# Is There Causation in Fundamental Physics? New Insights from Process
Matrices and Quantum Causal Modelling

Emily Adlam The Rotman Institute of Philosophy, 1151 Richmond Street, London
N6A5B7<EMAIL_ADDRESS>
Bertrand Russell famously argued that causation plays no role in science: it
is ‘ _a relic of a bygone age, surviving, like the monarchy, only because it
is erroneously supposed to do no harm._ ’[1] Cartwright[2] and later writers
moderated this conclusion somewhat, and it is now largely accepted that in a
macroscopic setting causal concepts are an important part of the assessments
we make about possible strategies for action. But the view that causation in
the usual sense of the term is not present in fundamental physics, or at least
that not all fundamental physical processes are causal, remains prevalent[3,
4] \- for example, Norton writes that ‘ _(causes and causal principles) are
heuristically useful notions, licensed by our best sciences, but we should not
mistake them for the fundamental principles of nature_ ’[5]. Furthermore, many
influential philosophical analyses of causation posit that causation arises
only at a macroscopic level, as a result of the thermodynamic gradient[6, 7],
interventions[8, 9], the perspectives of agents[10], or some such feature of
reality which plays no role in fundamental physics.

In light of this widespread orthodoxy, it may seem surprising that in recent
years a significant literature around causation has sprung up within quantum
foundations. Thus it is important to understand the nature of the recent
renaissance of causal talk in fundamental physics: do these research
programmes offer a counterexample to the claim that causal notions do not
appear in fundamental physics, or are they simply using the word ‘cause’ in a
non-standard way? In this article, we address this question in the context of
two research programmes in quantum foundations: the process matrix formalism,
and the causal modelling approach.

Woodward[11] distinguished between several different philosophical projects
concerning our understanding of causation - a descriptive project, a
metaphysical project, and a functional project. This article spans all three
of these projects. We begin with the descriptive project, aiming to understand
what physicists working in the process matrix formalism and the causal
modelling approach mean by their use of causal terminology; thus we review a
debate between proponents of the two approaches over the significance of an
experiment called ‘the quantum SWITCH,’ which brings into sharp focus the
differing conceptions of causation employed in these two frameworks. We show
that the process matrix programme has correctly identified a notion of _causal
order_ which plays an important role in fundamental physics, and argue that
this notion is weaker than the common-sense conception of causation because it
does not demand any kind of asymmetry.

We then take up the metaphysical project, arguing that causal order plays an
important role in grounding more familiar causal phenomena; thus we conclude
that Russell was correct in his observation that there is no causation in the
usual sense in fundamental physics, but nonetheless fundamental physics does
exhibit a phenomenon which is closely related to causation. We apply these
conclusions to assess the prospects of the causal modelling approach within
quantum foundations: since no-signalling quantum correlations cannot exhibit
causal order, we argue that certain quantum phenomena should not be analysed
using classical causal models. This resolves an open question about how to
interpret fine-tuning in classical causal models of no-signalling
correlations.

Finally we move to the functional project, where we observe that a quantum
generalization of causal modelling can play a similar functional role to
standard causal reasoning. Thus we conclude that quantum causal models have a
legitimate claim to be regarded as at least quasi-causal despite the absence
of causal order in the phenomena they describe; but we emphasize that this
functional characterisation does not entail that quantum causal models offer
novel explanations of quantum processes.

##  1 Background: Process Matrix Formalism

To understand the purpose of the process matrix formalism, let us begin by
considering the classical case. Using purely classical physics we can (in
principle) construct processes which lack a unique causal order by putting our
laboratories on a closed timelike curve. For example, suppose we have a single
lab on a closed timelike curve, with its output connected to its input via a
channel $C$, as shown in figure 1. An obvious problem with this kind of setup
is that it can lead to apparent logical contradictions. For example, suppose
that $C$ is the identity channel whilst $T$ is a bit flip operation; then
necessarily the output of the lab equals both $0$ and $1$ at once, which most
people would consider to be both logically and physically impossible. The
problem can be avoided if we say that the agents in the lab are simply not
able to perform a bit flip operation, but this is unappealing since we
typically think that classical agents should be able to perform any
classically allowed local operation within an isolated lab.

TC Figure 1: Schematic diagram of a channel around a closed causal loop with
one laboratory

The classical process function formalism[12] has been formulated to address
this kind of problem. First, we consider a set of agents in separate
laboratories who are free to perform any operation which is permitted by
classical physics; then we define a ‘process function’ to describe the
classical dynamics between the labs, without making any assumptions about the
causal ordering of the labs. The set of valid process functions is defined in
such a way as to include all possible dynamics which will never give rise to
contradictions no matter which operations the individual agents choose to
perform. For example, the identity channel from a lab to itself not one of the
allowed dynamics, because it can lead to the kind of contradiction we have
just described. In fact, there are no valid process functions on a CTC
involving one or two labs, apart from trivial dynamics which output a constant
or random value. However it can be shown that for three or more labs on a CTC
there are nontrivial process functions, thus demonstrating that there are
possible classical dynamics on a CTC which allow agents complete local freedom
whilst avoiding contradictions (here the term ‘possible’ should be understood
in the sense of ‘logically possible’ - it is not yet known whether dynamics
such as these can be implemented in our actual world).

The process matrix formalism[13, 14, 15, 16, 17] is simply the quantum version
of this formalism. We define a ‘process’ to be composed of a set of agents in
separate laboratories who are free to perform any local operation which is
permitted by quantum mechanics, together with a ‘process matrix’ which
describes the quantum dynamics between the labs, without making any
assumptions about the causal ordering of the labs. The process matrix operates
as a higher-order map from a set of quantum operations to another operation -
i.e. it maps the quantum operations performed by the agents in the
laboratories to an operation representing the net effect of performing all of
those operations in the context of the particular process. The set of valid
process matrices is defined in such a way as to include all possible dynamics
which will never give rise to contradictions no matter which operations the
individual agents choose to perform. Again we find that there are possible
dynamics which don’t have a well-defined causal order but which allow agents
to perform all operations allowed by quantum physics; but in the quantum case
we have more possibilities for non-causal structures - we could still have
CTCs, but we could also have superpositions of different causal structures,
which are often characterised as ‘indefinite causal order.’

Process matrices are classified in terms of the degree of causal structure
that they exhibit, as shown in figure 2. First, we have the process matrices
which are _causally ordered_ , i.e. they are compatible with the existence of
a strict partial order such that the probabilities for the outcomes in some
laboratory depend only on the settings in that laboratory or in experiments
which precede it in the partial order. Note that here and throughout this
article, ‘depend’ refers to statistical dependence, i.e. it encodes the
possibility of signalling - the kinds of relationships that may occur between
no-signalling laboratories, as in the Bell correlations, do not count for the
purpose of these definitions. Second, we have process matrices which are
_causally separable_ , i.e. they can be written as a convex decomposition of
causally ordered process matrices. Third, we have process matrices which are
neither causally ordered nor causally separable, but are still _causal_ in the
sense that we can write down a well-defined probability distribution over
possible strict partial orders such that in each possible case, the choice of
setting in a local experiment does not affect the occurrence of events which
are earlier in the order, nor the strict partial order on the set of the
events and the experiment in question[18].111 More precisely, according to ref
[18] a process matrix is causal iff there exists a probability distribution
over possible strict partial orders on the set of local experiments and
outcomes of those experiments such that for every local experiment $A$, every
subset $X$ of the other local experiments, and every choice of strict partial
order, the joint probability distribution over the outcomes for the
experiments $X$ and the possible strict partial orders over $A,X$ (obtained by
summing over only those orders for the full process such that $A$ does not
precede any experiments in $X$) are independent of the settings in the
experiment $A$. Finally, we have the process matrices which are _non-causal_ ,
meaning that they are not compatible with a distribution over strict partial
orders which obeys these requirements. We can define ‘causal inequalities,’
somewhat analogous to the Bell inequalities which can only be violated by non-
causal processes[19]. Although a number of non-causal process matrices have
been formulated and studied, all processes currently known to be realised in
nature (without post-selection) are causal.

Ref [20] gives greater insight into the reasons why all processes that we have
so far been able to implement are causal: in fact, any process made up of
localised events which is implemented in a fixed acyclic background spacetime
in a way which satisfies relativistic causality (i.e. no superluminal
signalling) can be rewritten using a ‘fine-graining’ process which splits
individual laboratories into several distinct laboratories in order to produce
a fixed acyclic causal structure. This makes sense, since after all the
acyclic background spacetime necessarily imposes a global strict partial order
on any processes performed in that spacetime, if those processes are required
to respect relativistic causality. In particular, even if the process involves
quantum systems in a superposition of different causal histories, relativistic
causality entails that ‘ _an agent has the potential to intervene at any of
the spacetime locations to verify the probability of detecting the particle
there … (and) if agents choose to perform such interventions, this will not
enable them to signal outside the future of the spacetime,_ ’ and it is this
requirement which ensures that these processes can always be rewritten with a
fixed causal order. Ref [20] therefore argues that no experiment in a well-
defined spacetime can demonstrate ‘indefinite causal order’ in a strong sense
- in their view, the only way to exhibit genuine ‘indefinite causal order’
would be to do some kind of quantum gravity experiment where the spacetime
background is no longer fixed. However, this still leaves open questions about
how to interpret processes performed with a fixed background which at least
superficially seem to involve superpositions of causal orders; we will return
to this question in section 3.

causallyorderedcausallyseparablecausalnon-causal processes(not yet observed in
reality) Quantum SWITCH  Figure 2: Schematic diagram of types of process in
the process matrix formalism

####  1.0.1 Causal terminology

Often in the process matrix framework the term ‘causal’ seems to refer simply
to the transmission of information between systems, so for example the
simplest kind of ‘causally ordered’ processes are just one-way signalling
processes - in effect $A$ is regarded as being a cause of $B$ if and only if
$p(B|A)\neq p(B)$222Some proponents of the process matrix formalism seem
inclined to identify causation with signalling[18], but others take a more
nuanced approach - for example, ref [20] acknowledges that signalling is not a
necessary condition for the presence of a causal relation, as one can imagine
classical cases where two causal influences exactly cancel one another out, so
there is no observable signalling effect even though there are cause-effect
relations. Thus ref [20] merely uses signalling as a sign that causation is
present, rather than a definition of causation.. This information-theoretic
way of thinking about causation is essentially a version of Lewis’ simplest
probabilistic counterfactual theory of causation[21], where we say that $A$ is
a cause of $B$ if and only if, if $A$ had not occurred, the chance of $B$
occurring would be less than its actual chance. In the process framework we
don’t particularly care about whether $A$ makes $B$ more or less probable as
long as it changes the probability of $B$, but it is clear that the approaches
are similarly motivated and will have similar advantages and disadvantages. In
particular, it is well-recognised[22] that the simplest counterfactual account
is not asymmetric and hence must rely on contingent external considerations
(e.g. thermodynamics) if it is to recover the asymmetry of causation, and
evidently the process matrix formulation faces a similar problem, since a
relation defined in terms of the possibility of signalling between two
variables is evidently not inherently asymmetric.

Approaches to causality aiming to reproduce its asymmetry often do so by
appealing directly to agency or agents - for example, interventionists argue
that a variable A causes a variable B if and only if B would change if A were
manipulated by an agent in an appropriate way[8, 9]. The interventionist
approach may seem like a good fit for the process matrix formalism, since the
operations performed by the agents in the laboratories can be regarded as
interventions. However, problems arise from the fact that the literature on
process matrices has a tendency to equivocate around the nature of the
‘laboratories’ in which the quantum operations are performed. On the one hand,
the experiments are described in terms of agents - that is to say, they are
usually said to be performed by entities named ‘Alice’ and ‘Bob,’ and readers
could be forgiven for assuming that Alice and Bob are supposed to be human
beings. But in fact, in practical implementations of indefinite causal order,
such as the quantum SWITCH (see section 3), the roles of ‘Alice’ and ‘Bob’ are
played by quantum gates or similar quantum instruments - and indeed, certain
interpretations of quantum mechanics, like the spontaneous collapse
approach[23] 333We use the term ‘interpretation’ in the popular sense here, as
referring to any possible solution to the measurement problem, e.g. Everett,
de Broglie-Bohm, GRW collapse and so on. It might be argued that not all of
these approaches are really ‘interpretations’ - de Broglie Bohm and GRW, for
example, add something to standard quantum mechanics and thus should perhaps
be referred to as ‘alternative hypotheses.’ However we use the term
‘interpretation’ for consistency with the existing literature. have the
consequence that it is effectively impossible for an agent to be in a
superposition, which would entail that a ‘quantum SWITCH’ and other similar
processes could _never_ be implemented with a real agent. And this distinction
is not trivial, because if we want to interpret the ‘causal structures’
involved here in the interventionist sense, then we must insist that the
operations performed by Alice and Bob are genuinely ‘free’ interventions; but
quantum gates evidently do not exercise freedom of choice, so it is less clear
that the causal interpretation is still valid when these experiments are
implemented using quantum gates.

Furthermore, in order to employ an interventionist account of causation we
will need to be able to assign probabilities to measurement outcomes
conditional on the actions taken by various possible agents; and although it
is straightforward to write down such probabilities, it is less clear what the
probabilities actually mean, because if we to keep the agents in coherent
superposition then we will not be able to invoke a collapse interpretation to
make sense of them. Perhaps the most well-developed extant approach to making
sense of quantum probabilities in a no-collapse setting is the Deutsch-Wallace
decision-theoretic analysis[24, 25] developed in the context of the Everett
interpretation; but this approach may not be applicable in the context of
experiments probing indefinite causal order, because it’s designed to be
applied in the kind of regime where different branches of the wavefunction
have effectively decohered and thus interference between branches is
impossible, and this is not the case in experiments like the quantum SWITCH
because the point of these experiments is that the branches are kept in
coherent superposition and ultimately recombined (otherwise we could not
verify that the superposition of causal structure has taken place). For
example, one important axiom in the decision-theoretic analysis is ‘
_Branching Indifference: An agent doesn’t care about branching per se: if a
certain measurement leaves his future selves in N different macrostates but
doesn’t change any of their rewards, he is indifferent as to whether or not
the measurement is performed_ ’[25] \- this axiom seems very reasonable if
branches can’t interfere, but not so much in the case where agents remain in
coherent superposition, because in that case branching brings with it the
possibility of phenomena like interference and recombination which could lead
to the agents’ memories being altered or wiped, and these certainly seem like
things they might care about! So it is unclear that we can define a sensible
notion of probability for agents involved in experiments like the quantum
SWITCH.

In virtue of these concerns, it should not be assumed that terms like
‘causally ordered’ and ‘causal’ in the process matrix framework line up
exactly with the common-sense usage or with any standard philosophical
analysis of causation. Thus for the moment terms like ‘causally ordered’ and
‘causal’ should be understood in a purely formal way. One of our aims in this
article is to determine the connections between these technical notions and
the more general concept of causality, so we will eventually have something
more specific to say about the sense in which ‘causal order’ really pertains
to ‘causation’ in the ordinary sense.

##  2 Background: Causal Modelling

The classical causal modelling framework, whose developers and proponents
include Pearl[8] and Spirtes, Glymour, and Scheines[26], is a methodology for
making inferences about causal relationships from statistical data. A causal
model for a set of variables is a directed graph whose edges represent causal
relationships between the variables $\\{X_{i}\\}$, together with a set of
conditional probabilities $\\{p(X_{i}|Pa(X_{i}))\\}$ giving the probability
distribution for the variable $X_{i}$ conditioned only on the set $Pa(A_{i})$
of its parents in the directed graph (where X is a parent of Y iff there is an
arrow from X to Y in the graph). The probabilities are supposed to obey a
‘Markov condition,’ which states that the joint probability distribution over
all the variables can be written as the product of each of the individual
probability distributions $\\{p(X_{i}|Pa(X_{i}))\\}$. It is usually assumed
that the directed graph must be acyclic, although in some circumstances this
assumption is relaxed[27, 28]. The causal modelling framework is closely
linked to the interventionist account of causation[8, 9], as causal models can
be regarded as encoding the results of possible interventions.

The causal modelling approach was originally intended for the study of
macroscopic causal relationships, such as the consequences of medical
interventions. In this setting it is uncontroversial that the variables stand
in causal relationships, since the macroscopic world exhibits many causal
features which we must take seriously if we wish to characterise ‘ _an
‘objective’ distinction between effective and ineffective strategies’_[2].
However, in recent years there has been increasing interest in applying the
tools of causal modelling to the microscopic world, and in particular to
quantum mechanics. One might question whether techniques designed to study
macroscopic causality can be sensibly translated to this context, given the
aforementioned doubts about whether causation features in fundamental physics,
so one of our aims in this article will be to reach a verdict on this matter.

Causal modelling has been used to analyse various quantum phenomena. For
example, one significant result in this framework is due to Wood and
Spekkens[29], who showed that any classical causal model which reproduces the
Bell correlations must be ‘fine-tuned’ in the sense that the underlying causal
model postulates causal connections between events which appear causally
unconnected at the level of the empirical statistics. To see how this works,
note that the proof of the Bell inequality can be regarded as telling us that
any causal model for correlations violating the Bell inequality must postulate
a causal connection between the choice of measurement on one particle and the
outcome of the measurement on the other particle, but the quantum mechanical
no-signalling theorem ensures that at the statistical level there will be no
dependence of the outcome on the measurement choice, so if we wish to
represent these statistics by a causal model we must carefully ‘fine-tune’ the
parameters of the model to ensure that the underlying causal influences
exactly cancel out so as to be invisible at the level of the empirical
statistics.

There also exists a generalisation of Pearl’s schema to quantum mechanics
which replaces classical causes with quantum ones. Various versions of this
approach exist[30, 31, 32], but here we will follow the approach of ref [33]
which takes ‘ _causal relations to be relations of influence between quantum
nodes mediated by unitary evolution._ ’ In this approach, a quantum causal
model is defined on a set of quantum nodes - each node is a pair of Hilbert
spaces, one associated with an incoming system, and one associated with an
outgoing system, and the idea is that an agent could potentially perform an
operation $O_{i}$ at a node $A_{i}$, thus implementing a transformation on the
incoming system to produce the outgoing system. Thus a quantum causal model
constitutes a directed acyclic graph on a set of quantum nodes, together with
a set of quantum channels $\\{\rho_{A_{i}|Pa(A_{i})}\\}$ such that the
incoming system for node $A_{i}$ is given by the output of the channel
$\rho_{A_{i}|Pa(A_{i})}$ acting on the set $Pa(A_{i})$ of its parents in the
DAG, and every channel $\rho_{A_{i}|Pa(A_{i})}$ commutes with every other
channel $\rho_{A_{j}|Pa(A_{j})}$. The process as a whole can be described by a
‘process operator’ which maps the set of operations $O_{i}$ implemented by
observers at nodes to an operation which represents the net effect of
performing all of these operations in the context of the particular process;
usually it is assumed that the channels should obey a ‘quantum Markov
condition,’ which requires that the process operator can be written as the
product of each of the individual channels $\rho_{A_{i}|Pa(A_{i})}$. It can be
seen that this approach has much in common with the process matrix framework -
the ‘nodes’ play the role of laboratories where agents freely choose their
operations, and the ‘process operator’ plays a similar role to the process
matrix. However, the emphasis here is on defining a specific causal structure
rather than allowing indefinite causal orders.

If we assume that a set of operationally defined statistics is produced by a
quantum causal structure taking this specific form, we can then use
statistical tools to make inferences about the nature of the underlying
quantum structure - for example, given a process operator we can use the
algorithm set out in ref [33] to find out if the operator obeys the Markov
condition for some particular DAG on the quantum nodes, thus determining
whether that DAG represents a candidate quantum causal structure for the
process. Evidently there are a number of practical uses for this formalism -
for example, given an experimental procedure where we are not sure our
apparatus is doing exactly what we expected, we can apply quantum causal
inference to determine if the observed statistics are consistent with our
desired underlying quantum structure. However, it has also been suggested that
there may be foundational applications of this formalism - it serves to ‘
_establish an account of causality in quantum theory’s own terms, without
assuming a separate realm of classical systems or measurement outcomes._ ’[33]
One of our aims in this article will be to evaluate these claims.

##  3 The Descriptive Project: The Quantum SWITCH

Let us now begin with the descriptive project, seeking to understand what
exactly physicists working in the process matrix research programme mean by
their use of causal terminology. To do this, we will examine the discussion
around an experiment known as the quantum SWITCH, which is a causally non-
separable but still causal process originally proposed as an example of
indefinite causal structure in the process matrix formalism[34, 17]. In this
process, we use a control qubit to determine the order in which two gates $A$,
$B$ are applied to some other qubit: if the control qubit is in the state
$|0\rangle$ the gate $A$ is applied first and the gate $B$ follows, whereas if
the control qubit is in the state $|1\rangle$ the gate $B$ is applied first
and the gate $A$ follows. Evidently then if the control qubit is prepared in a
superposition of the state $|0\rangle$ and the state $|1\rangle$ we will get a
superposition of two orderings - in one branch the gates will be applied in
the order $A,B$ and in the other the gates will be applied in the order $B,A$.
This scenario is interesting particularly because it can be implemented with
current technology, thus it has already been demonstrated in the lab[17].

The conceptual significance of the quantum SWITCH has been questioned by
physicists working on the causal modelling approach. In particular, in ref
[35] MacLean et al make two objections to the claim that the quantum SWITCH
represents a genuine superposition of two different causal orders. The first
is that the two branches do not really contain copies of the same events in
different orders; the events occur at different times in each of the branches,
so really we should think of them as completely distinct events. Guérin and
Brukner[36] respond to this by invoking diffeomorphism invariance to argue
that events cannot be identified across branches by their spacetime location,
since that is not a diffeomorphism-invariant fact - instead they must be
identified by the intrinsic properties of the event, which are the same in
each branch of the quantum SWITCH. Now, this argument might succeed if it
these events were occurring in complete isolation so that there were no way to
identify them other than either spacetime location or the intrinsic properties
of the event. But in this case the events are not isolated: they are
controlled by an external system and thus they stand in well-defined relations
to that external system, meaning that there are many ways they can be
identified by means of their relations to other external events.
Diffeomorphism invariance is thus not really a relevant consideration in this
kind of experiment.

However, although the diffeomorphism argument does not succeed, there may be
other reasons to conclude that we should think of the two branches as
containing ‘the same events.’ To get clearer on this question, it is helpful
to consider in more detail what is actually achieved by the quantum SWITCH. On
the one hand, the results of refs [37, 20] underline that the quantum SWITCH
does not exhibit the strongest and most interesting kind of ‘indefinite causal
order’ that is possible within the process matrix framework444Ref [37]
demonstrates that the quantum SWITCH is not equivalent to the gravitational
SWITCH, which implements a similar circuit but uses superpositions of
different spacetime geometries to determine the two different orders; this
indicates that the ordinary quantum SWITCH does not exhibit indefinite causal
order in the same unambiguous way as the gravitational SWITCH. Meanwhile, the
results of [20] show that at the purely operational level it would be possible
to come up with a ‘fine-grained’ version of the quantum SWITCH process which
can be assigned a well-defined causal order, so operationally speaking the
SWITCH is not achieving anything that could not be achieved with a fixed
causal order.. On the other hand, if Maclean et al were right that in fact
completely different events occur in each branch, then one would naturally
expect the quantum SWITCH to be equivalent to a superposition in which the
qubit goes down two different paths, one containing the gate $A$ and the gate
$B$, and another containing a completely distinct copy of the gate $B$ and
then a completely distinct copy of the gate $A$. However, as shown in ref
[38], the quantum SWITCH is not equivalent to this superposition - rather it
is equivalent to a superposition in which the two paths are correlated. That
is to say, the quantum SWITCH produces the same results as a scenario where we
have one spacetime region containing a system $S_{A}$ which implements gate
$A$ and another spacetime region containing a system $S_{B}$ which implements
gate $B$, and in one branch the qubit first visits the $A$-region and then the
$B$-region, while in the other branch the qubit first visits the $B$-region
and then the $A$-region. In this case it is easy to see that the reason the
paths become correlated is precisely because the copy of gate A that the qubit
visits is really _the same gate_ in each branch: in the branch where the qubit
visits the $A$-region first, it interacts with $S_{A}$ and undergoes gate $A$,
and meanwhile the state of $S_{A}$ is presumably altered, but in the other
branch nothing happens during that period of time and instead the qubit visits
the $A$ region later, finding $S_{A}$ still in its original quantum state so
that gate $A$ is implemented. So this equivalence seems to uphold the
conclusion of Guérin and Brukner that the quantum SWITCH is in some sense a
superposition of different causal orders.

The second objection made by MacLean et al is that even if we agree that the
same events occur in the two branches, nonetheless we don’t really have
indefinite causal order, because the causal structure is the same in both
branches. That is to say, in both branches the causal structure is ‘$C$ causes
both $A$ and $B$,’ where $C$ is the control qubit. And indeed, this does seem
to be a correct description of the experiment, so this objection seems like it
could be a serious problem for the indefinite causal order programme - and
even more so because Costa has recently shown that under quite broad
conditions it is not possible to have a valid process in which we have a pure
superposition of causal orders without any external control[39], so the same
objection will hold even in more complex examples of indefinite causal order.

However it does not seem that the proposers of the quantum SWITCH ever meant
to claim that this experiment represents a superposition of different causal
_structures_ in the sense in which MacLean et al use this term. Rather they
argued that it demonstrates a superposition of different _causal orders_[34].
In fact there is some confusion here because the language used by researchers
in this programme equivocates between ‘causal order,’ ‘causal structure’ and
‘causal relations.’ In some cases it seems to be implied that all three are
the same: for example ref [40] seems to be making an identification between
‘causal order’ and ‘causal relations’ in their reference to ‘ _realization of
indefinite causal order (ICO), a theoretical possibility that even causal
relations between physical events can be subjected to quantum superposition._
’ But on further examination, it is clear that these things are not identical.
The term ‘causal relations’ is naturally understood as referring to ‘cause-
effect relations’ of the kind that we are familiar with from our macroscopic
experiences, and the term ‘causal structure,’ usually seems to refer to causal
structures as analysed in the causal modelling framework, i.e. structures made
up out of these ‘cause-effect relations.’ _Causal order_ , on the other hand,
does not invariably refer to cause-effect relations. The quantum SWITCH is a
good example: for after all, even if we ignore the control qubit it is clearly
not the case that in one branch $A$ _causes_ $B$ and in the other $B$ _causes_
$A$ \- the experiment is not set up in such a way that the application of the
first gate is ‘the cause’ of the application of the second gate. But
nonetheless the order in which these events occur is meaningful, otherwise the
two branches would be identical and we would not have a superposition at all.
And moreover, we have seen that there are indeed good reasons to think that it
really is the _order_ of events which differs between these branches in
significant ways. So it seems reasonable to say that these events stand in a
physically meaningful _causal order_ even though they do not bear any cause-
effect relation to one another.

###  3.1 Causal Order

The observations of the previous section provide good reason to think that
‘causal order’ plays an important role in the process matrix formalism. But
what exactly is causal order, and how does it differ from the more familiar
notion of cause-effect relations? Though it may be tempting to elide causal
order with temporal order, this is not correct, because the process matrix
framework is specifically designed to study processes in the absence of a
fixed background structure and so the notion of causal order that emerges out
of that framework cannot simply be temporal order, although of course in real-
life examples we expect causal order and temporal order to coincide. Rather
‘causal order’ here simply refers to the fact that in many processes, the
operations occurring within that process can be arranged in a well-defined
order which is a property of the process itself: for example, if a process
involves passing a single system between four laboratories, with an operation
being performed on it in every laboratory, the order in which these operations
take place is encoded in the process matrix and can be inferred from the
subsequent state of the system, since the net effect of performing the
operations $A,B,C,D$ (in that order) will in general be different from the net
effect of performing the operations $A,C,B,D$ (in that order). Thus there is
no need to appeal to the temporal ordering of the operations, and indeed it is
at least logically possible that the order defined by the process matrix
should fail to coincide with the temporal order (for example this could happen
in a world where retrocausality is possible, or where there exist closed
timelike curves).

The distinction between causal order and ordinary cause-effect relations is
less clear in the context of classical physics, because classical operations
commute, and therefore in classical physics the order in which some set of
operations are performed on a system $S$ makes a difference to the final state
only if the choice of later operations depends on earlier operations in some
way (for example, if we decide which operation to perform on $S$ conditional
on the outcome of a measurement on $S$) - and in that case we can just say we
have cause-effect relations in the usual sense. So there don’t seem to be many
examples in classical physics where the order of operations is important in
and of itself. On the other hand in quantum physics operations do not in
general commute, and thus the order in which the operations are performed
makes a difference to the final state, even if later operations do not
causally depend on earlier operations. So in quantum physics it is
particularly important to distinguish between causal order and cause-effect
relations, because the former can be significant even when the latter are not
present: we can have ‘pure’ causal order instantiated simply by a sequence of
non-commuting operations without outcomes.

Thus we propose a formal definition of causal order. First, we define the
‘signal requirement’ as a condition on the strict partial order assigned to a
set of laboratories which requires that for any possible choice of local
operations by the agents in the laboratories, the outcomes obtained in each
laboratory should depend only on the settings in that laboratory or the
settings for experiments which occur earlier in the order (note that this is
the same requirement used in the definitions of causally ordered and causal
processes, as described in section 1). We recall that the word ‘depend’ here
refers to statistical dependency of the kind that could be used for
signalling, hence the name ‘signal requirement.’ Then we have the following
definition:

######  Definition 3.1.

A subset $\\{X\\}$ of the laboratories $\\{P\\}$ belonging to a process $P$
exhibits causal order iff there is some choice of strict partial order $O$ for
the remaining laboratories $\\{P\backslash X\\}$ such that after conditioning
on the order $O$ for the laboratories in $\\{P\backslash X\\}$, a) there
exists at least one choice of strict partial order for the laboratories in $X$
which is compatible with the signal requirement, and b) there exists at least
choice of strict partial order for the laboratories in $X$ which is _not_
compatible with the signal requirement.

The point of this definition is that for many processes, some strict partial
orders are compatible with the signal requirement and some are not, and thus
these processes can be regarded as having some inherent order, which is what
we refer to as ‘causal order.’ For example, a process composed simply of an
error-free channel exhibits causal order because the only strict partial order
compatible with the signal requirement is the one in which the laboratory
containing the input to the channel is earlier than the laboratory containing
the output to the channel. On the other hand, there are two obvious ways in
which a process can fail to exhibit causal order (for simplicity, here we
focus on processes composed of just two laboratories). First, a _non-causal_
process composed of two laboratories cannot exhibit causal order, because
there is _no_ choice of strict partial order for these laboratories which is
compatible with the signal requirement. Second a causal process composed of
two laboratories with no signalling from one lab to the other also cannot
exhibit causal order, because _any_ choice of strict partial order for these
laboratories is compatible with the signal requirement. More generally, a
subset of laboratories in a given process can exhibit causal order only if the
process allows signalling between at least two laboratories in that subset,
since the signal requirement never places any constraints on the relative
order of laboratories with no signalling between them. So a process can fail
to exhibit causal order either because no possible order for it is compatible
with the signal requirement, or because all possible orders for it are
compatible with the signal requirement - either way, there is no ‘inherent
order’ which follows from the definition of the process together with the
signal requirement.

One might worry that because this definition refers crucially to outcomes, it
does not cover the motivating case of pure causal order without outcomes.
However, recall that according to the definition of a process, agents involved
in the process should be free to perform any operations allowed by quantum
mechanics, so we can always consider an instance of the same process where the
agents perform operations which _do_ have outcomes, and then in general we
will indeed be able to find some subset of the laboratories where the signal
requirement places constraints on the strict partial order. Thus in a sense,
‘causal order’ is similar to ‘causal structure’ in the context of Minkowski
spacetime, since it expresses not standard cause-effect relations but some
kind of higher order modal structure - it tells us where there _could be_
cause-effect relations if the operations required to obtain outcomes are
carried out. We will have more to say about this connection later.

###  3.2 Asymmetry

Is the observation that causal order plays an important role in quantum
mechanics in tension with the common view that there is no causation in
fundamental physics? To answer this question, we must examine more closely the
reasons usually offered for this view. Russell has a number of arguments, but
perhaps his most enduring point is that causation is usually supposed to be
asymmetric - if $A$ causes $B$ then $B$ cannot also be the cause of $A$ \-
whereas the fundamental laws of physics do not seem to be asymmetric or
directed in any way: ‘ _The law makes no difference between past and future:
the future“determines” the past in exactly the same sense in which the past
“determines” the future._ ’[1] This point continues to appear in the work of
current philosophers - for example, Frisch suggests that the apparent time-
symmetry of fundamental physical looks like a serious problem for causal
fundamentalists, because ‘ _the (fact that) the notion of causation is
asymmetric … may even be thought to constitute the core of our notion of
cause,_’[41]555As we will see in section 3.3, Frisch subsequently offers a
putative counterexample to the claim that fundamental physics is not in fact
time-symmetric, and so argues that causal fundamentalism can be rescued. and
Field points out that this argument still succeeds even if the fundamental
laws are not time-symmetric: all that is needed to make it is that the laws of
physics ‘ _have essentially the same character in both (temporal) directions,_
’[4] which does seem to be the case for most of our fundamental physical
laws666Field does point out that certain interpretations of quantum mechanics
seem to violate this criterion - most obviously collapse interpretations,
since systems evolve differently after the collapse of the wavefunction.
However we will not dwell on these cases here, as our purpose is to show that
there can be a well-defined notion of causal order in fundamental physics even
if it does not exhibit any directedness or asymmetries.. So there is
apparently nothing intrinsically directed in fundamental physics which could
play the role of causation, and therefore it seems natural to conclude that
there is no causation in fundamental physics - causation arises at a
macroscopic level, perhaps because of the thermodynamic gradient[42] or due to
the presence of agents who can play a role in an interventionist[8, 9] or
perspectival[10] analysis.

The presence of ‘causal order’ in fundamental physics might seem like a
counterexample to the claim that there is nothing intrinsically directed in
fundamental physics, but in fact it is not, because causal order does not
require a notion of asymmetry or directedness. Here, a distinction must be
made between _directedness_ and _order_ \- we can define an order on a set of
objects which stand in relations to one another even if the relations are not
intrinsically directed. Consider, for example, the rungs on a ladder which is
perfectly symmetric, i.e. it looks the same if we flip the whole thing about
an axis through the middle parallel to the rungs. Due to the symmetry there
cannot be any ‘intrinsic direction’ to the ladder, and so, given two rungs,
there is no fact of the matter about which rung comes after the other; but
nonetheless if we start at one end and go towards the other end without
crossing any segment twice, we will end up with a well-defined total order for
the rungs. This order is not the objectively correct order, and we could get
another equivalent order by starting from the other end, but in either
direction we get a well-defined total order. In light of this distinction, in
this article we will use the term ‘bi-order’ to refer to an order where the
direction of the arrows does not matter: that is to say, a bi-order is the
equivalence class consisting of an order and the corresponding order obtained
by simply reversing the direction of all the relations making up the order.
Thus, for example, a strict partial bi-order is a bi-order such that each of
the two orders in the equivalence class is a strict partial order (note that
reversing the direction of all of the relations in a strict partial order will
always give rise to another strict partial order).

We now observe that ‘causal order’ requires only a bi-order, not an ordinary
order with an intrinsic direction. To see this, it will be helpful for us to
use terminology due to Pienaar[43, 44] distinguishing between two ways of
describing a process. First, the ‘observational scheme,’ (OS) specifies the
process by giving a joint probability distribution over the full set of
variables, thus allowing us to choose any set of variables and make
predictions for the other variables conditional on the values of the variables
in that set. Second, the ‘interventionist scheme,’ (IS) begins by choosing
inputs and outputs and then specifies the process by giving probabilities for
the outputs conditional on the inputs. The idea is that the ‘observational
scheme’ encodes only the fundamental physical relationships between variables
that constitute the process, while the interventionist scheme consists of
various possible descriptions of this process that come from embedding it in
macroscopic scenarios where agents have control over certain variables and not
others777The claim that the ‘observational scheme’ is more fundamental than
the ‘interventionist scheme’ is tantamount to the idea that joint probability
distributions over histories are more fundamental than conditional
probabilities which predict one event in terms of previous ones; this idea has
been discussed in another context by Wharton and Liu in ref [45].. So for
example, if we have an error-free channel between variables $A$ and $B$, in
the OS we would describe this process by the symmetric probability
distribution $p(A=B)=1$, whereas in the IS we could pick $A$ as the input and
thus describe the process by the conditional distribution $\forall x\
p(B=x|A=x)=1$, or we could pick $B$ as the input and thus describe the process
by the conditional distribution $\forall x\ p(A=x|B=x)=1$ \- notice that the
IS descriptions look superficially asymmetric even though the underlying OS
process postulates a perfectly symmetric relation between $A$ and $B$.

With this terminology in hand, let us now consider a process in which agents
in separate labs perform the non-commuting invertible quantum operations
$A,B,C,D$ consecutively on a single quantum system. Nothing in the fundamental
physics of these operations singles out one particular direction: if we treat
the input to $A$ as the initial state then we can write the resulting IS
description in the form $ABCD$, while if we treat the output of D as the
initial state then we can write the resulting IS description in the form
$D^{-1}C^{-1}B^{-1}A^{-1}$, but if we throw away external information about
time direction and the involvement of agents it would seem that there is no
deep distinction between these two descriptions: that is to say, $ABCD$, and
$D^{-1}C^{-1}B^{-1}A^{-1}$ can be regarded as two different IS descriptions
which correspond to the same OS joint probability distribution. However,
although the direction does not matter, evidently the effect of $ABCD$ is not
the same as the effect of $ACBD$: we are free to write the operations in
either direction in the IS, but since the operations don’t commute, whatever
direction we choose is associated with a unique correct order, and if we
change the order then the IS description will no longer correspond to the same
OS process. In this sense this example exhibits causal order even though it is
reversible, and thus it corresponds to a bi-order - in this case a total bi-
order.

What about the case where the agents perform operations that have nontrivial
outcomes? Again, if the underlying physics is truly time-reversal invariant
then in principle it would seem that given an IS description we should be able
to write a time-reversed IS description which will correspond to the same
underlying OS process. This time-reversal will involve inverting all the
operations performed inside the individual laboratories, thus turning input
variables into output variables and vice versa (see appendix A for more
details about time-reversal). So suppose we have an IS description which
exhibits causal order in the simplest sense, i.e. it has an output $O_{1}$
which is correlated with an input $I_{1}$, so when we try to write down the
‘inherent order’ of this process in accordance with the signal requirement, it
must be the case that $I_{1}$ occurs before $O_{1}$. Under time-reversal,
$O_{1}$ becomes $I_{2}$ and $I_{1}$ becomes $O_{2}$, and if the reversal
operation preserves the underlying OS joint probability distribution, then it
must be the case that $O_{2}$ now depends on $I_{2}$. Moreover, if we simply
flip the directions of all the arrows in the original strict partial order,
this yields a new strict partial order according to which $I_{2}$ occurs
earlier than $O_{2}$, so we obtain a new ‘inherent order’ which remains
compatible with the signal requirement. Thus in at least some cases, reversing
an IS description of a process which exhibits causal order will yield another
process exhibiting causal order in the opposite direction, and this second
process is ex hypothesi an alternative description of the same underlying OS
process. This demonstrates that the definition of causal order does not
require that there is any asymmetry built into the underlying physics: to
identify causal order we need only a strict partial bi-order rather than a
strict partial order, since we can have processes compatible with two
different IS descriptions for which the inherent orders associated with the
signal requirement are oppositely directed. And thus although the objections
of Russell, Field and others provide compelling reasons to think there are no
ordinary relations of cause and effect in fundamental physics, they certainly
do not rule out _causal order_ in the sense used in the process matrix
framework, since that notion does not require any intrinsic asymmetry.

But does this mean that the distinction between processes which exhibit causal
order and processes which do not is purely conventional, depending entirely on
our choice to regard some variables as inputs and others as outputs? No,
because the question at issue is not which ordering is correct, but whether
there is _any_ possible ordering. Thus we can define causal order at the level
of the OS description without specifying a choice of inputs and outputs at all
- that is, we can say that an OS process exhibits causal order iff there
exists at least one IS description of this process which exhibits causal
order, whereas an OS process does not exhibit causal order if there is _no_ IS
description of it which exhibits causal order. So there is a well-defined,
non-conventional distinction between processes which exhibit causal order and
processes which do not, even if all the processes we are dealing with are
symmetric and reversible.

###  3.3 Causal order in physics

We will henceforth refer to cause-effect relations in the everyday sense as
‘strong causation,’ in contrast with the weaker notion of ‘causal order.’ We
will not offer a precise definition of strong causation here; instead we
suggest that readers should fill in the details according to their preferred
analysis of causation - the idea is that strong causation captures whatever
intuitions about causality one considers as essential to the concept, whereas
causal order requires only to the presence of well-defined order and may not
satisfy other intuitions about causality. For example, ‘strong causation’
presumably requires some form of asymmetry or intrinsic directedness, and it
has also been argued that causation should be deterministic[5], local[4, 5],
or should involve only a small set of causally relevant factors[4], so one
might insist that strong causation should have these features.

This distinction now allows us to offer greater clarity on the meaning of the
term ‘causal’ in the terms of _causally ordered_ , _causally separable_ and
_causal_ in the process matrix formalism: since the signal requirement appears
in the definitions for all of these terms, and meanwhile the definitions say
nothing about asymmetry, determinism, locality etc, it is clear that these
definitions should be understood as referring to causal order rather than
strong causation. Moreover, having observed that causal order need not involve
asymmetry, we can provide an answer to the problems for the process matrix
formalism described in section 1.0.1. For the concerns we raised there about
the role of agents hold only if one is determined to describe process matrices
as encoding _strong causation_ ; if instead we understand the object of study
of the process matrix formalism to be _causal order_ , then we don’t need to
invoke interventionist or perspectival considerations to ensure that the
causal structure has some asymmetry, and therefore we have no need to
introduce any agents.

More generally, we suspect that equivocation between causal order and strong
causation is the source of some of the ongoing disputes over the role of
causality in fundamental physics, and thus distinguishing carefully between
these concepts can resolve a number of outstanding questions. For example,
Frisch[46] argues that causal notions are intrinsic to dispersion relations,
because the derivation of the dispersion relation relies on a condition that
many physicists[47, 48] describe as ‘causality,’ stipulating that ‘ _no output
can occur before the input._ ’ As Frisch notes, there is ‘ _no purely formal
or mathematical reason_ ’ for this requirement but it is nonetheless accepted
as physically well-founded and indeed treated as ‘ _a general constraint on
all physically plausible models_.’ The causal ordering account explains why,
for this ‘causality’ condition is essentially a version of what we have called
‘the signal requirement’ and thus there is a very good physical reason why it
should be obeyed: because to the best of our current knowledge, all physical
processes are causal (and arguably they are required to be so in order to
avoid logical contradictions - see section 4.2). Thus although the causality
condition used here is superficially time-asymmetric, as Frisch emphasizes,
nonetheless it does not indicate the presence of any intrinsic asymmetry, as
we have seen that well-defined causal order does not depend on asymmetry. So
Frisch’s observation that causal notions play an important role in the
dispersion relations is not in tension with the fact that the physics involved
in these relations seems to be time-symmetric, since it is causal order and
not strong causation that is relevant here.

Similarly, although physicists very frequently refer to ‘causal structure’ in
the context of special relativity, there has been controversy over whether
special relativity really has anything to do with causation: for example,
Nerlich[49] argued that causality plays no fundamental role in the theory, for
after all the lightcone structure can be identified purely in terms of the
behaviour of light, without any reference to ‘causation.’ Nerlich also
demonstrated that attempts by Zeeman[50], Malament[51] and Winnie[52] to
construct special relativity on the basis of a causal relation fail to show
convincingly that the relation in question is really ‘causal’ in any strong
sense. But this controversy can be resolved by appeal to the distinction we
have just made: lightcone structure in special relativity defines _causal
order_ , because it ensures that spacetime points are arranged in a strict
partial order or bi-order which defines the possibilities for signalling
between them, but it does not necessarily say anything about _strong
causation_. To have strong causation we need some kind of asymmetry, and we
will not get this from special relativity alone, since the time-reverse of a
special-relativistic spacetime is another valid special-relativistic
spacetime. So spacetime structure is genuinely causal in the sense of ‘causal
order,’ but it does not instantiate _strong_ causation and in this sense
Nerlich’s criticisms are well grounded.

Furthermore, notice that spacetime structure and causal order as we have
defined it are structurally similar: neither one really depends on asymmetry
or directedness, and both ultimately instantiate a strict partial order (which
is really a bi-order). And indeed, the connection between them has often been
recognised. One possible approach to that connection is explored in ref [20],
which sets up a framework to study the relationship between ‘causality’ in the
spacetime structure sense and ‘causal order’ in the process matrix sense. This
approach involves embedding a causal structure defined using the process
matrix formalism (or another similar information-theoretic formalism) in a
spacetime, and then imposing compatibility constraints - for example, a system
should only be able to signal to other systems in its future lightcone - so
that ordering requirements for processes can be understood as consequences of
relativistic causality. An alternative approach is taken by the causal set
programme[53], which builds spacetime up out of a set of causally related
events which are required to stand in a strict partial order. From this point
of view, rather than trying to embed a causally ordered structure in a
spacetime, we can simply allow the spacetime to emerge from the causally
ordered structure. Moreover, the methodology of the process matrix formalism
adds a new nuance to this construction, because it allows us to see that the
‘strict partial order’ requirement need not be imposed by fiat but instead
could be derived from the need to avoid contradictions (see section 4.2), and
thus from this point of view, spacetime structure could potentially be
regarded as a consequence of basic consistency requirements for valid
processes.

##  4 The Metaphysical Project: Strong causation versus causal order

We now move to the metaphysical project, where we will argue that the notion
of causal order identified in the previous section offers new insight into the
nature of causation. For although causal order does not fully satisfy our
everyday intuitions around causality, nonetheless it clearly bears some
relation to strong causation. Indeed, once we select a preferred direction
along the strict partial bi-order associated with a process exhibiting causal
order, it then becomes an ordinary strict partial order, which can be
interpreted as encoding strong causation as in the causal modelling framework.
Moreover, in many cases there is indeed a natural choice of direction along
the bi-order arising from the particular perspective that we have on it - for
example, usually we experience ourselves as having control over certain
variables and not others, so it is natural to regard the controllable
variables as ‘inputs,’ and as we have seen, specifying a choice of inputs for
a causal process singles out a choice of direction for the associated bi-
order.

In this connection, it is important to emphasize that choosing a direction for
a bi-ordered graph is not the same as adding a direction to an undirected
graph. If you simply take an undirected graph and assign a direction to one of
those edges, that choice does not define a direction for all the other edges;
you are free to assign directions to the other edges in many different ways,
and in a sufficiently complex graph it will usually be possible to do this in
such a way that the resulting directed graph fails to be acyclic. Whereas in a
bi-ordered graph there is a well-defined ‘flow’ through the graph, although
the direction of the flow is immaterial; so there are exactly two valid ways
of assigning directions to the edges, and one is the reverse of the other,
such that choosing a direction for any one edge is equivalent to selecting one
out of the two valid assignations of directions. Therefore in this case we are
not free to simply create loops: if the bi-order is acyclic, then any choice
of direction will lead to an acyclic graph. So it is specifically the fact
that the underlying process exhibits causal order which ensures that invoking
interventionist or perspectival considerations to select a direction leads us
naturally to a structure which can be represented by an acyclic causal model
and which can therefore be regarded as encoding strong causation. This idea is
explored in more detail in ref [54].

Indeed, causal order is clearly crucial to any account of strong causation,
although this point is not often articulated in accounts of the emergence of
macroscopic causality. Invoking a perspective or an intervening agent provides
a local direction of causation, but this will not yield well-defined
macroscopic causal structures more globally unless there is some underlying
causal order that ensures consistency of causal direction throughout the
network of related events. Often the importance of causal order goes
unmentioned because there is an implicit assumption that we are working on a
fixed spacetime background, so temporal order can be substituted for causal
order - but in fact, this is not enough to guarantee a well-defined causal
structure unless retrocausality is ruled out by fiat, and if the aim is to
account for the emergence of causality it is hard to achieve this in a non-
question-begging way. So to arrive at a non-circular account of strong
causation we must start from a process which exhibits some form of causal
order, and thus when we add perspectival or interventionist considerations we
will arrive at a well-defined global causal structure without loops or other
pathologies.888 Of course, adding a direction to the causal order associated
with quantum mechanical systems will only yield strong causation in the
context of the description of certain sorts of highly specific quantum
experiments; to move from fundamental physics to the kinds of causal laws that
feature in our everyday life, like ‘Exercise prevents heart disease,’ would be
an extremely complex problem requiring us to deal with various confounding
factors and to settle the question of how to reduce the classical to the
quantum. So for most instances of strong causation in the macroscopic world
there is no straightforward way to explicitly derive the causal laws from
quantum causal order; but nonetheless it is clear that we would not be able to
arrive at macroscopic causal laws if it were not the case that macroscopic
events exhibit some kind of causal order, and it seems reasonable to think
that the causal order of the macroscopic world has its origin in the fact that
certain sorts of processes in the underlying microscopic physics also exhibit
causal order, even if we can’t at present perform that derivation explicitly.

###  4.1 Causal Modelling

This claim about the origins of causality has consequences for the causal
modelling programme in quantum foundations. For we have emphasized that no-
signalling processes cannot exhibit causal order999It should be emphasized
that this claim about no-signalling processes applies only to _genuinely_ no-
signalling processes, like the Bell correlations: there is nothing anyone can
do to make the Bell correlations transmit a signal, so they are simply not
subject to ordering requirements. On the other hand, in cases where we have
‘no-signalling’ only because two classical causal influences cancel out, a
small change in the initial conditions will turn the no-signalling process
into a signalling one, and thus we would expect that these sorts of influences
will still be required to stand in a well-defined order, which in turn
justifies the original instinct to refer to them as ‘causal.’, so if it is
accepted that causal descriptions are typically grounded by underlying causal
order, it seems that one should _not_ expect that causal notions will be a
suitable way of analysing no-signalling processes. And in particular, since
measurements on entangled quantum particles are no-signalling, we shouldn’t
expect to be able to find a natural causal account of phenomena involving
measurements on entangled particles, such as the Bell correlations. Of course
there is nothing to stop us from writing down causal models for such
processes, but it’s unclear that such models can be regarded as describing any
real feature of the processes, since ex hypothesi causal phenomena arise from
causal order and there is no causal order involved here.

Moreover, this conclusion exactly matches what we observe in reality: no-
signalling correlations from quantum entanglement can be exhibited between
measurements at spacelike separation, in which case there is no natural
temporal order to assign to them, whereas signalling processes can never be
performed at spacelike separations, and therefore they are always associated
with a temporal order. That is, in quantum mechanics we find that signalling
processes are always ordered (temporally) whereas no-signalling processes
don’t usually seem to have any inherent order, which is what we would expect
based on our argument that no-signalling processes do not exhibit causal
order.

This observation raises questions about certain conclusions that have been
drawn using causal models in quantum foundations. For example, the fact that
any causal model which reproduces the Bell correlations must be fine-tuned is
sometimes taken to mean that there is some mystery about the Bell correlations
which urgently requires explanation[29], but the line of reasoning we have
pursued here tells us that it’s not surprising that we are unable to find a
satisfying causal model for the Bell correlations, because the Bell
correlations are simply not the right kind of thing to analyse with a causal
model. Note that this is true even though the Bell correlations can be
described in interventionist terms, with the preparation and measurements
playing the role of the ‘loci of possible interventions’ - an interventionist
analysis can succeed in transforming a bi-order into a directed acyclic graph
only if there is some well-defined bi-order present, so in the case of the
Bell correlations, since these processes need not instantiate causal order,
there is no reason to expect that an interventionist analysis will yield a
sensible causal model. Of course, results like that of Wood and Spekkens are
still very important to help us understand the specific ways in which standard
causal models fail in the context of quantum mechanics, but the point is that
we should not be too distressed to find that no non-fine-tuned causal model is
possible for these phenomena.

In response to this argument, proponents of the causal modelling approach
could of course reject the account we have suggested here of the origins of
causal phenomena, and maintain that there exists a set of ontic variables
underlying quantum mechanics which stand in genuinely asymmetric causal
relations, so strong causation is in fact a feature of reality at the most
fundamental level. However, we would then be required to accept the following
story:

> _Nature contains asymmetric causal relations at the most fundamental level.
> This fundamental causal structure gives rise to quantum mechanics, which is
> time-symmetric 101010As shown recently in ref [55] quantum mechanics is
> time-symmetric even if we include the Born rule., so the underlying
> asymmetries have become invisible. Then from this temporally symmetric
> physics there arises a higher-level macroscopic description which contains
> asymmetric causal relations, but these macroscopic asymmetries have nothing
> to do with the fundamental asymmetries - they may perhaps be explained as a
> consequence of the second law of thermodynamics, or simply given a
> perspectival account based on the fact that agents perceive themselves as
> having some temporal orientation, but crucially, at no point do fundamental
> asymmetric causal relations play any part in this story. So the causal
> features of the macroscopic world are not connected in any way to the causal
> structure that underlies quantum mechanics._

The problem with this story is that if macroscopic causal features have
nothing to do with the putative underlying causal structure, then macroscopic
causal features cannot count as evidence for the existence of that underlying
causal structure, so it’s unclear why we should believe in this putative
underlying causal structure. After all, ex hypothesi these fundamental
asymmetric causal relations impact macroscopic reality only via the role they
play in giving rise to time-symmetric quantum mechanics, so macroscopic
reality would exhibit exactly the same causal features even if there were no
underlying asymmetries. Indeed, it would surely be quite a strange coincidence
were the causal asymmetries of the macroscopic world to be reflected so
precisely in an underlying fundamental asymmetry, even though this fundamental
asymmetry apparently has nothing whatsoever to do with the asymmetry of the
macroscopic world - is it not more likely that this hypothesis is simply the
result of erroneously extrapolating the causal structure of our macroscopic
experience?

To push back against this criticism, the proponents of the causal modelling
framework might observe that at least for the moment it seems that not all
possible process matrices actually occur in reality - at present we only know
how to implement the causal ones. Therefore one might be tempted to argue that
the reason we see only causal process matrices is that reality is
fundamentally ‘causal’ in the sense of strong causation, and thus the only
processes which can occur are the ones which have an underlying causal
structure. This would seem to justify the methodology of seeking to come up
with an underlying causal structure for the quantum mechanics, including the
Bell correlations. However, it remains possible that the non-causal processes
are in fact physically possible, and simply occur only in special regimes,
such as in quantum gravity[56] or on closed timelike curves[13] (after all,
CTCs do in fact occur in some solutions to the Einstein Field Equations[57]).
Moreover, even if it is true that only causal processes occur in nature, it
remains the case that there doesn’t appear to be intrinsic asymmetry in these
processes, so there is no reason to invoke specifically _strong_ causation to
explain the absence of non-causal processes. At best we have reason to
hypothesize that nature is fundamentally causal in the sense that all
physically possible processes can be assigned a strict partial order which
obeys the signal requirement, and since the signal requirement places no
restrictions on no-signalling processes, _this_ hypothesis does not justify us
in seeking to come up with an underlying causal structure for the Bell
correlations.

###  4.2 Why causal order?

In addition to offering a novel account of the origins of causal phenomena,
the process matrix framework takes a further intriguing step: it offers a
possible explanation for the _necessity_ of causal phenomena. For the whole
point of the process matrix framework is that it is defined in such a way as
to avoid contradictions, and therefore it directly incorporates a set of
consistency conditions on dynamics which must be obeyed if agents locally have
the freedom to perform operations and compose processes in any way they like.
And as we have just noted, this places strong limits on processes which are
signalling. In particular, a large class of process matrices (indeed, all
those process matrices known to be physically possible) avoid contradictions
specifically because they are causal, meaning that there is at least one
choice of strict partial order for the laboratories involved which obeys the
signal requirement: evidently this rules out scenarios like the one depicted
in figure 1, where signalling processes are used to give rise to a
contradiction. Moreover, any causal processes which allow signalling between
different laboratories will necessarily instantiate causal order, and thus the
consistency conditions encoded in the process matrix formalism help to explain
the phenomenon of causal order.

On the other hand, we have emphasized that causal order cannot be instantiated
by laboratories which can’t signal to one another. This makes sense if we
think of the signal requirement as a kind of consistency condition, because
no-signalling processes cannot give rise to contradictions in the style of
figure 1, and therefore there is no need for no-signalling processes to have
any inherent order. This underlines our previous point about the
inappropriateness of applying causal modelling to no-signalling correlations:
not only is it the case that no-signalling correlations can’t exhibit causal
order, they are not even subject to the kinds of consistency requirements
which one might regard as the underlying reason for causal order.

Thus the consistency conditions encoded in the process matrix formalism offer
a partial explanation for the prominence of causal order in the physical
world, as well as the apparent absence of causal order for processes like
measurements on entangled particles. However it is only a partial explanation,
because we can also use the process matrix formalism to write down _non-
causal_ processes which obey the consistency conditions, so something more
than just consistency conditions is needed to explain why it seems that all
physical processes are causal, or at least why the regimes we have so far
probed contain only causal processes. For example, ref [15] proposes that all
physical processes should be ‘purifiable,’ i.e. they should be expressible as
part of a pure process in a larger space, where a pure process is one that
induces a unitary transformation from the past to the future whenever unitary
transformations are also applied in the local laboratories. It is shown that
this rules out a large class of non-causal processes, although not all of
them. Several other arguments are presented in ref [54]. First, if we allow
non-causal processes, then the dynamics of the allowed processes cannot be
defined in a local way, since certain sorts of dynamics are possible outside
closed timelike curves and exotic causal structures but are not always
possible inside such structures - e.g. we saw in section 1 that in the
classical process function approach the identity operation around a closed
timelike curve is not possible, but evidently the identity operation is
possible outside closed curves. One might argue that this novel kind of
nonlocal dependence on the global structure of spacetime is not permitted by
the laws of our actual world, and thus non-causal processes are not physically
possible. Second, the notion that non-causal processes are possible relies
crucially on the idea that there is a fundamental distinction between local
operations in laboratories, which can be freely chosen, and dynamics between
the laboratories, which are fundamentally uncontrollable - for example, the
framework must have the consequence that if we try to implement the identity
operation around a closed curve we will always fail. And yet in a sense one
can think of dynamics as being just a sequence of chained local operations, so
one might feel that the distinction being made here is too strong: if we have
free choice over local operations, we should also be able to freely construct
dynamics on a CTC by means of chains of local operations. This line of
reasoning might be taken to suggest that these sorts of non-causal processes
are not in fact physically possible - or at least they should not be possible
in the kinds of regimes where we can in principle have very good control over
the dynamics we implement, which would explain the fact that we have not
observed any of them in the regimes we have thus far probed. In any case, if
some such argument is accepted, we can then argue that the necessity for all
signalling processes to exhibit causal order follows directly from the need to
avoid contradictions, and thus the causally ordered nature of the physical
world can be regarded as a simple consequence of basic consistency conditions.

This way of thinking allows us to bring some additional nuance to the approach
to causality adopted in the process matrix framework. As we have noted, this
approach largely identifies causal influences with signalling, or at least
uses signalling as a proxy for causation, whereas most people consider that
signalling is at best a sufficient condition for the presence of a causal
relation[20]. However, the understanding that causal ordering is related to
consistency requirements allows us to avoid presupposing the association of
causation with signalling, and rather to derive it from more fundamental
considerations. That is, suppose we accept that ordinary causal phenomena
arise from an underlying causal order together with a choice of direction.
Then recall that consistency requirements impose conditions only on signalling
processes, not on no-signalling ones. It follows that causal order, and hence
causal phenomena, will typically be associated with signalling processes and
not with no-signalling ones, so the association of ‘causation’ with
‘signalling’ in this framework can thus be retrospectively justified.

###  4.3 Modal Structure

One might use the considerations we have presented here to argue for a kind of
causal fundamentalism, where one would argue that the phenomenon of causal
order demonstrates that there _is_ in fact causation in fundamental physics,
even if it does not involve any temporal asymmetries. Ney advocates a position
of this kind in ref [58], suggesting that at the fundamental level, ‘ _There
is still causation, because there is still physical determination. But the
distinction between what is the cause and what is the effect may not be
fundamental._ ’ That is to say, she holds that we can identify causation with
fundamental symmetric relationships and allow that the asymmetry of causation
only emerges at a higher level. Evidently Ney’s take on causal fundamentalism
fits nicely with the picture we have advocated here, where causal order exists
at the fundamental level but causal asymmetry appears only when we use higher-
level considerations to select a direction through the bi-order.

Alternatively, one could parse our approach as a causal non-fundamentalist
viewpoint, where the causally ordered nature of underlying fundamental
processes is not itself regarded as a form of causation but instead is taken
to be a partial truth-maker or grounds for higher-level causal claims. In this
sense, our account could be regarded as one possible way of realising the
vision set out by Norton, who argued that ‘ _a causal character can be
recovered from the science as looser, folk sciences that obtain in restricted
domains_ ’ - one could interpret this as a description of the way in which
strong causation emerges from causal order in special domains where there is a
natural way of choosing a direction for the bi-order. However, we emphasize
that since causal order is essentially a kind of modal constraint (determining
which processes can and cannot exist, and how they can and cannot be arranged
in spacetime) this position still requires a kind of _modal_ fundamentalism -
that is to say, it does not eliminate objective modality altogether, but
rather reduces causation to a deeper modal structure which is more consistent
with known physical law.

The idea that fundamental physics involves causal order but not strong
causation may raise some concerns amongst the subset of scientific realists
who argue that realism is necessarily requires causal explanation[59, 29].
This kind of position seems to be related to the common idea that there is a
contrast between (bare) correlation and causation, as epitomized by the slogan
‘correlation is not causation,’ which comes with the implication that
‘causation’ expresses something stronger than bare correlation. And indeed, if
‘(strong) causation’ and ‘bare correlation’ were our only options for
describing the character of fundamental physics, it would seem reasonable to
say that the realist must be committed to a causal description, since
committing only to ‘bare correlations’ would lead to a collapse into
constructive empiricism[60]. However, (strong) causation and bare correlations
are not in fact the only possible options. In particular, we would argue that
causation is stronger than correlation not because it is _asymmetric_ (after
all, bare correlations themselves can also be asymmetric) but because it is
_modal_. It is precisely this modal force which is captured in the
interventionist account of the difference between causation and correlation -
‘ _only some and not all correlational relationships are potentially
exploitable for purposes of manipulation and control; we regard those
relationships that are so exploitable as causal,’_[61] \- since modal
relations are required to ground counterfactuals describing what would be the
outcome of various kinds of manipulation and control. So we can still give a
realist account of fundamental physics without invoking strong causation,
provided that we are willing to make some other kind of modal claim, for
example by postulating _symmetric_ modal relations.

What exactly does a ‘symmetric modal relation’ look like? Ney bases her
account on a symmetric modal relation that she refers to as _determination_ ,
defined as ‘law-governed co-instantiation,’ which could be expressed
quantitatively in the form $p(A=B)=1$. More generally, we could also imagine
other ‘law-governed’ modal relations expressed in the form $p(A=B)=x$ for any
$x\in(0,1)$. These sorts of modal relations are less familiar to us than
cause-effect relations, as evidenced by the fact that there does not seem to
be any word for them in ordinary non-technical language, but they are in fact
ubiquitous in science. For in Cartwright’s terminology, many laws that appear
in science are not causal laws but rather ‘ _laws of association … (which)
tell how often two qualities or quantities are co-associated_ ’[2] \- and
indeed this observation was indeed another of Russell’s arguments for his
claim that causation does not appear in fundamental science[1]. For example,
consider a relation like the Maxwell-Faraday equation, $\nabla\times
E=-\frac{\partial B}{\partial t}$. This equation is not directed: it is not
the case that the state of the electric field _causes_ the magnetic field to
have a certain rate of change or vice versa, since there is no fact of the
matter about which ‘comes first.’ But nonetheless the relation between $E$ and
$\frac{\partial B}{\partial t}$ is not just ‘bare correlation’ - it is an
instance of Ney’s ‘law-governed co-instantiation,’ which is to say it is
nomically necessary that the two must stand in this symmetric relationship.
Moreover, although we tend to think of dynamical laws as being intrinsically
directed, Ismael emphasizes that dynamical equations are not really different
in form from these co-instantiation laws: ‘ _Dynamical laws … are constraints
on the relationships between states at different times, but … there is nothing
in the law itself to say either determines the other._ ’[69] So in fact, there
are good reasons to suppose that we should think of dynamical equations in the
same way as equations like the Maxwell-Faraday equation - they simply express
a symmetric, undirected modal relationship between states at different times.
Thus realism need not be tied to causal explanation, as we can offer a
legitimately realist account of fundamental physics which replaces causation
with symmetric modal relations linking events at different times.

These observations allow us to bring additional nuance to our argument that
the Bell correlations should not be given a causal explanation, for we can now
emphasize that this does not mean they must be regarded as simply ‘bare
correlations.’ Instead we can say that the Bell correlations are ultimately to
be explained by symmetric modal relations, just like the relations involved in
ordinary laws of association and arguably even dynamical laws. The only
difference between the Bell correlations and more ordinary cases of signalling
correlations is that the signalling correlations are subject to consistency
conditions which require that they must have some inherent order (or rather,
bi-order), but the bi-order requirement is not the explanation of the
signalling correlations - it only explains the way in which the events that
instantiate them are arranged. In both the signalling and the no-signalling
case the correlations are ultimately explained by appeal to underlying
symmetric modal relations, so the Bell correlations are just as explicable as
signalling correlations.

##  5 The Functional Project: Non-Classical Causal Models

We have reached a broadly negative verdict on the applicability of classical
causal modelling to quantum mechanics based on the observation that the usual
grounds for a causal description is not present in all quantum processes. But
one might still hope to offer a justification for causal modelling in quantum
mechanics in the context of the functional project, which, in Woodward’s
words, ‘ _takes as its point of departure the idea that causal information and
reasoning are sometimes useful or functional in the sense of serving various
goals and purposes that we have._ ’ With this motivation in mind, one might
argue that causal modelling in quantum makes sense because it can serve an
important functional role in the description of quantum phenomena, even if the
usual physical underpinnings of causation are absent.

In this context, we note Woodward’s warning that ‘ _“cause” might not turn out
to be like “entropy” in the sense that it is a notion that it is usefully
applicable to physical situations with a certain structure when analyzed at a
certain level of description, and not usefully applicable elsewhere,_ ’[11].
This suggests a possible objection to the functional programme - one might
worry that this ‘certain structure’ which makes causal notions ‘usefully
applicable’ is simply _causal_ structure, in which case the functional
analysis of causation apparently cannot be given in a non-circular way. But
the approach we have taken here suggests a natural answer to this objection:
the relevant kind of structure is in fact _causal order_ , since the presence
of causal order entails that there is a uniquely correct way to understand the
situation in causal terms once we select a direction for the underlying bi-
order. Thus the functional analysis can be pursued without risk of circularity
if we regard it as an analysis of _strong causation_ and we understand the
requisite underlying structure in terms of the independent concept of causal
order. And if this is accepted, it seems that we should not expect causal
models in the traditional sense to be ‘usefully applicable’ in the description
of no-signalling quantum processes, since these phenomena do not have the
right kind of structure.

However, this argument does not rule out the possibility that there could be
some kind of generalized ‘quasi-causal’ description which might play a similar
kind of functional role to a traditional causal model in the context of
phenomena which fail to exhibit causal order. Indeed, insofar as these
phenomena are to some degree under the control of macroscopic agents, it must
surely be possible to arrive at a description of them which characterises the
way in which they can be used to achieve practical goals and purposes, just as
a traditional causal description would do. And this is exactly what is
achieved by the formalism of quantum causal models, as presented in section 2.
For quantum causal models do not appear to be attempting to describe an
‘underlying reality’ for quantum mechanics, but rather the reverse - they
provide a ‘top-down’ description of the laboratory procedures that are
implemented on the relevant systems, so for example ref [59] suggests that
quantum causal models are suitable for the case where ‘ _one typically does
not have a direct description of the causal mechanisms, but rather only a very
coarse-grained description of them in terms of laboratory procedures that are
implemented on the relevant systems_ ’ and ref [33] notes that if this
methodology is applied to characterise a quantum circuit, it will often simply
yield the connectivity graph representing the way the gates are literally
wired together in the laboratory111111Or it may yield a subgraph of this
connectivity graph, because there may turn out to be nodes which do not
influence one another even though they are physically connected.. Essentially,
the quantum causal structure provides a set of instructions for the minimum
possible laboratory operations which would be needed to produce the desired
effects. Thus these models serve a similar function to causal descriptions in
that they show us how to construct quantum circuits which are suitable to
achieve various goals, so from the functional point of view the idea that
these models have a claim to be thought of as quasi-causal seems very
reasonable.

However, the literature also contains some suggestions about the foundational
significance of these models which might be interpreted as going beyond these
operational claims. For example, ref [33] suggests these models can be used ‘
_to establish an account of causality in quantum theory’s own terms, without
assuming a separate realm of classical systems or measurement outcomes. The
account provides its own answer to many of the basic questions concerning
causality in a quantum universe._ ’ The idea is that if we redefine ‘cause’ in
the quantum regime to refer to quantum causal models rather than classical
ones, then we can after all have ‘ _causal explanations of quantum processes_
’[70]. But some caution is required here - the fact that ‘quantum causal
structure’ offers a practically useful way of describing certain laboratory
phenomena does not necessarily entail that it is providing a meaningful
_explanation_ of those phenomena. After all, ‘quantum causal structure’ is not
causation in the traditional sense, so even if one takes it that a traditional
causal account is automatically explanatory, something more must be done
before we can accept that this kind of structure is similarly explanatory. And
since it is a truism that it is possible to give an operational description of
laboratory procedures, the existence of such a description for some phenomenon
should not be taken as revealing any deep fact about the phenomenon beyond the
fact that it can be implemented in a laboratory. In particular, we caution
that the use of quasi-causal concepts in operational descriptions of phenomena
like the Bell correlations should not be allowed to obscure the very important
point that the Bell correlations and other nonlocal phenomena do not exhibit
causal order, which as we have seen in this article can be regarded as an
important clue about the right way to understand the Bell correlations and how
they relate to spacetime causal structure. This observation is entirely
consistent with the claim that the Bell correlations can be embedded in a top-
down ‘quantum causal model’ describing the operational procedures needed to
produce them: the point is simply that this kind of operational description
does not necessarily tell us anything about the nature of the Bell
correlations.

Of course, proponents of quantum causal models might be inclined to insist on
functional grounds that there is actually no distinction between a ‘causal
explanation’ and the kind of operational account offered by quantum causal
models. We have no particular objection to this position, but we emphasize
that taking such a stance would have the consequence that a ‘causal
explanation’ for quantum phenomena would no longer confer any particularly
novel insight into those phenomena. For example, what the quantum causal
modelling framework tells us about the Bell correlations is simply what we
already knew, i.e. that these correlations can be produced in the lab by an
agent who prepares a joint entangled state in the causal past of both
measurements. There may be good functional reasons to refer to this
description as a ‘(quantum) causal model’ but the name alone does not confer
on the description any additional explanatory power - for example, those who
previously believed that the Bell correlations must be given a local account
by means of retrocausality or superdeterminism will surely not be swayed from
their view by linguistic or metaphysical arguments concerning what counts as a
‘cause.’ 121212On a different note, in light of the preceding discussion it is
interesting to consider the symmetry properties of quantum causal structures.
Quantum causal models replace causal relations with unitary transformations,
and as noted in ref [33] ‘ _the causal structure of a unitary transformation
is reversible in a strong sense, which goes beyond the obvious fact that
unitaries are reversible transformations: if the causal structure of a unitary
transformation U is represented by drawing arrows from inputs to outputs …
then the causal structure of $U^{-1}$ is given by inverting the arrows. In
case this seems obvious, note that a similar thing is not true for classical
reversible functions._’ So ‘ _any temporal asymmetry … in the framework of
quantum causal models does not arise in the specification of the causal
relations themselves_ ’[33]. Hence the quantum causal modelling framework in
fact contains no intrinsically asymmetric ‘cause-effect relations,’ but
instead instantiates exactly the notion of causal order that we have set out
in this article, which may be regarded as further evidence that the physics of
quantum mechanics instantiates causal order but not strong causation.

###  5.1 Other Nonclassical Causal Theories

In light of the operational character of existing quantum causal models, there
have been proposals that we should seek more radical versions of nonclassical
causal theories to characterise the reality underlying quantum mechanics. For
example, ref [59] sets out to find a way of unscrambling the purely
inferential features of quantum mechanics and the features which ‘ _describe
realities of nature,_ ’ where the latter are assumed to be described by a
‘realist causal theory.’ Here it is argued that classical causal models aren’t
viable because of fine-tuning issues and no-go theorems, and that quantum
causal models are not sufficiently realist, so we need to find some other kind
of nonclassical causal theory to represent the realities of nature. Moreover,
ref [59] places great emphasis on the idea that this representation must still
be sufficiently causal, noting that ‘ _Any attempt to provide a nonclassical
generalization of the notions of causation and inference, however, is highly
constrained insofar as it will need to preserve those features of these
notions which one judges to be essential_ ’ and arguing that the nonclassical
theory should contain ‘ _analogues of most, if not all, of the standard
notions that arise in the framework of classical causal models: common causes,
causal mediaries, d-separation, evaluation of counterfactuals._ ’

However, given the position we have set out here, the justification for
seeking a specifically _causal_ model to represent the realities underlying
quantum mechanics seems somewhat unclear. After all, if one agrees that
‘causality’ in the usual sense arises from causal order plus a choice of
direction, then not only is there no reason to expect a classical causal model
to work in the context of the Bell correlations and other nonlocal quantum
phenomena, there is no obvious reason to expect any other kind of causal
description to work either - the conditions which typically give rise to the
appearance of causal phenomena are absent, and in their absence why should we
expect that the underlying reality should have causal features at all? Thus
this emphasis on explaining quantum mechanics in terms of a ‘nonclassical
theory of causation’ seems too limiting: for example, both classical and
quantum causal models are required to obey a Markov condition, so presumably
any further ‘nonclassical theory of causation’ would also obey such a
condition, but if we are really seeking to understand the reality underlying
quantum mechanics it’s not very clear why we should restrict our attention to
models with this feature, and indeed several authors have argued that
realistic models for quantum mechanics should _not_ obey a Markov
condition[71, 72]. Similar points can be made for other features of causation
which ref [59] would wish to preserve in a nonclassical causal theory; while
it could certainly turn out to be the case that the reality underlying quantum
mechanics does in fact have some of these features, and one might reasonably
decide pragmatically to begin the search for possible models by looking at
models with these features, nonetheless we should not rule out other
possibilities just because they don’t look sufficiently ‘causal’ to our
classical intuition.

Similarly, ref [20] argues that ‘ _in light of Bell experiments, we are forced
to update our prior understanding of the interface between information-
theoretic and spacetime notions of causality and events,_ ’ and thus sets out
to present a framework which allows us to disentangle the information-
theoretic and spacetime notions and reconnect them more carefully. However,
the line of argument pursued in this article suggests that some quantum
phenomena should not be associated with an information-theoretic notion of
causality at all - that is simply the wrong way to describe them.

Another interesting line of research involves extending quantum causal models
to allow for cyclic causal structures[70, 73]. In light of the arguments we
have made here, this seems like a promising approach, since one way to model
symmetric relationships which can’t be decomposed into a strict partial order
would be to write them as cyclic structures in the context of a broader causal
description. However, we would still caution that the expectation that the
underlying structure should be _causal_ (albeit not acyclic) should be treated
with care, since these sorts of phenomena do not involve causal order and
therefore there is no obvious reason to expect them to be best explicable in
the context of a causal formalism, even one that has been generalized in
certain ways. For example, ref [73] continues to maintain that the Bell
correlations should be understood in terms of ‘ _causation without signalling
(i.e., fine-tuned causal influences)_ ’ whereas we have argued here that this
is not the right way to think of the Bell correlations. This does not mean
that research on cyclic causal structures should not be pursued, but it does
mean that we should be prepared to be quite flexible about these kinds of
models: assumptions that they must have certain prototypically causal features
should be carefully examined, and discarded if they are not working well.

###  5.2 Possible defences of the causal approach

In support of the idea that the ‘realities of nature’ should be described by a
(suitably generalised) causal theory, it has been argued that the purpose of
quantum foundations is to provide an interpretation for quantum physics which
retains as many concepts from classical physics as possible, so we should aim
to retain the common-sense notion of causation. For example, ref [33]
motivates its use of causal modelling for quantum mechanics with the
observation, ‘ _Reasoning in causal terms is omnipresent, from fundamental
physics to medicine, social sciences and economics, and in everyday life._ ’
However, before accepting this argument we should first try to understand what
reasons we might have for aiming to retain concepts from classical physics in
our interpretation of quantum mechanics. One possible answer is that we do so
purely on pragmatic grounds, because we are unable to understand
interpretations which don’t retain such concepts. And indeed, it may well be
the case that we are cognitively incapable of grasping concepts which are too
distant from the classical reality to which our brains are adapted, but this
doesn’t really seem like a danger in this instance: the notions of symmetrical
modal relations and causal order may be unfamiliar but they are surely not
incomprehensible. Another possible answer is that we should try to retain
concepts from classical physics in our quantum interpretations in order to
help explain why these concepts appear in the classical limit. But this kind
of argument is valid only if the presence of the relevant concept in the
interpretation of quantum mechanics actually plays some role in explaining the
appearance of the corresponding concept in the classical limit, and that does
not seem to be the case here: as we observed in section 4.1, the existence of
asymmetric relations underlying quantum mechanics would apparently be
completely unrelated to the emergence of macroscopic causality, so the latter
gives us no good reason to believe in the former.131313See ref [74] for a
discussion of a similar error that crops up in discussions of locality in the
interpretation of quantum mechanics

An alternative version of this defence involves the assertion that scientific
realism is in some way intrinsically linked to causal explanation: for
example, ref [59] states that ‘ _In our view, the most constructive way of
defining ‘realities of Nature’ is as causal mechanisms acting on causal
relata._ ’ There is a Kantian element to this line of argument, since it seems
to be proposing that causal concepts are an indispensable feature of any
attempts we might make to formulate an idea of ‘reality.’ Indeed, Nagel has
expressed a more explicitly Kantian version of this view, writing that
causation ‘ _is an analytic consequence of what is commonly meant by
‘theoretical science’ … it is difficult to understand how it would be possible
for modern theoretical science to surrender the general ideal expressed by the
principle without becoming thereby transformed into something incomparably
different from what that enterprise actually is_.’[75] But history has shown
that we should be cautious about assenting to these kinds of transcendental
claims: after all, at one time Euclidean geometry[76] and locality[77] were
also both regarded as indispensable features of our thought, but science has
since dispensed with both in certain regimes, without thereby becoming
unintelligible. So we should not be too quick to assume that causation is an
essential feature of our attempts to describe reality.

And finally it should be recognised that causal models for quantum mechanics
retain the classical notion of ‘causation’ only in a somewhat superficial way.
For if there exists something fundamentally causal underlying quantum
mechanics, it is not really like classical strong causation at all - for
example, if the underlying fundamental causal structure is asymmetric, that
asymmetry cannot be understood in interventionist terms, nor in perspectival
terms, nor by appeal to the thermodynamic gradient, and therefore the causal
relations involved must be inherently asymmetric in a way that we have
apparently not seen in any other natural phenomena. So far from preserving a
classical concept, this approach in fact introduces something radically new,
and therefore it cannot really be justified by appeal to conceptual
conservatism.

##  6 Conclusion

We have seen that the emerging research programme around the process matrix
formalism helps clarify an important philosophical distinction between _causal
order_ and _strong causation_. Unlike strong causation, causal order can be
defined without appeal to any notion of asymmetry, so it is possible for
causal order to be present in fundamental physics even if it is perfectly
time-symmetric, and indeed there is good evidence that causal order is
instantiated by all physically possible singalling processes, or at least all
signalling processes that are possible in the regimes we can currently access.

This observation also suggests an account of the physical origins of strong
causation: once we choose a direction, the bi-order associated with causal
ordering becomes a directed acyclic graph, so causal order can be combined
with a perspectival or interventionist approach in order to explain the
appearance of strong causation at the macroscopic level. Without underlying
causal order, the relationships between events would in general be too chaotic
for any global macroscopic causal structure to appear. However, it must be
emphasized that causal order is not the same as strong causation and it is
important to keep the two notions separate to avoid confusion. Indeed, we
would even be inclined to advocate that some word other than
‘causal/causation’ should be used to describe what we have in this article
referred to as ‘causal order,’ since there are a number of important
disanalogies between causal order and our everyday concept of macroscopic
causation, and this distinction is currently being obscured by the fact that
the word ‘causal’ is used throughout the literature to refer to both of these
notions.

Finally, these results have important consequences for the way we should think
about the foundations of physics. In particular, if we accept the account
offered here for the origins of macroscopic strong causation, it follows that
we should not expect to find natural causal accounts of the Bell correlations
and other no-signalling quantum processes, since such processes are not
subject to any ordering requirements and thus they do not instantiate causal
order. This casts some doubt on the strategies pursued in various causal
modelling approaches to quantum foundations, because it is unclear that we are
justified in assuming that quantum mechanics has an underlying causal
structure.

We finish with the following question: if there can be no causal account of
the Bell correlations, how exactly should one think about the nature of
‘nonlocality’ in quantum physics? A detailed exploration of this topic must be
left for future work, but roughly speaking we would expect that ‘locality’
would emerge in a similar way to ‘strong causation.’ For if we accept that all
signalling processes must instantiate causal order, then since causal order
largely coincides with the lightcone structure of spacetime (indeed, we
suggested in section 3.3 that the lightcone structure might emerge from the
causal structure), it follows that once we choose a direction through the
underlying bi-order, we will find that chains of signalling processes will
necessarily be associated with sequences of variables lying along a timelike
or lightlike path in spacetime. Therefore chains of signalling processes will
appear to be locally mediated from our macroscopic point of view, and yet this
form of ‘locality’ need not be regarded as a fundamental feature of reality,
but simply as an illusion which arises from the need for signalling processes
to exhibit causal order.

##  7 Acknowledgements

Thanks to Pascal Rodriguez Warnier for very helpful comments on a draft of
this article. This publication was made possible through the support of the ID
61466 grant from the John Templeton Foundation, as part of the “The Quantum
Information Structure of Spacetime (QISS)” Project (qiss.fr). The opinions
expressed in this publication are those of the author and do not necessarily
reflect the views of the John Templeton Foundation.

##  Appendix A  Time Reversal

Our description of time reversal has been quite schematic, so let us now be
more precise. In fact the technical details of how to implement an appropriate
reversal operation are non-trivial, because this requires making a clean
separation between the time-symmetric underlying physics and the asymmetries
introduced by assumptions about the role of agents, and these features can be
difficult to distinguish. Currently, it is known that at least some causal
processes are time-reversal invariant in a sense made precise by Pienaar[43].
Given an IS description of a quantum process, Pienaar shows how to write down
the corresponding OS process using symmetric informationally complete (SIC)
quantum instruments, and then proves that if the directed acyclic graph G
associated with the process in the IS description has a certain structure (a
‘layered’ structure) and the process matrix W associated with the IS
description is unbiased (inputting the maximally-mixed-state to the process
produces the maximally-mixed state as output), then the process is causally
reversible, i.e. the OS process is compatible with another IS descrption whose
graph is obtained by reversing the direction of all the arrows in G. That is
to say, for at least some causal processes it is indeed the case that we can
reverse the IS description to obtain an alternative IS description which
corresponds to the same underlying OS process, and moreover this reversed
description will still be causal. It’s unclear whether some result of this
kind could be extended to all causal processes, but we do know that all
physically possible processes can be obtained from ‘pure’ processes by tracing
out an ancilla system[79], and ‘pure’ processes are those which induce a
unitary (and hence reversible) transformation from the past to the future
whenever unitary transformations are also applied in the local laboratories;
so it appears that causal processes can indeed be derived from time-reversal
invariant underlying physics, and therefore it would seem reasonable to
conjecture that all causal processes should be reversible according to an
appropriate formulation of operational time reversal, or at least can be
extended using an ancilla to a process which is reversible in this way.

That said, we emphasise that our argument does not depend on this kind of
conjecture. Our aim here is to address the question of whether there can be
causation in fundamental physics even if it is the case that, as many
physicists currently believe, fundamental physics contains no intrinsic
temporal asymmetries. So the point we want to make is simply that the notion
of ‘causality’ employed in the definition of ‘causal order’ and ‘causal
process’ does not _need_ an underlying asymmetry: processes may instantiate a
well-defined causal order regardless of whether or not they can be reversed to
yield another causally ordered process. Pienaar’s result demonstrates that at
least some causal processes can indeed be reversed in this way, thus verifying
that the definition of ‘causal’ does not depend in any crucial way on
asymmetry.

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