[]axiom

# Impossibility of superluminal signalling in Minkowski space-time does not
rule out causal loops

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

###### Abstract

Causality is fundamental to science, but it appears in several different
forms. One is relativistic causality, which is tied to a space-time structure
and forbids signalling outside the future. A second is an operational notion
of causation that considers the flow of information between physical systems
and interventions on them. In [Vilasini and Colbeck, Phys. Rev. A. 106, 032204
(2022)], we propose a framework for characterising when a causal model can
coexist with relativistic principles such as no superluminal signalling, while
allowing for cyclic and non-classical causal influences and the possibility of
causation without signalling. In a theory without superluminal causation, both
superluminal signalling and causal loops are not possible in Minkowski space-
time. Here we demonstrate that if we only forbid superluminal signalling,
superluminal causation remains possible and show the mathematical possibility
of causal loops that can be embedded in a Minkowski space-time without leading
to superluminal signalling. The existence of such loops in the given space-
time could in principle be operationally verified using interventions. This
establishes that the physical principle of no superluminal signalling is not
by itself sufficient to rule out causal loops between Minkowski space-time
events. Interestingly, the conditions required to rule out causal loops in a
space-time depend on the dimension. Whether such loops are possible in three
spatial dimensions remains an important open question.

Introduction.—Understanding cause-effect relations is central to the
scientific method, yet there are several inequivalent notions of causality.
Often, it is defined with respect to a background space-time structure, after
which causal structure and space-time structure are treated synonymously. An
alternative is to define causality operationally and independently of space-
time. One way to do this is through causal models, which are based on
intervening on physical systems and analysing the resulting correlations
Pearl2009 ; Spirtes2001 . This is the approach we take here. Causal models
have been extensively applied to situations involving classical variables,
being used for instance for medical testing Kleinberg2011 ; Raita2021 ,
economic predictions Spirtes2005 ; Pearl2009 , and machine learning Maya2014 ;
Kusumawardani2020 ; Liu2021 .

Bell’s theorem Bell  demonstrates that classical causal models cannot explain
quantum correlations within the causal structure that is naturally associated
with a Bell experiment Wood2015 . This has fuelled several approaches for
providing causal explanations to quantum and more general non-classical
correlations. One approach is to develop causal models Tucci_1995 ;
Leifer_2006 ; Laskey2007 ; Leifer_2008 ; Leifer2013 ; Henson2014 ; Wood2015 ;
Pienaar2015 ; Ried_2015 ; Costa2016 ; Fritz_2015 ; Allen2017 ; WC_review ;
Barrett2020A ; Pienaar_2020 ; Schmid2021  that go beyond classical random
variables and allow quantum or even post-quantum systems Barrett07 ;
WC_causal_GPT  to be causes. Other approaches suggest modifying the causal
structure itself without necessarily considering non-classical causes, such as
allowing for additional causal influences that go outside the future light
cone (e.g., non-local hidden variable theories Bohm1952 ) or against the
direction of time (retro-causality Wharton2019 ). Such causal influences must
remain hidden in order to prevent superluminal signalling at the observed
level. More radical approaches lean towards giving up the standard
understanding of causation as being acyclic, and replacing it with a suitable
notion of logical consistency Baumeler_2016 ; Baumeler2017 ; Baumeler2018 .
While these alternatives correspond to different descriptions of the
underlying causal model, they are all compatible with the impossibility of
superluminal signalling in Minkowski space-time Note1 .

In our associated paper VilasiniColbeckPRA , we have developed a general
framework for causation that can describe non-classical and cyclic causal
influences as well as causation without signalling. We do so by keeping the
direction of causation and the order of events in space-time distinct. The
former is modelled operationally and we characterize when the two are
_compatible_ with each other i.e., when we can assign space-time locations to
the random variables in the causal model without leading to signalling outside
the space-time’s future (we call this assignment an embedding). Here we ask:
does the ability to compatibly embed a causal model in an acyclic space-time
(such as Minkowski space-time) imply that the operational predictions of the
causal model can be reproduced within an acyclic causal structure? If so, it
would not be necessary to consider cyclic causation.

Within relativistic physics, causal influences are taken to flow within the
light cone, making both causal loops and superluminal signalling impossible in
Minkowski space-time. Here we relax this assumption, and require only that
observable signalling is limited by the light cone structure. In scenarios in
which there is causation without correlation (i.e., a _fine-tuned_ influence),
answering the above question is more challenging since fine-tuned influences
can act outside the lightcone without leading to signalling. Our framework
allows treatment of both correlations and interventions on physical systems in
general scenarios with cyclic causal influences. It can model fine-tuned
causal influences as well as latent quantum and post-quantum causes, and be
used to characterise conditions under which a causal model can be compatibly
embedded in a space-time. Here, we apply this framework to demonstrate the
mathematical possibility of causal loops between events in (1+1)-Minkowski
space-time where these loops can be operationally detected without
superluminal signalling, providing an explicit example. We also show that the
observable predictions of this loop cannot be reproduced in any acyclic causal
model, answering the aforementioned question in the negative. Interestingly,
the same example fails to be embeddable in (3+1)-Minkowski space-time. In the
associated paper VilasiniColbeckPRA , we have characterised a large class of
operationally detectable causal loops within our framework. Whether any of
these may be embeddable in (3+1) dimensions remains an interesting open
question.

Fine-tuned causal explanations are often undesirable as fundamental
explanations of physical phenomena Note2 , but can be crucial in practical
information processing tasks. For instance, cryptographic protocols (such as
the one time pad) rely on fine-tuned correlations, and the security of
relativistic cryptographic protocols Kent_RBC ; CK1  combines both
relativistic notions of causality and information-theoretic concepts. Hence,
understanding the extent to which compatibility with Minkowski space-time
restricts the possible operational causal models also has practical
implications.

Causal models: correlations, interventions and fine-tuning—We begin by
reviewing the essentials of the causal modelling framework (see
VilasiniColbeckPRA  for details). A _causal structure_ is modelled as a
directed graph $\mathcal{G}$ whose nodes correspond to observed or unobserved
systems and directed edges  denote causation between these systems Note3 . The
set of observed nodes (denoted $S_{\mathrm{obs}}$) comprises classical random
variables (RVs) such as measurement settings or outcomes, while unobserved
nodes may be classical, quantum or post-quantum systems (such as those from a
generalised probabilistic theory).

Implicitly the nodes are associated with _causal mechanisms_ that specify how
information from their incoming edges is processed. For instance, in the
classical case this processing corresponds to a function $f_{N_{i}}$ from the
parents of the node and possibly an additional, parentless Note4  RV
$E_{N_{i}}$ (to allow for situations where $N_{i}$ is not deterministically
dependent on its parents) to the node variable $N_{i}$ itself. “$A$ is a
direct cause of $B$” then corresponds to the function $f_{B}$ having $A$ as a
(non-trivial) argument. [In the non-classical case, these functions would be
replaced by valid maps between systems in the theory (e.g., CPTP maps in
quantum theory)].

Often these causal mechanisms are not explicitly known and hence our treatment
has to work at the level of observed probability distributions rather than the
causal mechanisms. The causal structure imposes constraints on the possible
distributions that may arise over the observed nodes. One set of such
constraints can be expressed using the notion of _d-separation_. For two
disjoint subsets $X$ and $Y$ of observed nodes $S_{\mathrm{obs}}$ of a causal
structure $\mathcal{G}$, $X$ and $Y$ are said to be d-separated, denoted
$(X\perp^{d}Y)_{\mathcal{G}}$, if there are no directed paths between
variables in $X$ and $Y$ and if no variables in $X$ and $Y$ have common
ancestors in $\mathcal{G}$. More generally, d-separation
$(X\perp^{d}Y|Z)_{\mathcal{G}}$ is defined for three disjoint subsets $X$, $Y$
and $Z$ of the observed nodes (see Section A of the Supplemental Material for
details).

We say that a distribution $P_{\mathcal{G}}(S_{\mathrm{obs}})$ _satisfies the
d-separation property_ with respect to a causal structure $\mathcal{G}$ if
whenever we have d-separation between observed nodes in $\mathcal{G}$, then we
have a corresponding conditional independence in the observed distribution
$P_{\mathcal{G}}(S_{\mathrm{obs}})$ i.e., $(X\perp^{d}Y|Z)_{\mathcal{G}}$
implies $P_{\mathcal{G}}(XY|Z)=P_{\mathcal{G}}(X|Z)P_{\mathcal{G}}(Y|Z)$. If
the converse holds for all disjoint subsets of observed nodes, then the
distribution is said to be _faithful_ or, equivalently, not _fine-tuned_. In
the present work, we adopt a minimal definition of a causal model which
corresponds to a directed graph $\mathcal{G}$ and an observed distribution
$P_{\mathcal{G}}(S_{\mathrm{obs}})$ that satisfies the d-separation property
with respect to $\mathcal{G}$ Note5 . In this work we allow fine-tuned causal
models in which conditional independences can occur in
$P_{\mathcal{G}}(S_{\mathrm{obs}})$ without the corresponding d-separation in
$\mathcal{G}$ (i.e., we allow causation without correlation).

In cases where we know the causal mechanisms, a causal model can be specified
by a causal structure $\mathcal{G}^{C}$, causal mechanisms
$\\{f_{N_{i}},P_{\mathcal{G}^{C}}(E_{N_{i}})\\}_{i}$ and an associated
observed distribution $P_{\mathcal{G}^{C}}(S_{\mathrm{obs}})$ for
$S_{\mathrm{obs}}\subseteq\\{N_{i}\\}_{i}$. Here the observed distribution
must be consistent with relationships specified by the causal mechanisms.
Methods for modelling operational causal structures beyond the classical case
are proposed in our associated paper VilasiniColbeckPRA . In the present work,
we restrict to the classical case which suffices to illustrate our main
claims.

So far, we have only discussed the possible correlations that fit with a
causal structure. Inferring causation requires more, and the concept of an
intervention has been introduced to deal with this Pearl2009 . If intervening
on $X$ changes the distribution on $Y$, then we can deduce that $X$ is a cause
of $Y$.

In the case where $X$ is parentless in a causal structure $\mathcal{G}$,
correlation between $X$ and another variable $Y$ i.e.,
$P_{\mathcal{G}}(Y|X)\neq P_{\mathcal{G}}(Y)$ suffices to conclude that $X$ is
a cause of $Y$. More generally, an intervention on $X$ corresponds to forcing
$X$ to take a certain value, $x$, irrespective of its parents. This results in
a post-intervention causal structure $\mathcal{G}_{\text{do}(X)}$ obtained
from $\mathcal{G}$ by removing all the incoming edges to $X$, while
maintaining the causal mechanisms for $\mathcal{G}_{\text{do}(X)}$ from
$\mathcal{G}$, except the mechanism for the intervened node $X$, which is
replaced by $X=x$. We say that $X$ _affects_ $Y$ if there exist values $x$ and
$y$ such that

$P_{\mathcal{G}_{\text{do}(X)}}(Y=y|X=x)\neq P_{\mathcal{G}}(Y=y).$ (1)

In general $P_{\mathcal{G}_{\text{do}(X)}}(Y|X)$ is not the same as
$P_{\mathcal{G}}(Y|X)$ (but they are equal when $X$ is parentless in
$\mathcal{G}$).

Acyclic and cyclic causal models embedded in Minkowski space-time—The affects
relation defined in (1) naturally extends to joint interventions on sets of
observed nodes. In the presence of fine-tuning, a set of RVs can jointly
affect another set without any affects relations between individual pairs of
elements in the sets. We will illustrate three such examples here and use
these to establish our main claim. All three causal models have observed nodes
$S_{\mathrm{obs}}:=\\{A,B,C\\}$ which we will take to be classical and binary.
The same observed correlations will be used in all three cases. The
distinguishing feature at the observed level will be the affects relations
which model what happens under intervention. The causal structures and
compatible space-time embeddings for the three examples are given in Figure 1.

(a)  (b)  (c)

Figure 1: Three examples of causal models and their compatible embeddings in
(1+1)-Minkowski space-time. In each case, the operational causal structure
associated with the model is given in black, circled variables are observed
nodes, while uncircled ones are not and the black arrows denote causation.
Space-time information is given in blue with time along the vertical and space
along the horizontal axis. The solid lines represent light-like surfaces and
the shaded region corresponds to the joint future of $A$ and $C$ in all cases.

######  Example 1 (The one-time pad).

Consider the causal structure of Figure 1a, $\mathcal{G}^{\mathrm{OTP}}$ with
the causal mechanisms $A=E_{A}$, $C=E_{C}$, $B=A\oplus C$, with $E_{A}$ and
$E_{C}$ being independent and uniformly distributed. Then, $B$ is uniformly
distributed and we have $P_{\mathcal{G}^{\mathrm{OTP}}}(B|AC)\neq
P_{\mathcal{G}^{\mathrm{OTP}}}(B)$,
$P_{\mathcal{G}^{\mathrm{OTP}}}(B|A)=P_{\mathcal{G}^{\mathrm{OTP}}}(B)$ and
$P_{\mathcal{G}^{\mathrm{OTP}}}(B|C)=P_{\mathcal{G}^{\mathrm{OTP}}}(B)$. Since
$A$ and $C$ are parentless, these statements are equivalent to $\\{A,C\\}$
affects $B$, $A$ does not affect $B$, and $C$ does not affect $B$ Note6 . A
Minkowski space-time embedding of the RVs that does not enable any signalling
outside the future is one in which $B$ is assigned a space-time location in
the joint future of the space-time locations of $A$ and $C$, as illustrated in
Figure 1a.

######  Example 2 (A simplified jamming scenario (cf. Grunhaus1996 )).

Consider the causal structure of Figure 1b, $\mathcal{G}^{\mathrm{jam}}$ with
the observed nodes $\\{A,B,C\\}$ and a classical unobserved node $\Lambda$.
Suppose we have the causal mechanisms $\Lambda=E_{\Lambda}$, $A=\Lambda$,
$B=E_{B}$, $C=B\oplus\Lambda$ with $E_{\Lambda}$ and $E_{B}$ uniformly
distributed. This gives the same observed correlations as the previous
example, with $B=A\oplus C$ and with all observed variables uniformly
distributed. Additionally, $P_{\mathcal{G}^{\mathrm{jam}}}(AC|B)\neq
P_{\mathcal{G}^{\mathrm{jam}}}(AC)$,
$P_{\mathcal{G}^{\mathrm{jam}}}(A|B)=P_{\mathcal{G}^{\mathrm{jam}}}(A)$ and
$P_{\mathcal{G}^{\mathrm{jam}}}(C|B)=P_{\mathcal{G}^{\mathrm{jam}}}(C)$. Since
$B$ is parentless, this is equivalent to $B$ affects $\\{A,C\\}$, $B$ does not
affect $A$ and $B$ does not affect $C$. A compatible Minkowski space-time
embedding in this case requires that the joint future of the space-time
locations of $A$ and $C$ is contained entirely within the space-time future of
$B$, as shown in Figure 1b. This is because $B$ affects $\\{A,C\\}$ can only
be verified when $A$ and $C$ are brought together, which is possible only in
their joint future. Since we have no pairwise affects relations, there is no
pairwise signalling between the RVs and no RV is required to be in the future
of any other. Note that the causal influence $B\leavevmode\hbox
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even though there is no superluminal signalling.

######  Example 3 (A fine-tuned causal loop).

Consider the causal structure of Figure 1c, $\mathcal{G}^{\mathrm{loop}}$,
with the same observed and unobserved nodes as the previous example, but with
the causal mechanisms $\Lambda=E_{\Lambda}$, $A=\Lambda$, $C=B\oplus\Lambda$,
$B=A\oplus C$ with $E_{\Lambda}$ uniformly distributed. Note that these causal
mechanisms do not admit a unique solution. Nevertheless, in Section B of the
Supplemental Material we apply a method proposed in VilasiniColbeckPRA  to
uniquely determine the observed distribution based on these mechanisms, and
show that the same distribution as the previous two examples is obtained. The
effect of interventions in this case cannot be directly inferred from the
observed correlations since none of the observed nodes are parentless. The
post-intervention causal structure
$\mathcal{G}_{\text{do}(AC)}^{\mathrm{loop}}$ is identical to
$\mathcal{G}^{\mathrm{OTP}}$ and the post-intervention causal structure
$\mathcal{G}_{\text{do}(B)}^{\mathrm{loop}}$ is identical to
$\mathcal{G}^{\mathrm{jam}}$. Applying (1), we find $\\{A,C\\}$ affects $B$
_and_ $B$ affects $\\{A,C\\}$ and no pairwise affects relations between $A$,
$B$ and $C$. The space-time embedding must satisfy the compatibility
conditions of both the previous examples, and Figure 1c illustrates an
embedding with these properties i.e., this causal loop can be compatibly
embedded in (1+1)-Minkowski space-time.

All three examples above lead to the same correlations $B=A\oplus C$ where $B$
is correlated jointly with $A$ and $C$ but not individually, hence there is
fine-tuning. Correlation between $B$ and $\\{A,C\\}$ implies (by the
d-separation property) that $B$ and $\\{A,C\\}$ must be d-connected i.e., $B$
is d-connected with $A$ and/or $C$. However, the RVs $A$, $B$ and $C$ are
pairwise uncorrelated so there must be a pair of variables that are
d-connected and yet independent, which constitutes fine-tuning (these
independences disappear for small changes in the distribution of one of the
variables, e.g., if $\Lambda$ is non-uniform in Examples 2 and 3).

The causal loop of Example 3 exhibits many curious features. It is an
operationally detectable causal loop i.e., any causal model that gives rise to
the affects relations of this example must necessarily be associated with a
cyclic causal structure. This is proven in Section C of the Supplemental
Material, but the intuition is relatively simple: Consider three parties in
possession of the 3 observed RVs $A$, $B$ and $C$ and two types of experiment:
E1) Alice and Charlie perform all possible interventions on $A$ and $C$ and
Bob observes $B$ without intervening; E2) Bob performs all possible
interventions on $B$ while Alice and Charlie observe $A$ and $C$ without
intervening. After both experiments the parties can get together to verify
whether $B=A\oplus C$ holds. Here, “all possible interventions” on a variable
corresponds to running through all possible values of that variable, setting
these independently of their parents – see (1), and collecting statistics for
each choice. These statistics differ between the three causal models, as they
have different sets of affects relations. These interventions do not enable
the parties to signal outside the space-time future as the affects relations
of all these causal models are compatible with the given space-time embedding.
They nevertheless allow the parties to operationally verify the existence of a
causal loop as we show in the Supplemental Material.

In Example 3, experiment E1 shows that $A$ and $C$ are causes of $B$ while E2
shows that $B$ is a cause of at least one of $A$ and $C$. Given the space-time
embedding from Figure 1c, these interventions would enable the agents to
operationally detect retro-causation. By contrast, for the first model, E2
would correspond to a post-intervention scenario with no edges and therefore
lead to no correlations between the RVs, while in the second model, E1 would
also lead to no correlations. In other words, these two experiments enable the
parties to operationally distinguish between the causal models of Examples 1-3
in spite of them having the same observed correlations.

The mathematical possibility of an operationally detectable causal loop being
embedded in Minkowski space-time without signalling necessarily involves fine-
tuning (see VilasiniColbeckPRA ) since in the absence of fine-tuning,
signalling and causation are equivalent. Note that in Example 3 the locations
of the random variables in Minkowski space-time have to be carefully chosen to
allow compatibility: $B$ and $C$ must be light-like separated, and arbitrarily
small adjustments to the location of either $B$ or $C$ will remove
compatibility. This requires $B$ to be embedded exactly at the earliest
location in the joint future of $A$ and $C$. In a sense this is another kind
of fine-tuning, but at the level of the space-time embedding. Furthermore,
such an embedding is not possible in (3+1)-Minkowski space-time because the
intersection of two lightcones is not itself a cone and consequently there
does not exist a frame-independent earliest location in the joint future of
two given points, unlike in (1+1)-D Note7 .

Beyond these examples, a causal model may give rise to a complicated set of
affects relations between various subsets of the observed nodes, making
characterising compatibility with a space-time a more complicated task. For
instance, if $A$ affects $B$ and $\\{A,C\\}$ affects $B$ but $C$ does not
affect $B$, does $B$ have to be embedded in the joint future of $A$ and $C$,
or only in the future of $A$? The novel causal modelling concept required to
answer such questions in fine-tuned models is the notion of a _higher-order
affects relation_ , which we introduce in our framework VilasiniColbeckPRA .
Here we have illustrated one example, but there is a general class of physical
theories involving cyclic causal structures that are compatible with no
superluminal signalling in Minkowski space-time as discussed in our associated
paper VilasiniColbeckPRA .

Discussion and outlook.—In relativistic physics, a causal influence from one
space-time location to another implies that the latter is in the future light
cone of the former. This means that within relativistic physics, it is not
possible to have causal loops or closed timelike curves in space-times whose
light cone structures form a partial order. On the other hand, the approach
adopted here, where causal structure and space-time structure are distinct
notions, only requires observable signalling (and not causal influences) to
stay within the space-time future for compatibility. Maintaining a clear
separation between causation and space-time structure and characterising their
interdependence is useful for considering formulations of physics without a
fixed background space-time structure (e.g., in quantum gravity Oreshkov2012 ;
Zych2019 ), as well as practical information processing tasks in space-time.

We note that a previous work Horodecki2019  proposes a set of necessary and
sufficient conditions for ruling out causal loops in Bell-type scenarios. Our
framework VilasiniColbeckPRA  identifies implicit assumptions in this claim.
In particular, one of the claims in Horodecki2019  assumes that no
superluminal signalling in Minkowski space-time rules out causal loops, which
we have shown does not hold in general.

We have established that the principle of no superluminal signalling in
Minkowski space-time alone is insufficient to rule out causal loops, through
an explicit construction in (1+1) dimensions. We also found that the
conditions for ruling out causal loops can depend on the space-time dimension.
Previous works have established that logical consistency Deutsch1991 ;
Lloyd2011  or familiar quantum properties such as linearity, no-cloning
Araujo2015  etc. are not sufficient. Finding underlying principles that can do
so remains an interesting open problem, a pertinent question being whether the
principle of no superluminal signalling rules out causal loops in the case of
(3+1)-Minkowski space-time.

_Acknowledgements_ — VV acknowledges support from the Department of
Mathematics, University of York and the ETH Postdoctoral Fellowship from ETH
Zürich.

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  * (33) We reference Minkowski space-time here for concreteness, but our results also apply to any other partially ordered space-time.
  * (34) Vilasini, V. & Colbeck, R.  General framework for cyclic and fine-tuned causal models and their compatibility with space-time.  _Physical Review A_ 106, 032204 (2022).  link.
  * (35) Since one then has to explain how the fine-tuning comes about.
  * (36) Kent, A.  Secure classical bit commitment over finite channels.  _Journal of Cryptology_ 18, 313–335 (2005).  link.
  * (37) Colbeck, R. & Kent, A.  Variable Bias Coin Tossing.  _Physical Review A_ 73, 032320 (2006).  link.
  * (38) Such causal structures are formally independent of space-time, although, as we will later discuss, they may be embedded in a separate space-time structure.
  * (39) The functional dependences are typically described in terms of an augmented causal structure that is formed from the original causal structure by including $E_{N_{i}}\leavevmode\hbox to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}} {{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}} {}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}N_{i}$ for each node $N_{i}$. Note that the $E_{N_{i}}$ nodes have no other edges associated with them and are parentless in the augmented graph.
  * (40) Here, we make no assumptions about the causal mechanisms or the physical theory describing them, but instead derive the consequences of imposing d-separation at the observed level. We note that one may consider other top-down methods for describing causal structures without imposing d-separation, but this is yet to be explored.
  * (41) This is analogous to a one-time pad protocol where a message $M$ (a random classical bit) is encrypted using a uniformly distributed key $K$ to obtain a cipher text $C=M\oplus K$. Even though the message $M$ is a cause of the cipher text $C$, we have $P(C|M)=P(C)$ and $C$ contains no information about $M$ in the absence of the key. In causal modelling terms, $C$ and $M$ are independent even though they are d-connected; the model is fine-tuned.
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  * (43) Although we have focused here on Minkowski space-time, the framework of our associated paper VilasiniColbeckPRA  can be applied to consider compatible embeddings in any partially ordered space-time. Applying this framework, one can find other examples of embeddable causal loops in space-times (not necessarily Minkowski) where the property of having a frame-independent earliest location in the joint future of any two given points fails. We saw that this property holds in (1+1) but not in (2+1) or (3+1) Minkowski space-time.
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SUPPLEMENTAL MATERIAL

##  I Further details of the causal modelling framework

The d-separation property Pearl2009 ; Spirtes2001  was motivated in the main
text and can be used as a convenient tool to read off conditional
independences in the observed distribution $P_{\mathcal{G}}(S_{\mathrm{obs}})$
associated with a causal structure $\mathcal{G}$, from the topology of
$\mathcal{G}$. We provide the formal definition below.

######  Definition I.1 (Blocked paths).

Let $\mathcal{G}$ be a directed graph in which $X$ and $Y\neq X$ are nodes and
let $Z$ be a set of nodes not containing $X$ or $Y$. A path from $X$ to $Y$
(not necessarily directed) is said to be _blocked_ by $Z$ if it contains
either $A\leavevmode\hbox to17.47pt{\vbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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$A\leavevmode\hbox to17.47pt{\vbox
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$W$ nor any descendant of $W$ belongs to $Z$, where $A$ and $B$ are arbitrary
nodes in the said path between $X$ and $Y$.

######  Definition I.2 (d-separation).

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

For acyclic causal structures (both classical Pearl2009  and non-classical
Henson2014 ; Barrett2020A ), it is known that any distribution arising from
the causal mechanisms satisfies the d-separation property. This is also known
to hold in several classical cyclic causal models Forre2017  and our
accompanying paper VilasiniColbeckPRA  provides an example of a quantum cyclic
causal model where this holds as well. However, in contrast to the acyclic
case, in some classical cyclic causal structures these mechanisms do not imply
the d-separation property fails Neal2000 . Such examples are ruled out in the
definition of causal model we use which imposes the d-separation property. In
the classical case, Forre2017  introduces a more general graph separation
criterion ($\sigma$-separation) that applies to a larger class of cyclic
causal models and reduces to d-separation in the acyclic case, and one can in
principle generalise our definition of a causal model in terms of
$\sigma$-separation to account for this larger class of models. There
nevertheless remain classical cyclic causal models for which no graph
separation property is known, highlighting the counter-intuitive nature of
cyclic causal structures. Characterising the set of classical or quantum
cyclic models, defined from causal mechanisms, for which d- or
$\sigma$-separation holds is an interesting question for future work, and
Appendix C of our accompanying paper VilasiniColbeckPRA  outlines a possible
method for doing so.

Another useful framework for modelling cyclic quantum causal structures was
proposed in Barrett2020 . This follows a different, split-node approach to
causal modelling where all nodes are modelled quantum mechanically. This is a
bottom-up approach that describes the observed probabilities in terms of the
underlying quantum causal mechanisms. Unfaithful causal models, post-quantum
causes and space-time embeddings of the causal model were not considered here.
In contrast, our framework takes these into account but instead adopts a top
down approach where the causal model imposes minimal conditions given by the
d-separation property that relates the causal graph with a set of observed
distributions. This already suffices to solve the problem regarding space-time
embeddings of unfaithful, cyclic and non-classical causal models and
reproduces a number of results from the classical causal modelling literature
such as Pearl’s rules of do-calculus. Even though the causal models in our
framework are defined through the d-separation property, the techniques and
space-time compatibility condition we propose relies only on the set of
affects relations generated by a causal model and can hence also be applied to
cyclic causal models that do not satisfy the d-separation condition.

##  II A method to derive the distribution of Example 3

In acyclic (classical and non-classical) causal models, the observed
distribution is uniquely determined once all the causal mechanisms are
specified Pearl2009 ; Henson2014 ; Barrett2020A . However this is not the case
in cyclic models. The simplest example illustrating this is the 2-node cycle
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\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\hss}}\lxSVG@closescope\endpgfpicture}}Y$ and
$Y\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}X$ where the causal
mechanisms are simply $X=f_{X}(Y)=Y$ and $Y=f_{Y}(X)=X$. In this case any
distribution $P(XY)=P(X)=P(Y)$ agrees with the causal mechanisms, and using
the standard methods available in the previous literature on cyclic causal
models Forre2017 ; Bongers2021 , this distribution is not uniquely determined
by the mechanisms. In the associated paper VilasiniColbeckPRA  (Appendix C),
we have proposed a possible method for uniquely determining the observed
distribution in a class of classical and non-classical cyclic models. The
method can be applied to models where each directed cycle includes at least
one observed node, and involves splitting this observed node in each cycle
into two nodes, which results in a directed acyclic graph. We can then use the
known methods for acyclic models (which always yield a unique distribution),
post-select on the values of the split nodes being equal, and finally
renormalise the overall distribution to ensure that the final distribution for
the cyclic model is a valid normalised distribution.

We now apply this method to the cyclic causal model of Example 3 from the main
text, for which standard methods leave the observed distribution undetermined.
The directed cycle in this case is between the nodes $B$ and $C$ and we can
choose to split either one of these. Splitting $B$ into two nodes $B$ and
$B^{\prime}$ such that $B$ has all the same incoming arrows as $B$ (and no
outgoing arrows) while $B^{\prime}$ has all the same outgoing arrows as $B$
(and no incoming arrows), we obtain the following acyclic causal model:
$P(\Lambda)$ is uniformly distributed, $A=\Lambda$,
$C=B^{\prime}\oplus\Lambda$, $B=A\oplus C$. The corresponding causal structure
$\mathcal{G}^{\text{acyc}}$ is given in Figure 2a. Notice that the above
causal mechanisms for this acyclic graph which are induced by the original
cyclic model of Example 3 are not a complete set of mechanisms for the acyclic
case, since we are missing a specification of the distribution over the new
exogenous node $B^{\prime}$. Therefore, we can only calculate the conditional
distribution $P_{\mathcal{G}^{\text{acyc}}}(ABC|B^{\prime})$ associated with
the observed nodes of the acyclic model. We can then post-select on
$B=B^{\prime}$ and renormalise the distribution to obtain the observed
distribution $P_{\mathcal{G}^{\text{loop}}}(ABC)$ of the original cyclic
model. This is explicitly shown in Figure 2b, and one can see that this
procedure yields exactly the same distribution stated in the main text, and in
particular this is also the observed distribution obtained in Examples 1 and 2
of the main text. It is easy to verify that splitting the node $C$ instead and
applying this procedure also yields the same distribution. Generalising this
method to arbitrary non-classical cyclic causal structures is a subject of
future work.

$B$$B^{\prime}$AC$\Lambda$ (a)

$\Lambda$ | $A$ | $B$ | $B^{\prime}$ | $C$ | $P_{\mathcal{G}^{\text{acyc}}}(ABC|B^{\prime})$ | $P_{\mathcal{G}^{\text{loop}}}(ABC)$
---|---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{4}$
0 | 0 | 1 | 1 | 1 | $\frac{1}{2}$ | $\frac{1}{4}$
1 | 1 | 0 | 0 | 1 | $\frac{1}{2}$ | $\frac{1}{4}$
1 | 1 | 1 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{4}$
(b)

Figure 2: Calculating the observed distribution of Example 3 (a) Acyclic
causal structure $\mathcal{G}^{\text{acyc}}$ obtained by splitting the node
$B$ of the cyclic causal structure $\mathcal{G}^{\text{loop}}$ of Figure 1(b).
(b) Table showing the conditional probability distribution
$P_{\mathcal{G}^{\text{acyc}}}(ABC|B^{\prime})$ of the induced causal model
over (a). We then post-select on $B=B^{\prime}$ to recover the original cyclic
model and then ignore $B^{\prime}$. But this yields an unnormalised
distribution, and we must therefore renormalise the distribution to obtain the
required observed distribution $P_{\mathcal{G}^{\text{loop}}}(ABC)$ as given
in the last column.

##  III Operationally certifying the cyclicity of the causal model

We have seen that the cyclic causal model of Example 3 leads to the affects
relations $B$ affects $\\{A,C\\}$, and $\\{A,C\\}$ affects $B$, with no
pairwise affects relations between $A$, $B$ and $C$. We now show that it is
impossible for any acyclic causal model to give rise to the same affects
relations i.e., these affects relations operationally certify the cyclicity of
the causal model and hence the causal loop giving rise to these relations is
operationally detectable.

We show this by contradiction, suppose that there exists an acyclic causal
model with an associated causal structure $\mathcal{G}$, which is a directed
acyclic graph (DAG) that produces these affects relations. First consider the
affects relation $B$ affects $\\{A,C\\}$. We will show that this implies the
existence of a directed path from $B$ to $A$ or from $B$ to $C$ in
$\mathcal{G}$. Since every acyclic causal model satisfies the d-separation
property Pearl2009 ; Henson2014 , we can apply it to $\mathcal{G}$. If there
were no directed paths from $B$ to $\\{A,C\\}$ in $\mathcal{G}$, then $B$
would be d-separated from $\\{A,C\\}$ in $\mathcal{G}_{\text{do}(B)}$ i.e.,
$(B\perp^{d}AC)_{\mathcal{G}_{\text{do}(B)}}$. It is shown in
VilasiniColbeckPRA  that for any $\mathcal{G}$ satisfying the d-separation
property, $(X\perp^{d}Y)_{\mathcal{G}_{\text{do}(X)}}$ for any two disjoint
sets $X$ and $Y$ of observed nodes implies that $X$ does not affect $Y$.
Applying this result here gives $B$ does not affect $\\{A,C\\}$ which
contradicts the given affects relations, so we must have a directed path from
$B$ to $\\{A,C\\}$ in $\mathcal{G}$.

Next consider a causal model obeying the affects relation $\\{A,C\\}$ affects
$B$ along with $A$ does not affect $B$ and $C$ does not affect $B$. We will
now show that this implies the existence of a directed path from $A$ to $B$
and from $C$ to $B$ in $\mathcal{G}$. For this we will use the concept of a
_higher-order affects relation_ introduced in VilasiniColbeckPRA . For three
disjoint subsets $X$, $Y$ and $Z$ of the observed nodes, we say that $X$
affects $Y$ given do$(Z)$ if there exist values $x$ of $X$, $y$ of $Y$ and $z$
of $Z$ such that

$P_{\mathcal{G}_{\text{do}(XZ)}}(Y=y|X=x,Z=z)\neq
P_{\mathcal{G}_{\text{do}(Z)}}(Y=y|Z=z)$ (2)

Note that $\\{A,C\\}$ affects $B$, and $C$ does not affect $B$ imply the
higher-order affects relation $A$ affects $B$ given do$(C)$. Explicitly, the
first two conditions are

$\displaystyle\begin{split}P_{\mathcal{G}_{\text{do}(AC)}}(B|AC)&\neq
P_{\mathcal{G}}(B)\\\
P_{\mathcal{G}_{\text{do}(C)}}(B|C)&=P_{\mathcal{G}}(B).\end{split}$ (3)

When combined, this yields $P_{\mathcal{G}_{\text{do}(AC)}}(B|AC)\neq
P_{\mathcal{G}_{\text{do}(C)}}(B|C)$ which precisely expresses the higher-
order affects relation $A$ affects $B$ given do$(C)$. Similarly $\\{A,C\\}$
affects $B$ along with $A$ does not affect $B$ imply the higher-order affects
relation $C$ affects $B$ given do$(A)$. It was shown in VilasiniColbeckPRA
that $X$ affects $Y$ given do$(Z)$ implies that there is a directed path from
$X$ to $Y$ in any underlying causal structure $\mathcal{G}$ producing the
higher-order affects relations (and satisfying the d-separation property).
Applying this to the two higher-order affects relations obtained above
establishes the existence of directed paths from $A$ to $B$ and from $C$ to
$B$ in $\mathcal{G}$ (which we have assumed to be a DAG, and this assumption
implies that the associated model must satisfy the d-separation property).

To summarise, we have established above that there exists a directed path from
$A$ to $B$ and a directed path from $C$ to $B$ in $\mathcal{G}$ as well as a
directed path from $B$ to at least one of $A$ and $C$. This implies the
existence of a directed cycle in $\mathcal{G}$, contradicting the assumption
that it is a directed acyclic graph, and certifying the cyclicity of the
model.