 and VII.4
    4. B.4 Proof of Theorem VII.5
    5. B.5 Counterexamples used in the proof of Theorem VII.6
  11. C space-time positions compatible with jamming
    1. C.1 Lightcone notation
    2. C.2 $(1+1)$D
    3. C.3 $(2+1)$D

##  I Introduction

The nature of correlations arising from measurements on space-like separated
quantum systems has fuelled historical debates concerning the compatibility
between quantum theory and special relativity EPR ; Bohr ; Bohm1957 ; BellEPR
; Bell . The developments in quantum information theory in recent decades,
have revealed the power of quantum correlations for information processing
tasks such as computing, communication and cryptography. Notably, these
advances have also affirmed that, despite its non-relativistic as well as non-
classical nature (as highlighted by the existence of quantum correlations that
violate Bell inequalities Bell ; CHSH ), quantum mechanics nevertheless aligns
harmoniously with relativistic principles, such as the impossibility of
superluminal signalling. This has placed our understanding of quantum
entanglement, quantum correlations, and their information-processing
possibilities on solid theoretical and conceptual ground.111An overview of
these advances and results can be found in any textbook on quantum information
theory, e.g., NielsenChuang .

More generally, this raises the question: when can a theory of information
processing be deemed compatible with relativistic causality principles in
space-time? In Bell scenarios, where space-like separated parties locally
measure subsystems of a shared quantum system, obtaining measurement
statistics adhering to the non-signalling constraints (which demand that the
measurement outcome of one party remains uncorrelated with the measurement
setting of another party) is usually taken as necessary to prevent
superluminal signalling. Quantum correlations satisfy these constraints, as do
more general post-quantum theories PR1994 , also known as generalized non-
signalling theories (GNST) Barrett07 . In Bell scenarios with three or more
parties, relaxations of the standard non-signalling constraints have been
proposed Grunhaus1996 ; Horodecki2019 . These relaxations are permissive to a
broader range of theories than GNST, with the so-called _jamming non-local
theories_ Grunhaus1996  being examples. Existing literature suggests that
jamming theories are consistent with relativistic causality principles in
Minkowski space-time in that they do not give rise to superluminal signalling
or causal loops, and therefore go on to dub them as _relativistically causal
theories_ Horodecki2019 .

However, these claims are not made within a mathematical framework that
rigorously defines concepts such as _causality_ , _causal loops_ and
_superluminal signalling_ that feature in the arguments. Such a framework is
needed to put these concepts on a solid footing and clearly define the
conditions under which a (possibly non-classical) theory adheres to or
violates them. There are at least two notions of causality at play in
information-processing protocols within space-time: the _information-theoretic
causal structure of the protocol_ and the _causal structure of space-time_.
While the two are distinct, in physical experiments, they interact in a
compatible manner. Therefore, in order to formalise relativistic principles of
causation a framework needs to distinguish and relate these concepts.
Moreover, to deal with causation distinct from correlation, a suitable
framework should account for general interventions that agents can perform and
by means of which they may signal to other agents. Such a formalism,
addressing both these aspects in a theory-independent manner, has been
developed in our recent works VilasiniColbeckPRA ; VilasiniColbeckPRL .

The formalism of VilasiniColbeckPRA ; VilasiniColbeckPRL  adopts an approach
based on _causal modelling_ to formalize information-theoretic causal
structures. Causal modelling, originally developed in classical statistics and
data sciences Pearl2009 ; Spirtes2001 , is a powerful tool for linking
causation, correlations, and interventions with applications in various
fields, such as machine learning, medical testing, biology, and economics. In
light of the challenges posed by quantum correlations to classical causal
modelling frameworks Wood2015 , there has been significant progress in
developing quantum and non-classical causal models Leifer2013 ; Henson2014 ;
Costa2016 ; Allen2017 ; Barrett2020 . The formalism of VilasiniColbeckPRA ;
VilasiniColbeckPRL  builds on this progress and allows cyclic and non-
classical causal models to be described in quantum and post-quantum theories.
Specifically, the concept of _higher-order affects relations_ was introduced
to model general interventions in these settings. Relativistic causality
principles, such as the impossibility of superluminal signalling, can be
precisely formalized by embedding these causal models in a background space-
time and imposing graph-theoretic _compatibility conditions_ between the
higher-order affects relations coming from the causal model and the light-cone
structure coming from the space-time. Using this approach, it was demonstrated
that the absence of superluminal signalling in Minkowski space-time does not
preclude causal loops VilasiniColbeckPRL , highlighting that no superluminal
signalling and no causal loops are distinct principles that do not imply one
another. In the present work, we apply this formalism to analyze these
relativistic causality principles in Bell scenarios, and establish in
particular that theories admitting jamming non-local correlations in such
scenarios will inevitably violate principles of information-theoretic
causation (such as the principle of no fine-tuning, see below) as well as
relativistic causation.

###  I.1 Outline of paper and main contributions

The initial sections of the paper provide an overview of the relevant
background for our results. In Section II, we review the standard bi and
tripartite non-signalling constraints in the respective Bell scenarios.
Section III offers an overview of jamming non-local correlations and the
related claims about relativistic causality in previous literature. Section IV
provides a self-contained review of our recent causality framework
VilasiniColbeckPRA , which was motivated in the introduction. The main
contributions are summarized below:

  * •

Jamming requires fine-tuning. One reason why classical causal explanations of
correlations violating a Bell inequality are rejected is because they must be
fine-tuned Wood2015 , that is, they must be carefully chosen in order to
conceal causal influences from showing up in the observed correlations or
signalling relations. The usual quantum explanation need not be though. On the
other hand, in Section VI.1 we demonstrate that any setup exhibiting jamming
correlations must involve _causal fine-tuning_ whether or not the underlying
theory is classical. Fine-tuning is often considered an undesirable property
of a causal model because it challenges the ability to infer causal influences
from observed data and for this reason, is often used as a principle to reject
particular theories that are unable to explain observations without alluding
to fine-tuning (see e.g., Wood2015 ; Cavalcanti2018 ; Ying2023 ). Hence the
information theoretic causality principle no fine-tuning allows us to rule out
jamming correlations regardless of the physical theory describing the
underlying causes.

  * •

Jamming and violation of relativistic causality principles. Embedding the
tripartite Bell scenario in its natural space-time configuration proposed in
previous works, it follows that any theory that exhibits jamming correlations
in this scenario must involve superluminal causal influences – see Section
VI.1. Furthermore, in Section VI.2, we show that jamming theories can also
lead to superluminal signalling in this space-time configuration, contrary to
the original motivation for such theories Grunhaus1996 ; Horodecki2019 . We
find that to prevent superluminal signalling in jamming theories, it is
necessary that the common shared system of the Bell scenario remains
fundamentally inaccessible to the agents preventing it from being subject to
interventions (even if classical), and there are jamming scenarios where
superluminal signalling can be avoided in this manner. This result highlights
a necessary tension between jamming theories and _relativistic causality
principles_.

  * •

Necessary and sufficient conditions for relativistic causality in Bell
scenarios. In Section VII.1, we provide an equivalent formalization of the
standard and relaxed non-signalling constraints in terms of higher-order
affects relations. We then demonstrate through explicit examples that in bi
and tripartite Bell scenarios, neither the standard nor the relaxed non-
signalling constraints are sufficient to rule out superluminal signalling
(Section VII.2), in contrast to a previous claim Horodecki2019 . The analysis
reveals the precise reason for the failure of the previous claim, in essence:
non-signalling constraints only pertain to correlations, and when the settings
are freely chosen, they tell us about signalling through interventions on
setting variables. However, this does not fully encompass causation and does
not tell us whether more general interventions (on both settings and outcomes)
could be used for signalling. We employ causal modelling and affect relations
to establish a set of necessary and sufficient conditions for ruling out
superluminal signalling in Bell scenarios, when considering arbitrary
interventions. Finally, we discuss conditions for ruling out causal loops,
showing with counter-examples that the necessary and sufficient conditions for
no superluminal signalling (mentioned above) may not generally be sufficient
or necessary for ruling out causal loops in Bell scenarios.

These results show that previous claims Grunhaus1996 ; Horodecki2019
regarding no superluminal signalling and no causal loops in jamming theories
do that hold once the claims are formalised within our general framework. We
are currently unaware of any other framework within which our counter-examples
to the claims could be evaded and where the claims could be proven to be true,
but this remains a possibility. With these results we lay a foundation for
characterising relativistic causality principles (such as no superluminal
signalling and no causal loops) in Bell-type scenarios in arbitrary theories,
while accounting for correlations as well as interventions. More generally,
the framework of VilasiniColbeckPRL ; VilasiniColbeckPRA  can also be applied
to characterise relativistic causality principles in a theory-independent
manner in arbitrary information-processing scenarios, in a similar manner as
we have done for Bell-type scenarios in this work. Section VIII presents our
conclusions and offers an outlook for future research.

##  II The standard non-signalling conditions in Bell-type scenarios

Consider a bipartite Bell experiment involving two parties Alice and Bob who
share a common system $\Lambda$ (which may be a classical random variable, a
quantum system or more generally, a system described by a post-quantum
probabilistic theory). Alice and Bob perform measurements chosen freely using
the measurements settings $A$ and $B$ on their subsystems of the shared
system, and obtain corresponding measurement outcomes $X$ and $Y$ respectively
(where $A$, $B$, $X$ and $Y$ are classical random variables). Here, a variable
(such as $A$) is said to be freely chosen if it has no prior causes relevant
to the experiment (see Definition V.1 for further details on how we model free
choice in Bell experiments). Now, suppose that Alice and Bob’s measurements
are performed at space-like separated locations i.e., $A$, $B$, $X$, $Y$ and
$\Lambda$ are embedded in space-time as shown in Figure 2. Then, in order to
ensure that the parties cannot signal to each other superluminally, it is
necessary to require that each party’s outcome is independent of the
measurement setting of the other party.

Formally, taking $P(XY|AB)$ to be the joint conditional probability of the
outcomes given the settings, we have the following non-signalling conditions
on the correlations arising in such a bipartite Bell scenario. In the rest of
the paper, we use the small letter $x$ to denote a particular value of a
random variable denoted by the corresponding capital letter $X$.

######  Definition II.1 (Standard bipartite non-signalling conditions (NS2)).

$\displaystyle\begin{split}P(x|a)&:=\sum_{y}P(x,y|a,b)=\sum_{y}P(x,y|a,b^{\prime})\quad\forall
x,a,b,b^{\prime}\\\
P(y|b)&:=\sum_{x}P(x,y|a,b)=\sum_{x}P(x,y|a^{\prime},b)\quad\forall
y,a,a^{\prime},b\\\ \end{split}$ (1)

We refer to these as the standard bipartite non-signalling conditions or
(NS2). In short, we will denote the above conditions as $P(X|AB)=P(X|A)$ and
$P(Y|AB)=P(Y|B)$. If either one of these conditions is violated, then the
measurement outcome of one party will be correlated with the choice of
measurement of the other, space-like separated party allowing the latter party
to communicate information to the former by choosing different measurements on
their half of the joint state.

Similarly, consider a tripartite Bell experiment between the three parties
Alice, Bob and Charlie who share a tripartite system $\Lambda$ (which may
again be described by a classical, quantum or post-quantum theory), measure
their subsystem with freely chosen measurement settings $A$, $B$ and $C$ and
obtain corresponding measurement outcomes $X$, $Y$ and $Z$ respectively. If
the parties’ measurements are conducted within pairwise space-like separated
space-time regions, such as in the space-time configuration of Figure 3a, then
the standard tri-partite non-signalling conditions (which are also motivated
by the impossibility of superluminal signalling) require that the measurement
setting of each party cannot be correlated with the outcomes of any set of
complementary parties. Explicitly, this translates to the following
constraints on the joint conditional probability distribution
$P(X,Y,Z|A,B,C)$.

######  Definition II.2 (Standard tripartite non-signalling conditions (NS3)).

$\displaystyle\begin{split}P(x,y|a,b)&:=\sum_{z}P(x,y,z|a,b,c)=\sum_{z}P(x,y,z|a,b,c^{\prime})\quad\forall
x,y,a,b,c,c^{\prime}\\\
P(y,z|b,c)&:=\sum_{x}P(x,y,z|a,b,c)=\sum_{x}P(x,y,z|a^{\prime},b,c)\quad\forall
y,z,a,a^{\prime},b,c\\\
P(x,z|a,c)&:=\sum_{y}P(x,y,z|a,b,c)=\sum_{y}P(x,y,z|a,b^{\prime},c)\quad\forall
x,z,a,b,b^{\prime},c\\\ \end{split}$ (2)

In short, these conditions are expressed as $P(XY|ABC)=P(XY|AB)$,
$P(YZ|ABC)=P(YZ|BC)$ and $P(XZ|ABC)=P(XZ|AC)$. Notice that the above
conditions of Equation (2) imply the following conditions which are equivalent
to writing $P(X|ABC)=P(X|A)$, $P(Y|ABC)=P(Y|B)$ and $P(Z|ABC)=P(Z|C)$.

$\displaystyle\begin{split}P(x|a)&:=\sum_{y,z}P(x,y,z|a,b,c)=\sum_{y,z}P(x,y,z|a,b^{\prime},c^{\prime})\quad\forall
x,a,b,b^{\prime},c,c^{\prime}\\\
P(y|b)&:=\sum_{x,z}P(x,y,z|a,b,c)=\sum_{x,z}P(x,y,z|a^{\prime},b,c^{\prime})\quad\forall
y,a,a^{\prime},b,c,c^{\prime}\\\
P(z|c)&:=\sum_{x,y}P(x,y,z|a,b,c)=\sum_{x,y}P(x,y,z|a^{\prime},b^{\prime},c)\quad\forall
z,a,a^{\prime},b,b^{\prime},c\\\ \end{split}$ (3)

(a)  (b)

Figure 1: Typical causal structure associated with a bipartite Bell experiment
(a) This figure represents the typical information-theoretic causal structure,
which captures the flow of information in a protocol where Alice and Bob share
a bipartite system $\Lambda$, and locally perform freely chosen measurements
on their halves of this bipartite resource to generate outcomes. The settings
$A$, $B$, and outcomes $X$ and $Y$ are observed classical variables (circled)
and the resource corresponding to the common cause $\Lambda$ may be classical,
quantum or post-quantum. (b) The described protocol can be represented by a
circuit diagram as shown here, and one can see that the causal structure of
(a) captures the connectivity of this circuit diagram. Such diagrams involve
the causal mechanisms (systems, transformations, measurements) of a theory.
The meaning of the information-theoretic causal structure and causal
mechanisms are explained further in Section IV. Figure 2: Space-time
configuration associated with a bipartite Bell experiment Space-time points
are labelled by the variables of the information-theoretic protocol which are
embedded at that location, this can be understood as the space-time location
at which the corresponding measurement setting/outcome is generated. We
distinguish space-time embedded random variables from the random variable
itself using the caligraphic font for the former. Here, the measurement events
of the two parties are space-like separated. The diagonal lines represent
light-like surfaces. The blue region is the joint future of both measurement
outcomes, where the joint probability distribution $P(XY|AB)$ can be
evaluated.

Information-theoretic causal structure vs space-time configuration In the
above discussion of standard Bell scenarios, there are two distinct notions of
causality at play: the information-theoretic causal structure of the
experiment (Figures 1a and 3b) and the spatio-temporal causal structure in
which we embed the physical systems of the experiment (Figures 2 and 3a). The
former captures the flow of information between physical systems in the
experiment, for instance, if Alice does not communicate to Bob, then there
would not be any causal arrows from systems in Alice’s control to systems in
Bob’s control. This would be independent of whether or not Bob acts in the
spatio-temporal future of Alice i.e., we can naturally have scenarios where
Bob is in the future of Alice relative to the causal structure of the space-
time but not so relative to the information-theoretic causal structure (due to
absence of communication channels). However, through principles of
relativistic causality, the spatio-temporal structure constrains the possible
information-theoretic structures. For instance, if Alice and Bob are space-
like separated, we would expect, from relativistic principle of no
superluminal causation, that the information-theoretic causal structure also
does not have causal arrows between the variables describing Alice and Bob’s
measurements. Indeed, for the information-theoretic causal structures of the
Bell experiment (illustrated in Figures 1a and 3b) and the corresponding
space-time configurations (illustrated in Figures 2 and 3a), all (information-
theoretic) causal arrows flow from past to future in the space-time.

Moreover, through a causal modelling approach, one can show that the
information-theoretic causal structures of Figures 1a and 3b imply the non-
signalling constraints (NS2) and (NS3) as a consequence of their topology,
which highlights that these constraints are purely a property of the
information-theoretic structure.222More formally, this would be a consequence
of the d-separation property (cf. Definition A.3) applied to these causal
structures, which has been proven to hold for acyclic causal models in
classical, quantum and post-quantum theories Pearl2009 ; Henson2014 ;
Barrett2020A . However, when this structure is embedded in a space-time such
that Alice and Bob’s measurement variables are embedded are space-like
separated locations, then these constraints become relevant for avoiding
signalling outside the future light cone of the space-time. In Section IV we
review a framework that we have introduced in VilasiniColbeckPRA ;
VilasiniColbeckPRL  within which such statements linking information-theoretic
and spatio-temporal causal structures can be made mathematically precise in
rather general scenarios, while maintaining a clear conceptual separation
between these distinct notions of causation.

Free choice: Notice that in order to interpret the conditions (NS2) and (NS3)
as no-signalling conditions, the free choice of the settings $A$ and $B$ is
crucial. For instance, if the setting, $A$, was not freely chosen but rather,
determined by some event in the past, then it could naturally be correlated
with the outcome $Y$ of Bob. This is because an event in the common past
lightcone of $A$ and $Y$ could influence and correlate these variables without
having any causal influences from $A$ to $Y$ and without enabling Alice to
signal information to Bob using these correlations. However, $A$ being freely
chosen precludes any causes of $A$ that are relevant to the rest of the
experiment, in which case correlations between $A$ and $Y$ would indicate
signalling from $A$ to $Y$. Whenever $A$ and $Y$ are embedded at space-like
separated locations in space-time, this signal would be superluminal. The
condition that $A$, $B$ and $C$ are freely chosen is naturally captured
through the information-theoretic causal structure of the experiment, by
modelling them as parentless nodes in the causal graph. This is the case in
the typical information-theoretic causal structures used for the bipartite and
tripartite Bell experiment (Figures 1a and 3b).

(a)  (b)

Figure 3: A tripartite Bell experiment: (a) Space-time configuration for the
jamming scenario Grunhaus1996 ; Horodecki2019 : The measurement events of the
three parties are space-like separated such that the future of Bob’s input $B$
contains the entire joint future (in blue) of Alice and Charlie’s outputs $X$
and $Z$. Here, $B$ is allowed to signal to $X$ and $Z$ jointly (which can only
be verified in the blue region) but not individually. (b) Information-
theoretic causal structure for the usual tripartite Bell experiment: This
causal structure is typically used for modelling a tripartite Bell experiment
where correlations are constrained by the standard non-signalling conditions
(NS3). In order to model the more general conditions (NS3′) that permit
jamming, some modification is required. Figure 5 is a natural modification of
this causal structure which can admit jamming correlations.

##  III Jamming non-local correlations and claims about relativistic causality

The post-quantum phenomenon known as “jamming” was originally proposed in
Grunhaus1996  to argue that in a tripartite Bell scenario with pairwise space-
like separated measurements, the standard non-signalling conditions (NS3) are
not strictly necessary for ensuring no superluminal signalling. This defines a
more general set of non-local correlations in tripartite Bell scenarios that
go beyond correlations achievable in quantum theory or standard no-signalling
probabilistic theories, but are nevertheless expected to not lead to any
superluminal signalling. We now review the argument presented in the previous
literature on jamming Grunhaus1996 ; Horodecki2019 , while focussing our
attention on the tripartite Bell scenario, which is the simplest Bell scenario
that exhibits jamming.

The conditions (NS3) require, in particular, that the outcomes $X$ and $Z$ of
Alice and Charlie are independent of the setting $B$ of Bob, that is
$P(XZ|ABC)=P(XZ|AC)$. However, it was noted in Grunhaus1996  that in a space-
time configuration where the joint future of the space-time locations of $X$
and $Z$ is entirely contained in the future of the location of $B$ (as shown
in Figure 3a), a violation of $P(XZ|ABC)=P(XZ|AC)$ would not lead to
superluminal signalling, as long as $P(X|AB)=P(X|A)$ and $P(Z|BC)=P(Z|C)$
still hold. This is because whether or not $P(XZ|ABC)=P(XZ|AC)$ holds can only
be checked within the joint future of $X$ and $Z$. The logic is that a failure
of this condition only leads to signalling within $B$’s future in this space-
time configuration, as long as the marginals of the space-like separated
events, $P(X|A)$ and $P(Z|C)$ are independent of $B$, and therefore involves
no superluminal signalling. Nevertheless, this would allow Bob to non-locally
“jam” the correlations between Alice and Charlie, even though each of them are
space-like separated to him. Using this observation, Horodecki2019  propose
the set of relaxed tripartite non-signalling conditions (NS3′) which allows
for a larger set of correlations as compared to (NS3) by virtue of being less
constrained. The difference between (NS3) and (NS3′) lies in whether or not
$P(XZ|ABC)=P(XZ|AC)$ necessarily holds.

######  Definition III.1 (Relaxed tripartite non-signalling conditions
(NS3′)).

$\displaystyle\begin{split}P(x,y|a,b)&:=\sum_{z}P(x,y,z|a,b,c)=\sum_{z}P(x,y,z|a,b,c^{\prime})\quad\forall
x,y,a,b,c,c^{\prime}\\\
P(y,z|b,c)&:=\sum_{x}P(x,y,z|a,b,c)=\sum_{x}P(x,y,z|a^{\prime},b,c)\quad\forall
y,z,a,a^{\prime},b,c\\\
P(x|a)&:=\sum_{y,z}P(x,y,z|a,b,c)=\sum_{y,z}P(x,y,z|a,b^{\prime},c^{\prime})\quad\forall
x,a,b,b^{\prime},c,c^{\prime}\\\
P(z|c)&:=\sum_{x,y}P(x,y,z|a,b,c)=\sum_{x,y}P(x,y,z|a^{\prime},b^{\prime},c)\quad\forall
z,a,a^{\prime},b,b^{\prime},c\\\ \end{split}$ (4)

Notice that the first two of the above conditions imply the following
independence.

$P(y|b):=\sum_{x,z}P(x,y,z|a,b,c)=\sum_{x,z}P(x,y,z|a^{\prime},b,c^{\prime})\quad\forall
y,a,a^{\prime},b,c,c^{\prime}$ (5)

######  Definition III.2 (Tripartite jamming).

We say that a tripartite Bell scenario admits jamming if and only if the
corresponding joint distribution $P(XYZ|ABC)$ satisfies (NS3′) but not (NS3)
i.e., satisfies (NS3′) with $P(XZ|ABC)\neq P(XZ|AC)$.

Based on the above arguments, Grunhaus1996 ; Horodecki2019  claim that in the
space-time configuration of Figure 3a, (NS3′) and not (NS3) are the necessary
conditions on tripartite Bell correlations required for avoiding superluminal
signalling. Inspired by the special relativistic observation that the
possibility of superluminal signalling in Minkowski space-time enables the
possibility of signalling to the past, Horodecki2019  propose the following
definition for relativistic causality that forbids causal loops. Here,
measurement settings and outcomes are viewed as space-time random variables,
that is, random variables associated with a space-time location. The standard
notation $(\vec{r},t)$ with $\vec{r}\in\mathbb{R}^{3}$ and $t\in\mathbb{R}$ is
used to denote the space-time co-ordinates of a point in Minkowski space-time
in some inertial frame of reference $\mathcal{I}$. A measurement event
corresponds to a pair of random variables, a setting and an outcome variable.
Horodecki2019  associate the same location with both variables in a
measurement event, as an idealisation, and use the following quoted text as
their definition of relativistic causality.

Quotation from Horodecki2019  (Relativistic causality condition): “No causal
loops occur, where a causal loop is a sequence of events, in which one event
is among the causes of another event, which in turn is among the causes of the
first event.”

They then go a step beyond the previous claim and make the following stronger
claims for the bipartite and tripartite Bell scenarios.

######  Claim III.1 (Conditions for relativistic causality in bipartite Bell
scenario Horodecki2019 ).

In a bipartite Bell experiment the standard bipartite non-signalling
conditions (NS2) are necessary and sufficient for ensuring that no violation
of relativistic causality occurs.

######  Claim III.2 (Conditions for relativistic causality in tripartite Bell
scenario Horodecki2019 ).

Consider a tripartite Bell experiment performed in the space-time
configuration of Figure 3a, where in an inertial frame of reference
$\mathcal{I}$ the space-time locations $({\bf x}_{X},t_{X})$, $({\bf
x}_{Z},t_{Z})$ and $({\bf x}_{B},t_{B})$ associated with the random variables
$X$, $Z$ and $B$ are such that the joint future of $({\bf x}_{X},t_{X})$ and
$({\bf x}_{Z},t_{Z})$ is entirely contained in the future of $({\bf
x}_{B},t_{B})$. Then the relaxed tripartite non-signalling conditions (NS3′)
are necessary and sufficient for ensuring that no violation of relativistic
causality occurs.

###  III.1 Issues with these claims

While the argument that jamming is possible without superluminal signalling
makes intuitive sense, when attempting to understand the precise implication
of these stronger claims regarding necessary and sufficient conditions for
relativistic causality, a number of conceptual and technical issues arise. In
particular, this is because the claims are made in an intuitive manner rather
than within a framework with rigorous definitions of concepts such as “cause”,
“events”, “signalling”, “causal loops” which are used in the informal claims.
We outline the main problems with verifying the validity of these claims in
the absence of such a mathematical formalism.

1\. How is causation defined?
The relativistic causality condition proposed in Horodecki2019  hinges on the
concepts of “cause” and “events” but the precise meaning of these concepts is
left ambiguous. There are at least two distinct notions of causation at play
here: information-theoretic causation which can be defined as a directed graph
capturing the flow of information in a network of physical systems, and
spatio-temporal causation given by the light-cone structure of the space-time.
These works focus on Minkowski space-time whose light-cone structure forms a
partial order. If we interpret “events” here as space-time events and
“causation” as given by the light-cone structure, then a claim of no causal
loops holds by construction, irrespective of the correlations being considered
and thus (NS2) or (NS3′) would not be necessary for relativistic causality in
this case, which would be inconsistent with the claims.

On the other hand, if we interpret “events” and “causation” relative to the
nodes (physical systems) and directed edges (given by the structure of
information-theoretic channels) of the information-theoretic causal structure,
then the standard causal structure of a Bell scenario is a directed acyclic
graph and hence also has no causal loops. This is because the measurement
settings being free choices have no incoming arrows and the measurement
outcomes are typically assumed to not have any outgoing causal arrows (this
assumption also appears to be made in Horodecki2019 , in the proof of the
claims). Then again, the validity of the claims comes under question.
Therefore it is unclear if there exists a mathematically rigorous definition
of these concepts with respect to which Claims III.1 and III.2 hold true.

2\. How can correlation-based constraints be necessary and sufficient for a
statement about causation? The claims suggest that a condition on the
correlations is necessary and sufficient for ruling out causal loops. However,
because correlation does not imply causation, one would expect that further
assumptions relating to the causal structure are needed in order for these
claims to be true. As noted in Point 1 above, the notion of causal structure
featuring in the statements is ambiguous and consequently so are the
assumptions made on it. Generally, when inferring causation in the real-world
(e.g., when inferring whether a drug causes recovery from a disease), one must
consider interventions Spirtes2001 ; Pearl2009  in addition to correlations,
which form the basis of controlled trials and of the general methods by which
we infer causation from observed data in practice.

Interventions give the general method by which parties can signal between each
other, and allow us to infer the presence of causal influences. However,
signalling and causation are generally not equivalent, as it is possible that
underlying causal parameters are fine-tuned which can lead to certain causal
influences being undetectable at the level of observable signalling relations.
Previous works on relativistic causality have focused mainly on correlations,
and on causal models without such fine-tuned influences. However, as suggested
in VilasiniColbeckPRA ; VilasiniColbeckPRL , and shown in Section VI.1,
jamming correlations necessarily require fine-tuned causal explanations and if
we assume no fine-tuning (which would allow signalling and causation to be
equated), we can no longer study jamming correlations. It therefore becomes
important, when analysing these questions, to account for correlations and
interventions, while noting that signalling and causation are generally
different (but related) concepts.

3\. Superluminal signalling and causal loops do not necessarily imply each
other.
The connection between the non-signalling conditions (NS2) and (NS3′) and
superluminal signalling in bi/tripartite Bell scenarios is more intuitively
apparent from the original arguments of Grunhaus1996 , which were also
revisited in the previous sections. In Horodecki2019  this is taken a step
further by trying to link these non-signalling conditions to causal loops,
arguing that a violation of (NS2) or (NS3′) in the respective scenario would
enable superluminal signalling which can be used to signal to the past and
create causal loops. However, the possibility of superluminal signalling does
not by itself imply causal loops: if superluminal signalling was possible only
in a single preferred reference frame, this could not be used to create a
causal loop as discussed in the proof of the claims in Horodecki2019 .

Conversely, one may consider whether the existence of a causal loop always
enables parties to signal superluminally. In a recent work VilasiniColbeckPRL
, we have shown that this is also not the case using an explicit example of an
information-theoretic causal loop embedded in Minkowski space-time without
leading to superluminal signalling. This highlights that these two principles
(no causal loops and no superluminal signalling) which are both associated
with relativistic causality are distinct principles that do not imply one
another and one must therefore be careful to distinguish them when analysing
relativistic causality.

##  IV A framework for relating information-processing and relativistic
causality in general theories

We use the framework we developed in VilasiniColbeckPRA  which has the
following features, which together allow the problems identified in the
previous section to be addressed:333A formalism incorporating similar
desiderata was also developed in VilasiniRenner2022 , for the purpose of
addressing open problems relating to so-called indefinite causal order quantum
processes Oreshkov2012 ; Chiribella2013 . In this work we apply the framework
of VilasiniColbeckPRA ; VilasiniColbeckPRL  as we are interested in proving
theory-independent statements in scenarios with a well-defined (but possibly
cyclic) causal structure.

  * •

Clearly disentangles and formalises spatio-temporal and information-theoretic
causality notions

  * •

Distinguishes between correlation, causation and signalling in a theory-
independent manner

  * •

Allows for non-classical, fine-tuned and cyclic causal influences in the
information-theoretic causal structure

  * •

Connects the two causality notions in order to formalise different
relativistic causality principles

We proceed by outlining this framework starting with causal models and then
how they are embedded in space-time.

###  IV.1 Causal models, affects relations and causal loops

To formalise an information-theoretic notion of causality, our framework from
VilasiniColbeckPRA  adopts a causal modelling approach formulated under
minimal and theory-independent assumptions so as to be applicable to scenarios
with cyclic and non-classical causal influences. Here we present an overview
of the main ingredients of the formalism, illustrating the core physical
intuition through examples, and refer the reader to Appendix A and
VilasiniColbeckPRA ; VilasiniColbeckPRL  for further details.

Causal structures and observed correlations A _causal structure_ is modelled
as a directed graph $\mathcal{G}$ over a set of nodes $N$ with directed edges
that denote direct causal influence from one node to another. This directed
graph may either be cyclic or acyclic and is a priori independent of space and
time. Each node of the causal structure is labelled either as “observed” or
“unobserved”. The former are classical random variables (e.g., measurement
settings and outcomes) while the latter may be a classical, quantum or post-
quantum system, such as a system described by a generalised probabilistic
theory (GPT). A causal structure is said to be classical, quantum or post-
quantum depending on the nature of its unobserved nodes and the joint
probability distribution $P_{\mathcal{G}}(N_{\mathrm{obs}})$ over the set
$N_{\mathrm{obs}}\subseteq N$ of all observed nodes of a causal structure
$\mathcal{G}$ is known as the _observed distribution_ of $\mathcal{G}$. Which
observed distributions $P_{\mathcal{G}}$ are allowed by a causal structure
$\mathcal{G}$ will in general depend on the theory describing its unobserved
nodes. Whenever $\mathcal{G}$ is evident from context, we will simply denote
$P_{\mathcal{G}}$ as $P$.

We use minimal, theory-independent constraints to define when we can associate
a probability distribution over the variables $N_{\mathrm{obs}}$ with a causal
structure $\mathcal{G}$. The idea behind this condition is that the topology
of the causal graph $\mathcal{G}$, in particular whether or not there are
graph separation relations between some sets of observed nodes, will imply
corresponding conditional independences in any associated observed
distribution over those nodes. The graph separation criterion used for this
purpose is called _d-separation_ (where d stands for “directed”), which was
first introduced in the classical causal modelling literature Pearl2009 ;
Spirtes2001  and more recently considered in the context of quantum and GPT
causal models Henson2014 ; Barrett2020A . We use
$(X\perp^{d}Y|Z)_{\mathcal{G}}$ to denote that $X$ is d-separated from $Y$
given $Z$ in $\mathcal{G}$ (where $X$, $Y$, $Z$ are disjoint sets of observed
nodes $N_{\mathrm{obs}}$ of $\mathcal{G}$), and refer the reader to Appendix A
for the full technical definition. Then for a causal structure $\mathcal{G}$
with a set of observed nodes $N_{\mathrm{obs}}$ we say that a distribution $P$
on $N_{\mathrm{obs}}$ satisfies the _d-separation property_ with respect to
$\mathcal{G}$ iff for all disjoint subsets $X$, $Y$, $Z$ of
$N_{\mathrm{obs}}$,

$(X\perp^{d}Y|Z)_{\mathcal{G}}\quad\Rightarrow\quad(X\upmodels Y|Z)_{P},$ (6)

where $(X\upmodels Y|Z)_{P}$ denotes the conditional independence
$P(XY|Z)=P(X|Z)P(Y|Z)$. This provides a theory-independent condition for
associating a set of possible observed distributions with a graph
$\mathcal{G}$. The d-separation property implies certain intuitive features.
For instance, in the typical bipartite Bell causal structure of Figure 1a, the
non-signalling constraints (NS2) on the observed correlations $P(XY|AB)$,
along with the fact that $P(AB)=P(A)P(B)$ are implied by corresponding
d-separations in the graph.

Causal model The directed edges of $\mathcal{G}$ capture the flow of
information through the network of observed and unobserved systems associated
with its nodes. How this flow of information is mathematically modelled will
depend on how states, information-processing channels and measurements (which
we will broadly refer to as the _causal mechanisms_) are modelled in the
theory under consideration. For example, in classical deterministic theories,
where all states are described by classical variables and all evolutions in
terms of deterministic functions, then all the nodes of the causal structure
would be classical random variables and we would require there to exist, for
every node $N$, a deterministic function $f_{N}:\mathrm{par}(N)\to N$ from the
set par$(N)$ of all the parents of $N$, to $N$, when $N$ has at least one
parent, and the specification of a distribution $P(N)$ when $N$ is parentless.
In classical probabilistic theories, one would require conditional
distributions $P(N|\mathrm{par}(N))$ (or equivalently, stochastic maps) for
each node $N$ that describe classical informational channels from the parents
to the node. [Probabilistic models can be equivalently described as
deterministic models by including additional (possibly hidden) variables
associated with additional nodes.]

More generally, these causal mechanisms can be quantum channels or dynamics
described by a post-quantum theory. Previous approaches to causal modelling
(in the classical and quantum literature) adopt a bottom-up approach where
they start from assumptions on the causal mechanisms (possibly non-classical)
associated with a graph and derive conditions (such as the d-separation
property) on the observed correlations that they generate and these results
are often derived for the case where the causal graph is acyclic. There are
several such approaches to quantum and non-classical causal models (e.g.,
Leifer2013 ; Henson2014 ; Costa2016 ; Allen2017 ; Barrett2020 ). In our
framework VilasiniColbeckPRA , we adopt a top-down approach where we make no
assumptions on the causal mechanisms or about the acyclicity of the causal
structure. Instead we impose the d-separation property (Definition A.3) at the
level of the observed correlations and study what this implies for inferring
properties of the underlying causal structure (which may be fully or partially
unknown). This gives the following minimal working definition of a causal
model that does not make any assumptions about the underlying causal
mechanisms of the theory (not even that the theory must have some circuit-like
structure).

######  Definition IV.1 (Causal model).

A causal model is a pair $(\mathcal{G},P_{\mathcal{G}})$ where $\mathcal{G}$
is a directed graph over a set $N_{\mathrm{obs}}$ of observed nodes (or
equivalently random variables) and possibly additional unobserved nodes
corresponding to classical/quantum/GPT systems, while $P_{\mathcal{G}}$ is a
joint probability distribution on $N_{\mathrm{obs}}$ that respects the
d-separation property (Equation (6)) relative to $\mathcal{G}$.

Throughout this paper we will often use $S_{1}S_{2}$ as shorthand for the
union $S_{1}\cup S_{2}$ of two sets of variables.

(a)  (b)

Figure 4: If the original causal structure $\mathcal{G}$ between the observed
nodes $D$, $R$ and unobserved node $S$ is that of (a), then the causal
structure of (b) is the post-intervention causal structure
$\mathcal{G}_{\mathrm{do}(D)}$ associated with an intervention on the node
$D$.

Interventions and affects relations So far, we have discussed correlations,
which are symmetric (unlike causation which is an asymmetric relation) and are
not sufficient for inferring a causal structure. For instance, the existence
of correlation between taking a drug (modelled as a variable $D$) and recovery
from a disease (modelled as a variable $R$) in a population does not tell us
whether or not the drug causes recovery, because it is possible to have a
common factor $S$ (such as socio-economic background) that influences both an
individual’s access to a drug and their immunity to the disease, which could
explain the same correlations. In addition to correlations, considering free
interventions is crucial for deducing causal relations from observable data.
Such interventions play an important role in controlled trials that allow us
to infer whether a drug causally influences recovery from a disease.

Figure 4 illustrates the concept of interventions in causal models. To deduce
whether $D$ is a cause of $R$, we intervene on $D$ and this transforms the
original causal structure $\mathcal{G}$ (which may be unknown) to a post-
intervention causal structure $\mathcal{G}_{\mathrm{do}(D)}$ that is a sub-
graph of $\mathcal{G}$ where the incoming arrows to $D$ are removed (making it
parentless). An intervention then corresponds to fixing $D$ to take a certain
value, $D=d$ in the post-intervention causal structure. Intuitively, this
models the situation in a control trial where the values $d$ of $D$ would
determine whether or not the drug (or a placebo) is advocated in a sub-
population. Moreover, $D$ being parentless in the post-intervention causal
structure models the natural assumption that the members of control group for
the drug test and the placebo group are chosen freely and independently of
prior causes of $D$ (such as $S$). Then, to infer whether $D$ causally
influences another observed variable $R$, we check whether intervention on $D$
changes the distribution on $R$. In other words, whether there exists a value
$d$ of $D$ such that the distribution on $R$ in the post-intervention scenario
with $D$ fixed to $d$ differs from the original, pre-intervention distribution
on $R$. If so, we will say that $D$ affects $R$. The following definitions
formalise this intuition.

In previous causal modelling frameworks (prior to that of VilasiniColbeckPRA
), the post-intervention causal model and causal inference are typically
defined in terms of the causal mechanisms of the underlying theory. The
framework of VilasiniColbeckPRA  enables a more minimal definition of these
concepts based on the minimal Definition A.3 of a causal model, which is
independent of the theory or causal mechanisms.

######  Definition IV.2 (Post-intervention causal model).

Consider a causal model on a graph $\mathcal{G}$ specified by the graph
together with a distribution $P_{\mathcal{G}}$ satisfying Definition A.3. A
post-intervention causal model associated with interventions on a subset $X$
of the observed nodes of $\mathcal{G}$ is specified by a graph
$\mathcal{G}_{\mathrm{do}(X)}$ obtained from $\mathcal{G}$ by removing all
incoming directed edges to the set $X$, together with a distribution
$P_{\mathcal{G}_{\mathrm{do}(X)}}$ satisfying Definition A.3 relative to
$\mathcal{G}_{\mathrm{do}(X)}$. Further, suppose $Y$ is a (possibly empty)
subset of the observed nodes. Whenever another subset of the observed nodes,
$X$ (disjoint from $Y$), contains only parentless nodes in
$\mathcal{G}_{\mathrm{do}(Y)}$444If $Y$ is empty, then
$\mathcal{G}_{\mathrm{do}(Y)}=\mathcal{G}$., we require that

$P_{\mathcal{G}_{\mathrm{do}(XY)}}(S|XY)=P_{\mathcal{G}_{\mathrm{do}(Y)}}(S|XY),$
(7)

where $S$ is the set of all other observed nodes.

######  Definition IV.3 (Affects relation).

Consider a causal model over a set of random variables $S$ with causal graph
$\mathcal{G}$ and a joint distribution $P_{\mathcal{G}}$ associated with it.
For any disjoint and non-empty subsets $X$, $Y\subseteq S$, we say that $X$
_affects_ $Y$ if there exists a value $x$ of $X$ such that

$P_{\mathcal{G}_{\mathrm{do}(X)}}(Y|X=x)\neq P_{\mathcal{G}}(Y).$

Operationally, $X$ _affects_ $Y$ tells us that $X$ _signals to_ $Y$, as it
means that there is an operation (given by the choice of intervention) that
can be performed locally on $X$ which results in an observable change at $Y$.
In general, it has been shown in VilasiniColbeckPRA  that affects relations
are non-transitive ($X$ affects $Y$ and $Y$ affects $Z$ does not imply $X$
affects $Z$) and also that, correlations between $X$ and $Y$ in the post-
intervention (on $X$) model implies that $X$ affects $Y$ i.e.,

$P_{\mathcal{G}_{\mathrm{do}(X)}}(Y|X)\neq
P_{\mathcal{G}_{\mathrm{do}(X)}}(Y)\quad\Rightarrow\quad X\text{ affects }Y.$
(8)

Interestingly, the converse of the above implication is not true, as shown by
a counter-example in VilasiniColbeckPRA . Importantly, $X$ affects $Y$ allows
us to infer that $X$ causally influences $Y$, i.e., if $X$ and $Y$ are sets of
RVs, $X$ affects $Y$ implies that there exists $e_{X}\in X$ and $e_{Y}\in Y$
such that there is a directed path from $e_{X}$ to $e_{Y}$ in the underlying
causal structure VilasiniColbeckPRA . However, the converse is not true, we
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structure but have causal models on such a structure where $X$ does not affect
$Y$. This is possible by fine-tuning the causal mechanisms of the model so as
the hide the causal influence from being observable at the level of the
affects relations. More formally, fine-tuning relates to the converse
statement of the d-separation property (Equation (6)), it tells that there is
an absence of observed correlation (captured by conditional independence)
despite a corresponding causal connection (captured by d-connection).

######  Definition IV.4 (Fine-tuned causal model).

A causal model $(\mathcal{G},P_{\mathcal{G}})$ is said to be fine-tuned
whenever for some disjoint sets $X$, $Y$ and $Z$ of observed nodes of
$\mathcal{G}$, we have $(X\upmodels Y|Z)_{P_{\mathcal{G}}}$ but
$(X\not\perp^{d}Y|Z)_{\mathcal{G}}$.

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ with a causal model
where $X$, $Y$, $Z$ are binary, $Z$ is uniformly distributed and $Y=X\oplus
Z$, then $P(XY)=P(X)P(Y)$ would hold for any distribution over $X$, thus
making the causal model fine-tuned. Since $X$ is parentless, this also implies
that $X$ does not affect $Y$ (even though $X$ is a cause of $Y$). Due to fine-
tuning, a causal relationship $X\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ does not in general
imply a corresponding affects relation $X$ affects $Y$, which motivates the
following classification of the arrows  into solid $\longrightarrow$ and
dashed
$\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}$ ones.

######  Definition IV.5 (Solid and dashed arrows).

Given a causal graph $\mathcal{G}$, if two observed nodes $X$ and $Y$ in
$\mathcal{G}$ sharing a directed edge $X\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ are such that $X$
affects $Y$, then the causal arrow  between those nodes is called a _solid
arrow_ , denoted $X\longrightarrow Y$. Further, all arrows  between observed
nodes in $\mathcal{G}$ that are _not_ solid arrows are called _dashed arrows_
, denoted
$\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}$ i.e.,
$X\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}Y$ in $\mathcal{G}$ implies that $X$ does not affect
$Y$, for any observed nodes $X$ and $Y$.

More generally, when considering interventions on multiple nodes, we can
envisage protocols by which agents are able to signal to each other once
information about other interventions are also given to them. The concept of
_higher-order affects relations_ that we introduced in VilasiniColbeckPRA
captures such signalling relations.

######  Definition IV.6 (Higher-order affects relation).

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

$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X=x,Z=z)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z=z),$ (9)

which we denote in short as $P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z)$.

When $Z=\emptyset$, we refer to this as a _zeroth-order affects relation_ and
it coincides with Definition IV.3.

As with (zeroth order) affects relations, higher-order affects relations $X$
affects $Y$ given do$(Z)$ also allow us to deduce that $X$ causally influences
$Y$, or equivalently that _at least one_ variable $e_{X}$ in $X$ influences a
variable $e_{Y}$ in $Y$, as shown below.

$X\text{ affects }Y\text{ given do}(Z)\quad\Rightarrow\quad\exists e_{X}\in
X,e_{Y}\in Y,\quad e_{X}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}e_{Y}\text{ in
}\mathcal{G},$ (10)

where $e_{X}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}e_{Y}$ denotes a
directed path from $e_{X}$ to $e_{Y}$. Notice that in a causal structure with
$X_{1}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ and another
disconnected node $X_{2}$ which has no in or outgoing edges, if we have
$X_{1}$ affects $Y$, we will in general also have $X_{1}X_{2}$ affects $Y$.
However we know that the latter affects relation does not add any new
information beyond the former one since $X_{2}$ is completely irrelevant in
this affects relation. The concept of _reducibility_ of (higher-order) affects
relations formalises this idea just at the level of the observed affects
relation (without assumptions on the causal structure). A formal definition
can be found in Appendix A. Applied to this simple example, we can say that
$X_{1}X_{2}$ affects $Y$ is a reducible affects relation while $X_{1}$ is an
irreducible affects relation, and $X_{1}X_{2}$ affects $Y$ can be reduced to
$X_{1}$ affects $Y$. In the case that $X$ affects $Y$ given do$(Z)$ is an
irreducible affects relation, we have a stronger causal inference statement
that allows us to infer causation from _every_ element $e_{X}$ of $X$.

$X\text{ affects }Y\text{ given do}(Z)\text{
irreducible}\quad\Rightarrow\quad\forall e_{X}\in X,\exists e_{Y}\in Y,\quad
e_{X}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}e_{Y}\text{ in
}\mathcal{G},$ (11)

Higher-order affects relations have several useful implications and results
for causal inference in the presence of cyclic and fine-tuned causal
influences as well as latent non-classical causes. In VilasiniColbeckPRA  we
show that _Pearl’s rules of do-calculus_ Pearl2009  originally proven for
classical causal models hold in our general framework and follow from the
d-separation property alone. The results mentioned here, along with other
properties proven in VilasiniColbeckPRA , reveal the rich and highly non-
trivial structure of (higher-order) affects relations and their precise
relations to the concepts of causation and correlations. More generally, we
have _conditional higher-order affects relations_ which also account for post-
selection on non-interventional nodes, an interested reader can find further
details on this in Appendix A.

Causal loops We can now provide a definition of causal loops that is a priori
independent of space-time. A causal loop simply corresponds to a directed
cycle in a causal structure $\mathcal{G}$ and we only consider causal loops
involving observed nodes i.e., where for two observed nodes $X$ and $Y$ in
$\mathcal{G}$, there exist directed paths from $X$ to $Y$ and from $Y$ to $X$.
This itself is not a very useful definition because, as we have seen, due to
fine-tuning, causation does not imply the existence of signalling (i.e.,
affects relations). The latter provided motivation to classify causal arrows
(Definition IV.5) into solid and dashed based on the operational detectability
of these causal influences through corresponding affects relations. Similarly,
we can distinguish between different types of causal loops in our framework
depending on whether they can be operationally detected through their affects
relations.

######  Definition IV.7 (Affects causal loops (ACL)).

Any set of affects relations $\mathscr{A}$ that can only arise in a causal
model associated with a cyclic causal structure $\mathcal{G}$ are said to
contain an affects causal loop. In other words, affects causal loops certify
the cyclicity of the underlying causal structure through the observed affects
relations.

The simplest example of an ACL is $\mathscr{A}=\\{X\text{ affects }Y,Y\text{
affects }X\\}$, where $X$ and $Y$ are individual variables (see also Example
B.1 later). The former relations implies that $X$ is a cause of $Y$ and the
latter that $Y$ is a cause of $X$, implying that any causal model capable of
producing these affects relations must contain a directed cycle between the
nodes associated with $X$ and $Y$. More generally we can have affects causal
loops characterised by affects relations between sets of observed nodes, e.g.,
$\mathscr{A}=\\{AC\text{ affects }B,B\text{ affects }AC\\}$ where both are
irreducible (if they are not irreducible, this affects relations may not
necessarily form an affects causal loop555Irreducibility in this case implies
that $A$ is a cause of $B$ and $C$ is a cause of $B$, and $B$ is a cause of at
least one of $A$ and $C$ (cf. Equation (11)), which implies a directed cycle
between $A$ and $B$ or $C$ and $B$. However, if $AC$ affects $B$ was
reducible, then it can be that only one of $A$ or $C$ is a cause of $B$ while
$B$ is a cause of the other variable which does not cause $B$ (cf. Equation
(10)), which would not necessarily be cyclic.). In VilasiniColbeckPRL  we have
constructed an explicit causal model with these affects relations and shown
that there exists a way to “embed” this affects causal loop in
$(1+1)$-dimensional Minkowski space-time such that it does not lead to
superluminal signalling, which establishes the (mathematical) possibility of
operationally detectable causal loops that can be compatible with a partially
ordered space-time. This example is illustrated and further explained in
Figure 8. Definitions and characterisations of several further classes of
affects causal loops that arise in this framework can be found in
VilasiniColbeckPRA .

######  Definition IV.8 (Hidden causal loop (HCL)).

Given a causal model whose causal structure contains a directed cycle, we say
that this causal model contains a _hidden causal loop_ if the same set of
affects relations and same correlations as this model are also realisable in a
causal model on an acyclic causal structure.

HCLs can never be operationally detected, and, as they can be explained also
in an acyclic model, one would naturally prefer the simpler acyclic
explanation of the associated operational predictions (see Example B.2 and
VilasiniColbeckPRA  for examples of HCLs).

###  IV.2 Space-time embeddings and compatibility

Having defined causality operationally, without reference to a space-time
structure, we now bring space-time into the picture to define when a causal
model can be said to be compatible with a space-time structure. To define this
compatibility, we need a few concepts at hand. The first is that of an
_ordered random variable_ (or ORV) which can be seen as an abstract version of
space-time random variables. Here, we model space-time simply as a partially
ordered set where elements appearing earlier in the order denote space-time
points that are in the past of elements appearing later in the order, and
unordered elements correspond to space-like separated points. More formally,
we have the following.

######  Definition IV.9 (Ordered random variable (ORV)).

Each ORV $\mathcal{X}$ is defined by a pair
$\mathcal{X}:=(R(\mathcal{X}),O(\mathcal{X}))$ where $R(\mathcal{X})$
(equivalently denoted by the corresponding, non-caligraphic letter $X$) is a
random variable and $O(\mathcal{X})\in\mathcal{T}$ specifies the location of
$X$ with respect to a partially ordered set $\mathcal{T}$.

We then have the following definition of what it means to embed a set of RVs
in a space-time.666The definition of embedding provided here is a simplified
version of the original definition proposed in VilasiniColbeckPRA . There, the
embedding also involves an additional concept of an accessible region of an
ORV, which corresponds to the subset of space-time locations at which it
possible to find copies of that RV. Within standard relativity theories, the
accessible region corresponds to the future light cone of the ORV, but in our
general framework, this could be any subset of locations. For the results of
present paper, we set the accessible region of an ORV to be the future light
cone of the associated space-time location (while including that location).

######  Definition IV.10 (Embedding).

Given a set of RVs $S$, an _embedding of $S$_ in a partially ordered set
$\mathcal{T}$ produces a corresponding set of ORVs $\mathcal{S}$ by assigning
a location $O(X)\in\mathcal{T}$ to each $X\in S$. An embedding of a set of RVs
is called _non-trivial_ if no two RVs $X$ and $Y$ such that $X$ affects $Y$
are assigned the same location in $\mathcal{T}$.

_Notation:_ We use $\prec$, $\succ$ and $\nprec\nsucc$ to denote the order
relations for a given partially ordered set $\mathcal{T}$, where for $\alpha$,
$\beta\in\mathcal{T}$, $\alpha\nprec\nsucc\beta$ corresponds to $\alpha$ and
$\beta$ being unordered with respect to $\mathcal{T}$. This is not to be
confused with $\alpha=\beta$ which corresponds to the two elements being
equal. These relations carry forth in an obvious way to ORVs and we say for
example that 2 ORVs $\mathcal{X}$ and $\mathcal{Y}$ are ordered as
$\mathcal{X}\prec\mathcal{Y}$ iff $O(\mathcal{X})\prec O(\mathcal{Y})$. In
addition, $\mathcal{X}=\mathcal{Y}$ is equivalent to $X=Y$ and
$O(\mathcal{X})=O(\mathcal{Y})$.

######  Definition IV.11.

The _inclusive future_ of an ORV is the set

$\overline{\mathcal{F}}(\mathcal{X}):=\\{\alpha\in\mathcal{T}:\alpha\succeq
O(\mathcal{X})\\}.$

The inclusive future of a set $\mathcal{S}$ of ORVs is given as the
intersection of the inclusive futures of the ORVs in the set, and captures the
region where they can be jointly accessed777In the original framework
VilasiniColbeckPRA  a separate concept called the _accessible region_ of an
ORV is defined which is a priori distinct from the future of the ORV. Here, we
have set the accessible region to be equal to inclusive future, such that the
compatibility condition captures the impossibility of signalling outside the
space-time future.

$\overline{\mathcal{F}}(\mathcal{S}):=\bigcap_{\mathcal{X}\in\mathcal{S}}\overline{\mathcal{F}}(\mathcal{X}).$

Once we assign space-time locations to observed RVs in a causal model to
promote them to ORVs, a natural question that arises is whether the affects
relations of the model can be used to signal outside the space-time’s future.
For this, notice that a higher-order affects relation $X$ affects $Y$ given
do$(Z)$ (representing that $X$ can signal to $Y$ given information about
interventions performed on $Z$) between disjoint sets of RVs essentially
captures the most general way of signalling in such a causal modelling
framework888In our original paper VilasiniColbeckPRA  we also define
conditional higher-order affects relations of the form $X$ affects $Y$ given
$\\{\mathrm{do}(Z),W\\}$ where we have an additional conditioning on values of
some set $W$ of nodes which are not intervened upon, this captures signalling
from $X$ to $Y$ given an intervention on $Z$ and a post-selection on $W$. Our
compatibility condition also takes this into account. In this paper, these
conditional relations are not relevant and we therefore don’t consider them,
however they are defined in Appendix A for completeness.. When embedded in
space-time, this becomes an affects relation $\mathcal{X}$ affects
$\mathcal{Y}$ given do$(\mathcal{Z})$ between corresponding sets of ORVs. Now,
if Alice, who has access to $X$ and can intervene on it, wishes to signal to
Bob through this affects relation, Bob needs to jointly access $Y$ and $Z$ to
receive the signal from Alice. Once these variables are embedded in space-
time, in order to ensure that the affects relation does not enable signalling
outside the space-time future, we would require that the joint future of
$\mathcal{Y}$ and $\mathcal{Z}$ (which is where Bob can jointly access $Y$ and
$Z$ to receive the signal) must be contained in the future of $\mathcal{X}$
(where Alice performs her interventions). Further, we should only demand this
for irreducible affects relations, as we would otherwise be imposing
unnecessarily strong constraints. For instance in our example with
$X_{1}\leavevmode\hbox to17.47pt{\vbox
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in or outgoing edges, if we impose compatibility for the reducible relation
$X_{1}X_{2}$ affects $Y$ then we would be requiring $Y$ to be embedded in the
future of $X_{2}$ even though $X_{2}$ is completely causally disconnected from
$Y$. However, if we reduce all our reducible relations to their minimal
irreducible form, which in this case is $X_{1}$ affects $Y$, then
compatibility only requires $Y$ to be embedded in the future of the location
where we embed $X_{1}$, as we would expect. Therefore, the concept of
irreducibility is key for ensuring that our compatibility condition is both
necessary and sufficient for no superluminal signalling.

######  Definition IV.12 (Compatibility of a set of affects relations with an
embedding in a partial order ($\mathbf{compat}$)).

Let $\mathcal{S}$ be a set of ORVs formed by embedding a set of RVs $S$ in a
space-time $\mathcal{T}$ with embedding $\mathscr{E}$. Then a set of affects
relations $\mathscr{A}$ is said to be _compatible_ with the embedding
$\mathscr{E}$ if whenever $X$ affects $Y$ given do$(Z)$ belongs to
$\mathscr{A}$, and is irreducible with respect to the affects relations in
$\mathscr{A}$, then
$\overline{\mathcal{F}}(\mathcal{Y})\bigcap\overline{\mathcal{F}}(\mathcal{Z})\subseteq\overline{\mathcal{F}}(\mathcal{X})$
with respect to $\mathscr{E}$.

######  Definition IV.13 (Compatibility of a causal model with an embedding in
a partial order).

We say that a causal model over a set of RVs $S$ is compatible with an
embedding in a partial order if the set of affects relations $\mathscr{A}$
implied by the causal model are compatible with the embedding (cf. Definition
IV.12).

##  V Formal definitions of Bell scenarios and corresponding space-time
embeddings

Based on the framework reviewed in the previous section, we now provide a
formal definition of Bell scenarios, as well as the associated space-time
embeddings, that will be used in the rest of the paper.

######  Definition V.1 (Bi and tripartite Bell scenarios).

A bi (tri) partite Bell scenario refers to any experiment involving two
(three) parties Alice and Bob (and Charlie) who share a common system
$\Lambda$ and measure their respective sub-system as given by their
measurement settings $A$, $B$ (, $C$) obtaining the corresponding measurement
outcomes $X$, $Y$ (, $Z$) respectively. We assume that the settings are freely
chosen i.e., have no causes which are relevant to the experiment. Any causal
model associated with a bi (tri) partite Bell scenario will have the setting
variables $A$, $B$ (, $C$) as parentless nodes (capturing the free choice
assumption) and $\Lambda$ as a common cause of the outcomes $X$, $Y$ (, $Z$)
and itself being parentless. The settings and outcomes are always observed
classical nodes, while $\Lambda$ may be observed or unobserved and can thus be
described using a general, non-classical theory.

The above is a minimal definition of a Bell scenario which does not fix the
(information-theoretic) causal structure. In particular, the typical causal
structures associated with Bell scenarios (Figures 1a and 3b), satisfy this
definition. More generally, the definition also allows for causal structures
where the setting of one party (e.g., $B$) is the cause of the outcome of
another party (e.g., $X$), as well as cyclic causal structures where two
different outcome variables can mutually influence each other. Note that
although it would be natural to require that the settings are causes of the
respective outcomes, our minimal definition of Bell scenarios does not enforce
this, and allows situations where a party’s outcome is not caused by their
setting.

######  Definition V.2 (Space-time configuration for standard Bell scenarios).

A bipartite (or tripartite) Bell scenario is said to take place in a standard
space-time configuration if the random variables corresponding to the
measurement settings $A$, $B$ (, $C$) and outcomes $X$, $Y$ (,$Z$) are
embedded in space-time such that: $\mathcal{A}\prec\mathcal{X}$,
$\mathcal{B}\prec\mathcal{Y}$ (and $\mathcal{C}\prec\mathcal{Z}$) and the
remaining pairs of space-time embedded variables are space-like separated
(related by $\not\prec\not\succ$), where the order relations $\prec,\succ$ are
defined by the light cone structure of the space-time. Such a space-time
configuration is illustrated in Figure 2 (and Figure 3a).

Notice that our definition of a Bell scenario does not a priori forbid
superluminal causal influences when the Bell scenario takes place in the
standard space-time configurations. This will allow us to derive non-trivial
statements about relativistic causality principles in Bell scenarios without
assuming causal structures (such as those of Figures 1a and 3b) that already
enforce these principles once embedded in space-time. Further details on how
we define and model information-theoretic causal structures and their space-
time embeddings can be found in Section IV.

The jamming space-time configuration in a tri-partite Bell scenario, is
slightly different from the standard configuration as it involves as
additional condition, as defined below.

######  Definition V.3 (Space-time configuration for jamming tripartite Bell
scenario).

A tripartite Bell scenario is said to take place in a jamming space-time
configuration if the ORVs corresponding to the measurement settings $A$, $B$,
$C$ and outcomes $X$, $Y$, $Z$ are embedded in space-time such that:
$\mathcal{A}\prec\mathcal{X}$, $\mathcal{B}\prec\mathcal{Y}$,
$\mathcal{C}\prec\mathcal{Z}$,
$\overline{\mathcal{F}}(\mathcal{X})\bigcap\overline{\mathcal{F}}(\mathcal{Z})\subseteq\overline{\mathcal{F}}(\mathcal{B})$,
all other pairs of ORVs are space-like separated. Such a space-time
configuration is illustrated in Figure 3a.

Note that $X$, $B$ and $Z$ being spacelike separated is not enough to ensure
that
$\overline{\mathcal{F}}(\mathcal{X})\bigcap\overline{\mathcal{F}}(\mathcal{Z})\subseteq\overline{\mathcal{F}}(\mathcal{B})$,
even if the spatial coordinates of $X$, $B$ and $Z$ lie on a straight line
except in $(1+1)$ dimensional Minkowski space-time — see Appendix C for a
discussion of the possible space-time locations in $(1+1)$ and $(2+1)$D
Minkowski space-time.

##  VI Causal modelling analysis of jamming and implications for relativistic
causality

In this section, we analyse jamming correlations in a Bell scenario using a
causal modelling approach. We then show that in a Bell scenario where
$\Lambda$ is an observed node and which takes place in the space-time
configuration of Figure 3a, jamming correlations will lead to superluminal
signalling.

###  VI.1 Jamming requires fine-tuning and superluminal causal influence

Using our causal modelling framework formulated under minimal assumptions, the
following observations about correlation and causation in scenarios with
jamming follow immediately. Firstly, jamming correlations can only be
explained by a fine-tuned causal model; secondly, jamming in the space-time
configuration of Grunhaus1996 ; Horodecki2019  necessarily involves
superluminal causal influences (whether or not it involves superluminal
signalling).

######  Lemma VI.1.

In any tripartite Bell scenario (cf. Definition V.1) where $B$ is neither a
cause of $X$ nor of $Z$, it is impossible to generate jamming correlations in
any theory i.e., irrespective of the nature (classical, quantum, post-quantum)
of the common cause $\Lambda$.

######  Corollary VI.1.

The presence of jamming in a tripartite Bell scenario taking place in a
jamming space-time configuration (Definition V.3) implies the existence of
superluminal causal influence between at least one pair of space-like
separated parties. This is already the case when $\Lambda$ is classical.

######  Lemma VI.2.

Any causal model reproducing the jamming correlations must be fine-tuned,
irrespective of the theory describing the common cause $\Lambda$. In other
words, there exists no faithful causal model (classical or non-classical) for
a Bell scenario that generates jamming correlations.

Proofs of these results can be found in Appendix B.1.

Figure 5: Causal structure of the tripartite Bell scenario with jamming As
shown in Lemma VI.1, in order for a causal model over the settings and
outcomes of a tripartite Bell scenario to admit jamming correlations, the
original tripartite Bell causal structure of Figure 3b must be modified to
allow for causal influence from $B$ to $X$ and/or $Z$. Further, as shown in
Lemma VI.2, assuming that the settings $A$, $B$, $C$ are parentless nodes both
these causal influences must be fine-tuned, hence represented by dashed
arrows.

###  VI.2 Jamming can lead to superluminal signalling

Here we show that jamming can indeed lead to superluminal signalling i.e.,
that (NS3′) being satisfied in a tripartite Bell scenario is not sufficient
for ruling out superluminal signals (contrary to the claims reviewed in
Section III). We first illustrate the possibility of signalling by
constructing a causal model for the simplest jamming scenario, where Alice and
Charlie have no settings and Bob has no outcome. Here the only relevant
variables are Alice and Charlie’s outcomes $X$ and $Z$, and Bob’s setting $B$.
We show this simplest jamming scenario can already lead to superluminal
signalling in the space-time embedding considered in Grunhaus1996 ;
Horodecki2019  i.e., that of Definition V.3. This protocol captures the main
intuition behind our general Theorem VI.1 that will follow.

A causal model for jamming. Consider the causal structure shown in Figure 6
where $\Lambda$ and $B$ are direct causes of $X$ and $\Lambda$ is a direct
cause of $Z$. Suppose that $\Lambda$ is observed (and hence a classical RV)
and all four variables, $\Lambda$, $X$, $B$ an $Z$ are binary and uniformly
distributed. Then a causal model over this causal structure is given by
specifying functions from the parents of $X$ ($B$ and $\Lambda$) to $X$ and
from the parents of $Z$ (just $\Lambda$) to $Z$. We take $X=\Lambda\oplus B$
and $Z=\Lambda$. This immediately gives the correlation $B=X\oplus Z$ between
Bob’s setting $B$ and Alice and Charlie’s outcomes $X$ and $Z$. Moreover, we
can verify that these correlations do satisfy (NS3′) i.e., $P(X|B)=P(X)$ and
$P(Z|B)=P(Z)$ and also admit jamming since $P(XZ|B)\neq P(XZ)$. Since $B$ is
parentless, this is equivalent to: $B$ does not affect $X$ (hence the causal
arrow from $B$ to $X$ is dashed in Figure 6), $B$ does not affect $Z$ but $B$
affects $XZ$. These affects relations are compatible with the space-time
configuration of the figure.

However, since $\Lambda$ is observed, we additionally have the affects
relation $B\Lambda$ affects $X$ and $\Lambda$ affects $Z$ (hence the solid
causal arrow), along with $\Lambda$ does not affect $X$ (hence the dashed
causal arrow). The affects relation $B\Lambda$ affects $X$ is irreducible in
this model (since $B$ affects $X$ given do$(\Lambda)$ and $\Lambda$ affects
$X$ given do$(B)$) and would lead to superluminal signalling unless $X$ is
embedded in future of $B$ and of $\Lambda$. However, in the given space-time
configuration (which is the one proposed in Grunhaus1996 ; Horodecki2019 ),
the ORVs $\mathcal{X}$ and $\mathcal{B}$ are space-like separated and
therefore the aforementioned affects relation leads to superluminal signalling
in this space-time configuration, irrespective of the space-time location of
$\Lambda$. The following theorem (proven in Appendix B.1) generalises this
idea and establishes the possibility of superluminal signalling using an
observed $\Lambda$ in any tripartite Bell scenario with jamming.

(a)  (b)  (c)

Figure 6: A simple example of how jamming as described in Grunhaus1996 ;
Horodecki2019  can lead to superluminal signalling: (a) Space-time embedding
of the variables $X$, $B$, $Z$ and $\Lambda$ suggested in Grunhaus1996 ;
Horodecki2019 , where the joint future of the space-time variables
$\mathcal{X}$ and $\mathcal{Z}$ (blue region) is contained in the future of
$\mathcal{B}$. Here, $B$ corresponds to the input of a party, Bob while $X$
and $Z$ are the outputs of Alice and Charlie respectively. In this
configuration, Grunhaus1996 ; Horodecki2019  claim that there will be no
superluminal signalling whenever the correlations $P(XZ|B)$ satisfy (NS3′)
(where $A$, $C$ and $Y$ are taken to be trivial). (b) Causal structure in
which the classical causal model described in the text satisfies (NS3′) and
admits jamming, but leads to superluminal signalling in the space-time
configuration if $\Lambda$ is observed. This shows that the claim does not
hold in general, and it is necessary to assume that $\Lambda$ must be
unobserved and fundamentally inaccessible in any theory that allows for
jamming in this space-time configuration. (c) The correlations obtained from
the causal model, which satisfy (NS3′).

######  Theorem VI.1.

Consider a tripartite Bell scenario (Definition V.1) where the outcomes $X$,
$Y$ and $Z$ do not causally influence any systems relevant to the experiment
and thus are modelled as childless nodes in the causal structure. Suppose that
the scenario admits jamming (Definition III.2), and its causal model is
embedded in Minkowski space-time as per the jamming space-time configuration
(Definition V.3). Then, irrespective of the space-time embedding of $\Lambda$,
the causal model is incompatible with the space-time and leads to superluminal
signalling whenever $\Lambda$ is an observed system.

Non-classical causal models exhibiting jamming. The causal model for jamming
described above is entirely classical. Quantum and post-quantum extensions of
the same protocol which generate jamming correlations satisfying (NS3′) are
also possible. For example, consider Alice and Charlie who share a bipartite
state, measure it using the inputs $A$ and $C$ and obtain the outcomes $X$ and
$Z$, and Bob making an input $B$ that can causally influence at least one of
$X$ or $Z$. Suppose, depending on $B$, we would like Alice and Charlie to
share either correlations arising from measurements on the Bell state
$|\psi_{0}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ or those arising
from the same measurements on a different Bell state,
$|\psi_{1}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$. One causal
structure that could be used for this is an extension of the one in Figure 6b
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}X$ and
$\Lambda\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z$) with additional
causal influences $A\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
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}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}X$ and
$C\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
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}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z$ and where $\Lambda$
now represents an unobserved, quantum node. In the causal model, $\Lambda$
could correspond to a bipartite quantum system in the state
$\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, $X$ is generated from the first
subsystem of $\Lambda$, $A$ and $B$ by first applying a controlled not on the
subsystem of $\Lambda$ with $B$ as control, and then measuring the resulting
state of the subsystem in the basis choice given by $A$, while $Z$ is obtained
from the second subsystem of $\Lambda$ by directly measuring in the basis
choice given by $C$. Then the parties would measure $|\psi_{0}\rangle$
whenever $B=0$ and $|\psi_{1}\rangle$ whenever $B=1$ as required and generate
the required correlations. Similarly, $B$ can be used to decide whether Alice
and Charlie share one relabelling of a PR box or another and we can
analogously construct a post-quantum causal model that achieves
this.999Similarly, one can devise protocols to realise other types of jamming
correlations by allowing the underlying mechanisms (like the choice of local
operation prior to measurement) to depend on $B$ in a way that this does not
reflect in the local statistics. Further one can also consider jamming
correlations where Bob’s output $Y$ is non-trivially involved in the process,
which we do not analyse here as it is not directly relevant to the results of
this paper.

The causal model as a physical protocol vs dynamics of a post-quantum theory.
The causal models described above are purely information-theoretic, and have
been classified as classical, quantum or post-quantum depending on the nature
of the node $\Lambda$. On the other hand, whether such a causal model is
physically implementable in space-time, according to classical, quantum or
some other theory, depends not only on the causal model but also on the space-
time embedding. For instance, if every causal influence in the model flows
from past to future in the space-time (in particular, this would mean that
$\mathcal{B}$ is embedded in the past light-cone of $\mathcal{X}$), then the
model would not lead to any superluminal causal influences relative this
embedding and describes a physical protocol within classical, quantum or a
post-quantum theory (according to the nature of the model). Notice that
jamming correlations can be realised whether the model is classical, quantum
or post-quantum.

On the other hand, consider the classical causal model for jamming relating to
the causal structure in Figure 6b embedded in space-time according to to the
jamming space-time embedding illustrated in Figure 6a. In order to avoid
superluminal signalling in this case, it is necessary that $\Lambda$ is
unobserved or inaccessible to physical interventions (as implied by Theorem
VI.1). Withing our framework, the proposed causal model is compatible with the
space-time embedding of Figure 6 if and only if $\Lambda$ is unobserved: the
only if part is implied by Theorem VI.1 and the if part is immediate from
applying the compatibility condition to the affects relations of the example
(also shown explicitly in VilasiniColbeckPRA ; VilasiniColbeckPRL ). However,
the causal influence from $B$ to $X$ is then superluminal and goes outside the
future lightcone, which is not something we can realise within standard
quantum theory. More explicitly, the correlations between Bob’s setting $B$
and Alice and Charlie’s outcomes $X$ and $Z$ arising in the above mentioned
protocols cannot be realised in any tripartite quantum mechanical experiment
where the three parties are space-like separated101010This is due to the non-
signalling property of quantum theory which will ensure that measurements on
different subsystems in any tripartite Bell scenario always lead to
correlations that satisfy the stronger non-signalling condition (NS3) which
forbids jamming.. Therefore a theory that generates jamming correlations in a
jamming space-time configuration is truly post-quantum even though it may
admit a classical or quantum causal model.

Comparison to non-local hidden variable theories. Theorem VI.1 demonstrates
that post-quantum theories admitting jamming correlations in the space-time
configuration of Grunhaus1996 ; Horodecki2019  are similar to non-local hidden
variable explanations of Bell non-local correlations in that they involve
superluminal signalling at the level of certain underlying variables without
allowing superluminal signalling at the observed level, if these variables
remain inaccessible. Explicitly, in non-local HV models for bipartite Bell
correlations, we have the no-signalling conditions $P(X|AB)=P(X|A)$ and
$P(Y|AB)=P(Y|B)$ by construction even though one of
$P(X|AB\Lambda)=P(X|A\Lambda)$ and $P(Y|AB\Lambda)=P(Y|B\Lambda)$ necessarily
fails. If $\Lambda$ were entirely accessible, then Alice and Bob could exploit
this to signal to each other in a space-like separated Bell experiment.
Similarly, jamming correlations satisfy $P(X|ABC)=P(X|A)$ and
$P(Z|ABC)=P(Z|C)$ along with $P(XZ|ABC)\neq P(XZ|AC)$. However we have shown
that we must necessarily have at least one of $P(X|ABC\Lambda)\neq
P(X|A\Lambda)$ or $P(Z|ABC\Lambda)\neq P(Z|C\Lambda)$ whenever $\Lambda$ is
classical in order to have jamming. This implies that $\Lambda$ must be
unobserved even when it is classical, in order to avoid superluminal
signalling. Another point of similarity is that both non-local HV explanations
of Bell inequality violating correlations as well as causal explanations of
jamming correlations must necessarily be fine-tuned causal explanations. This
is established for non-local HV explanations in Wood2015  and for the jamming
in case in our Lemma VI.2. A crucial point of difference is however that in
the former case, there exists a faithful explanation of the correlations
through a quantum (or post-quantum) causal model while our results establish
that any explanation of jamming correlations, through classical quantum or
post-quantum causal model must necessarily be fine-tuned.

##  VII Relativistic causality in Bell scenarios: accounting for arbitrary
interventions

In this section, we apply our framework for causal inference and compatibility
with space-time to the bi- and tripartite Bell scenarios. The usual treatment
of these scenarios, including discussions on relativistic causality
violations, focuses on correlation constraints such as (NS2), (NS3) and
(NS3′). More generally however, we would like to rule out relativistic
causality violations even when arbitrary interventions are considered. Our
formalism allows us to systematically treat interventions and derive stronger
results for relativistic causality that can rule out superluminal signalling
under all possible interventions on observed variables. Proofs of all results
can be found in Appendix B.

###  VII.1 Non-signalling constraints and higher-order affects relations

We first start by casting the non-signalling constraints (NS2) and (NS3′)
(which are expressed in terms of correlations) in terms of higher-order
affects relations which we have introduced in our framework for capturing the
general interventions on observed nodes. Proofs of these theorems can be found
in Appendix B.2.

######  Theorem VII.1.

In any bipartite Bell scenario (Definition V.1), the following non-affects
relations are equivalent to the bipartite no-signalling conditions (NS2):

$\displaystyle\begin{split}B&\text{ does not affect }X\text{ given do}(A),\\\
A&\text{ does not affect }Y\text{ given do}(B).\end{split}$ (12)

######  Theorem VII.2.

In any tripartite Bell scenario (Definition V.1) the following non-affects
relations are equivalent to the relaxed tripartite no-signalling conditions
(NS3′):

$\displaystyle\begin{split}&C\text{ does not affect }XY\text{ given
do}(AB),\\\ &A\text{ does not affect }YZ\text{ given do}(BC),\\\ &BC\text{
does not affect }X\text{ given do}(A),\\\ &AB\text{ does not affect }Z\text{
given do}(C).\\\ \end{split}$ (13)

Correlation does not imply causation or signalling, however, correlation
constraints such as (NS2), (NS3) and (NS3′) capture the idea of no-signalling
between parties through interventions on freely chosen settings (Definition
V.1). Higher-order affects relations that we have introduced in
VilasiniColbeckPRA  capture signalling/no-signalling constraints in more
general scenarios beyond Bell-type situations, where we have no freely chosen
or parentless variables and where signalling cannot be always captured by the
correlations alone. The above theorems show that in the special case of Bell
scenarios, some of the higher-order affects relations are equivalent to the
correlation constraints (NS2) and (NS3′). In essence, this is because in any
causal model (according to Definition IV.1), correlation between a parentless
variable $A$ and another variable $X$ implies causal influence from $A$ to $X$
as well as an affects relation $A$ affects $X$.

###  VII.2 Conditions for preserving relativistic causality

Theorem VI.1 highlights that jamming correlations arising in tri-partite Bell
scenarios in the relevant space-time configuration, can lead to superluminal
signalling if the common cause is accessible to agents. This raises the
question of whether additionally taking $\Lambda$ to be unobserved is
sufficient for preserving relativistic causality. We analyse this question
within the causality framework reviewed in Section IV and find that in
general, superluminal signalling is possible in the standard space-time
configurations of a tripartite (/bipartite) Bell scenario even when we impose
(NS3′) (/(NS2)) along with the requirement that $\Lambda$ is unobserved.
Intuitively, this is because (NS2) and (NS3′) are constraints only on the
correlations but signalling and causation are concepts that are not fully
determined by correlations alone, but also require us to consider
interventions. Some of the previous analyses of jamming Horodecki2019  (cf.
Section III) focus on the relativistic causality principle of “no causal
loops”, here we consider both this and the related relativistic principle “no
superluminal signalling”. These principles do not imply one another in
general, as demonstrated by our recent work VilasiniColbeckPRL , which
highlights the importance of disentangling these different causality
principles.

####  VII.2.1 Conditions for ruling out superluminal signalling

The compatibility of a causal model with a space-time embedding (cf.
Definition IV.12) formalises the requirement of no superluminal signalling
while taking into account general interventions. Here, we apply this to
characterise conditions for no superluminal signalling in the bipartite and
tripartite Bell scenarios, and the relevant configurations in Minkowski space-
time. Proofs of all results of this subsection can be found in Appendix B.3.

Accounting for interventions on outcomes. In Bell scenarios where measurement
settings are freely chosen, the usual non-signalling constraints on
correlations capture the inability of different parties to communicate to each
other through interventions on the setting variables. More generally, one can
consider situations where it is impossible to signal between parties through
interventions on settings, but still possible to signal once we also consider
interventions on outcomes, which is not captured by the usual non-signalling
constraints.

By applying the causal inference formalism, we account for interventions on
all observed variables (settings and outcomes) of the scenario. If the causal
structure is one where the settings are free (parentless) then interventions
on settings need not be explicitly considered (see Definition IV.2).
Typically, interventions on outcomes are not considered because the causal
structure is taken to be of the form of Figures 2 or 3b where the outcomes are
assumed to be childless, i.e., to not be causes of any other variables
relevant to the experiment. However, these causal structures are acyclic, and
when embedded in Minkowski space-time in the usual way, all the causal arrows
in these structures flow from past to future in the space-time and do not
allow for superluminal signals. Thus relativistic causality violations such as
superluminal signalling and causal loops are ruled out by assumption. Here, we
have defined Bell scenarios without fixing the causal structure to be of this
form, but only requiring that the setting variables are parentless (to model
that they are freely chosen). This definition of Bell scenarios, and their
standard space-time embedding does not rule out superluminal signalling or
causal loops by construction. Furthermore, the following simple example
illustrates that imposing (NS2) and (NS3′) in such Bell scenarios is
insufficient for ruling out signalling.

######  Example VII.1 ((NS2)/(NS3) are insufficient for ruling out
superluminal signalling in Bell scenarios).

Consider a causal structure over variables $A$, $B$, $X$ and $Y$ where
$A\leavevmode\hbox to17.47pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ and
$X\leavevmode\hbox to17.47pt{\vbox
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{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ are the only edges
(Figure 7). Consider a causal model over this causal structure where all
variables are binary, $A$ and $X$ are uniformly distributed and $Y=A\oplus X$,
while $B$ can have any distribution (note that it is a dummy variable, which
is both parentless and childless). This causal structure satisfies our minimal
definition of a bipartite Bell scenario (Definition V.1), and the associated
model leads to probabilities $P(XY|AB)$ obeying the non-signalling conditions
(NS2) (since $A$ does not affect $Y$ given do$(B)$ and $B$ does not affect $X$
given do$(A)$, cf. Theorem VII.1). However we have $AX$ affects $Y$ (since $A$
and $X$ are parentless and correlated with $Y$), which enables Alice (who has
access to $A$ and $X$) to signal to Bob (who has access to $Y$). This signal
would be superluminal in any space-time configuration where Alice and Bob’s
measurements take place at space-like separated regions. The same construction
can be trivially embedded in a tripartite Bell scenario, by choosing $C$ and
$Z$ to be dummy variables, such that (NS3) (and hence (NS3′) is also trivially
satisfied and we still have the possibility of signalling from Alice to Bob.
Notice that $\Lambda$ is irrelevant to this example, so imposing that
$\Lambda$ is unobserved (which is necessary for avoiding superluminal
signalling as shown in Theorem VI.1) is insufficient to avoid superluminal
signalling in this example.

The above example emphasizes the need to account for interventions in addition
to correlation constraints such as (NS2) and (NS3) to rule out superluminal
signalling in Bell type scenarios in space-time, which we do in the following
theorems. In the following, when we say there are no affects relations
emanating from a set $S_{1}$ we mean that there are no (possibly higher-order)
affects relations $S_{1}$ affects $S_{2}$ given do$(S_{3})$ (cf. Definition
IV.6) with $S_{1}$ being the first argument.

######  Theorem VII.3.

Consider a bipartite Bell scenario according to Definition V.1 where $\Lambda$
is unobserved. Let the observed nodes $\\{A,B,X,Y\\}$ of the causal structure
be embedded in Minkowski space-time according to the standard bipartite
configuration of Definition V.2. Then the following conditions are necessary
and sufficient for the causal model of the scenario to be compatible with such
a space-time embedding.

  1. 1.

The correlations $P(XY|AB)$ satisfy (NS2).

  2. 2.

There are no affects relations emanating from any subset of $\\{X,Y\\}$.

In the above, we have provided necessary and sufficient conditions formulated
at the level of the observed correlations and affects relations in the Bell
scenario, which are operationally accessible quantities. However, often one
makes assumptions about the underlying causal structure (which may include
unobserved systems and may not always be directly operationally accessible).
Such assumptions at the level of the causal structure are often much stronger
than their counterparts at the level of affects relations, thus the above
theorem has the following corollary.

######  Corollary VII.1.

In any bipartite Bell scenario (Definition V.1) where $\Lambda$ is unobserved
and we assume that the outcomes $X$ and $Y$ are not causes of any variables
relevant to the experiment, the bipartite correlation constraints (NS2) are
necessary and sufficient for no-superluminal signalling in the standard space-
time configuration of such a scenario (Definition V.2).

The corollary follows because in any such causal structure, Condition 2 of