# Why Entanglement?

Huw Price and Ken Wharton University of Bonn and Trinity College, Cambridge,
UK; email<EMAIL_ADDRESS>of Physics and Astronomy, San José State
University, San José, CA 95192-0106, USA; email<EMAIL_ADDRESS>
(28th August 2024)

###### Abstract

In this piece, written for a general audience, we propose a mechanism for
quantum entanglement. The key ingredient is collider bias. In the language of
causal models, a collider is a variable causally influenced by two or more
other variables. Conditioning on a collider typically produces non-causal
correlations between its contributing causes. This phenomenon can produce
associations analogous to Bell correlations, in suitable post-selected
ensembles. Such collider artefacts may become real connections, resembling
causality, if a collider is ‘constrained’ (e.g., by a future boundary
condition). We consider the time-reversed analogues of these points in the
context of retrocausal models of QM. Retrocausality yields a collider at the
source of an EPR-Bell particle pair, and in this case constraint of the
collider is possible by normal methods of experimental preparation. It follows
that connections resembling causality may emerge across such colliders, from
one branch of the experiment to the other. Our hypothesis is that this
constrained retrocausal collider bias is the origin of entanglement. This
piece is based on a suggestion first made in [Price & Wharton 2021], and is an
ancestor of an essay now published online in Aeon magazine [Price & Wharton
2023a]. In an updated version of the argument in [Price & Wharton 2023b] we
(i) demonstrate its application in a real Bell experiment; and (ii) show that
we can do without an explicit postulate of retrocausality.

Keywords: Entanglement, collider bias, retrocausality

##  1 Introduction

Quantum entanglement has been in the news recently. The physicists Alain
Aspect, John Clauser and Anton Zeilinger won the 2022 Nobel Prize for, as the
citation puts it, ‘experiments with entangled photons,establishing the
violation of Bell inequalities and pioneering quantum information science.’

Within the foundations of physics community, the prize has also been welcomed
as a tribute to John Stewart Bell (1928–1990),the Irish physicist who argued
in the 1960s that quantum mechanics implies that the world is unavoidably
‘non-local’ – that Einstein’s ‘spooky action at a distance’ is unavoidable,if
quantum theory is correct. Clauser and Aspect won their Nobel Prizes for
pioneering work in the 1970s and 1980s to test the relevant predictions of
quantum mechanics. Such tests are now called _Bell experiments._ Many
increasingly sophisticated versions, including Zeilinger’s ’loophole-free’
experiments, have confirmed that the world does indeed behave as quantum
theory predicts.

But _why_ does the world behave this way? Bell experiments rely on quantum
entanglement, which was first identified by Erwin Schrödinger in 1935. He
called it ‘not _one_ but rather _the_ characteristic trait of quantum
mechanics.’ [Schrödinger 1935a] More recently, it has been described as ‘the
essential fact of quantum mechanics’ [Susskind & Friedman 2014], and ‘perhaps
its weirdest feature’ [Weinberg 2013]. The work of Clauser, Aspect, Zeilinger
and others confirms the reality of entanglement – as Aspect himself put it, in
his speech at the Nobel Prize banquet, ‘entanglement is confirmed in its
strangest aspects’ – but it doesn’t tell us what it is, or where it comes
from.

Our research [Price & Wharton 2021] suggests a surprisingly simple answer.
Entanglement may rest on a familiar statistical phenomenon known as _collider
bias._ In the light of collider bias, we think, entanglement is not really
mysterious at all. It is what we might have expected, if we’d taken seriously
the time-symmetry of the microworld.

Our proposal needs just three other ingredients, as well as collider bias – as
we said, it is a simple recipe. All these ingredients are available off the
shelf (though admittedly, in one case, from a niche corner of the shelf). But
as far as we know, it has not been noticed that they can be combined in this
way, to throw new light on the central puzzle of the quantum world.

##  2 What is collider bias?

Let’s start with the main ingredient. Collider bias was described in the 1940s
by Joseph Berkson (1899–1982), a Mayo Clinic physicist, physician and
statistician.111Berkson was not the first to notice the point. It dates back
at least to the Cambridge economist A C Pigou [Pigou 1911]. (We are grateful
to Jason Grossman and George Davey Smith here.) Berkson noted an important
source of error in statistical reasoning used in medicine. In some
circumstances, the selection of a sample of patients produces misleading
correlations between their medical conditions. Taken at face value, these
correlations can suggest that one condition prevents another. Berkson pointed
out that these apparent causal connections may not be real. They may be
artefacts of the way the sample has been selected [Berkson 1946].

Simplifying Berkson’s own example, imagine that all the patients admitted to
Ward C have similar symptoms, caused by one of two rare infections, Virus A or
Virus B. Ward C specialises in treating those symptoms, so all its patients
have at least one of these diseases. Some may have both, but everyone on the
ward who doesn’t have Virus A is certain to have Virus B, and vice versa.

Infection with Virus AAdmission to Ward CInfection with Virus B Figure 1: A
simple collider.

Berkson’s point was that this isn’t evidence that avoiding one virus leads to
infection with the other one. The patients on Ward C are a very biased sample.
In the general population, having a vaccine for Virus A won’t make you more
likely to catch Virus B.

This means that if a patient on Ward C with Virus A says to himself, “I’m on
Ward C, so if I hadn’t caught Virus A I would have caught Virus B”, then he’s
making a mistake. If he hadn’t caught Virus A then (most likely) he wouldn’t
have either virus, and he wouldn’t have been admitted to the ward.

This statistical effect is now called Berkson’s bias, or _collider bias._ The
term ‘collider’ comes from causal modelling, the science of inferring causes
from statistical data. Causal modellers use diagrams called Directed Acyclic
Graphs (DAGs), made up of nodes linked by arrows. The nodes represent events
or states of affairs, and the arrows represent causal connections between
those events. When an event has two independent contributing causes, like
being a patient on Ward C, it is shown in a DAG as a node where two arrows
‘collide’ – see Figure 1.

If we just look at a sample of cases in which the event at a collider happens,
we’ll often see a correlation between the two independent causes. It may look
like these causes are influencing one another, but they are not. It is a
selection artefact, as causal modellers say. That’s collider bias. The
correlation stems from the way in which the event at the collider depends on
the two causes – in our simple example it needed one cause or the other.

##  3 Rock-paper-scissors

We want to take collider bias in the direction of physics – ultimately, in the
direction of the kind of Bell experiments for which Clauser, Aspect and
Zeilinger won their Nobel Prizes. We want to propose an explanation for what
may be going on in those experiments, and other cases of quantum entanglement.

We’ll get there via a series of toy examples. For the first of them, imagine
that two physicists Alice and Bob play rock-paper-scissors, sending their
calls to a third observer, Charlie. Charlie makes a list of the results: Alice
wins, Bob wins, or it’s a draw.

Suppose that Charlie likes Alice and dislikes Bob. He throws away most of the
results when Bob wins. In the remaining ‘official’ results Alice wins a lot
more often than Bob. The correlation looks the way it would if Alice actually
had some influence over Bob’s choice – as though Alice choosing scissors makes
it a lot less likely that Bob will choose rock, and so on. If Alice and Bob
are far apart, this could look like spooky action at a distance. But there’s
no real Alice-to-Bob causation involved. It is just collider bias at work. The
event at the collider – whether Charlie retains or throws away the result – is
influenced both by Alice’s choice and by Bob’s choice, giving us the same kind
of converging arrows as in Figure 1.

Suppose that in a particular round of the game Alice chooses paper and Bob
chooses rock. As in the medical case, Alice would be making a mistake if she
says, “If I had chosen scissors instead, Bob would probably not have chosen
rock.” The right thing for her to say is, “If I had chosen scissors, then
Charlie would probably have discarded the result – so my choice didn’t make
any difference to Bob’s choice.”

##  4 Constrained colliders

Now to our second ingredient. It is the least familiar of all, though it, too,
is already on the shelf, if you know where to look. In the game just
described, Charlie could only favour Alice by discarding some results. Let’s
see what happens if we rig the game in Alice’s favour, _without throwing any
results away._

In our world this isn’t going to happen naturally, so for now, let’s imagine
it happening supernaturally. Suppose God also likes Alice more than Bob, so he
tweaks reality to give her an advantage. Perhaps he arranges things so she
never loses when she plays the game on Sundays. How does God do it? It doesn’t
matter for our story, which doesn’t need to be realistic at this point, but
here’s one possibility. In a deterministic universe everything that happens is
determined by the initial conditions at the very beginning of time. If God
gets to choose the initial conditions, and – relying on a Laplacian
calculation or simply Divine foreknowledge – knows exactly what follows from
them, then he can simply choose the initial conditions so that Alice never
loses on Sundays.

Readers who prefer a God-free version could imagine that Alice and Bob live in
a simulation, and that the superintelligence (AGI) that runs the simulation
favours Alice on Sundays. Some serious thinkers have suggested that we
ourselves may live in a simulation, so it would be hasty to say that this
version is inconceivable. But again, our example doesn’t need to be realistic
at this point. Later in our argument, when realism matters, we won’t have to
rely on God or simulations.

To invent some terminology, let’s say that God (or the AGI) _constrains the
collider_ – just on Sundays, in this version of the story. To see what
difference this makes, think again about a round of the game where Alice
chooses paper and Bob chooses rock. Is Alice still making a mistake if she
says, “If I had chosen scissors instead, Bob would not have chosen rock”? It
now depends what day of the week it is. This is still a mistake Monday through
Saturday. On those days, the right thing for Alice to say is, “If I had chosen
scissors, Bob would still have chosen rock (and I would have lost).” But
Sunday is different. On Sunday Alice never loses, so if she had chosen
scissors, Bob could not have chosen rock.

Let’s suppose that Alice knows that the game works this way. Perhaps she
figured it out after years of experiments, and now makes a comfortable living
as a gambler, working one day a week. From her point of view, it looks like
she can _control_ Bob’s choices, to some extent (and only on Sundays). By
choosing scissors she can _prevent_ Bob from choosing rock, and so on.

With a _constrained_ collider, then, we would have something that looks a lot
like real causality across the collider, from one of the pair of incoming
causes to the other. True, it would be a very strange kind of causality. For
one thing, it would work the other way, too, from Bob to Alice (though less
happily, from his point of view). By choosing rock on a Sunday Bob could
prevent Alice from choosing scissors, and so on.

For our purposes, it isn’t going to matter whether this would be real
causality, or even whether the question makes sense, in this case. If we press
too hard on a toy example like this, it is liable to fall apart at the seams.
Could we still speak of both Alice and Bob as making free choices, for
example, if the choices are linked in this way?

All we need from the example is the following lesson. If there were cases in
nature in which something restricted the options at a collider, we should
expect to find a new kind of dependence between the normally independent
causes that feed into that collider. To keep our terminology non-committal on
the question whether it would count as causality, we’ll call this new kind of
relation _connection across a constrained collider_ (CCC).

CCC is our second ingredient, and certainly the least familiar one, for most
readers. But it, too, is already on the shelf, in the sense that there’s at
least one place in physics where it has actually been proposed. It is the key
to a suggestion by Maldacena and Horowitz [Horowitz & Maldacena 2004] for
solving the so-called black hole information paradox (made famous by Stephen
Hawking’s bet with Kip Thorne and John Preskill).

The Maldacena-Horowitz hypothesis relies on the proposal that special ‘future
boundary conditions’ inside black holes constrain a collider (in our
terminology) at that point. Maldacena and Horowitz suggest that this creates a
zigzag causal path through time, along which information can escape from a
black hole.

Discussing the Maldacena-Horowitz hypothesis recently, the Cambridge physicist
Malcolm Perry says

> [t]he interior of the black hole is therefore a strange place where one’s
> classical notions of causality … are violated. This does not matter as long
> as outside the black hole such pathologies do not bother us. [Perry 2021, 9]

As we’ll explain, our proposal is going to be that such pathologies are
actually extremely common, if you know where to look. In the other direction
of time, they are the basis of quantum entanglement – and they don’t need
black holes.

So far, then, we have two ingredients on the table: collider bias itself, and
CCC – connection across a constrained collider. Before we introduce the two
remaining ingredients, let’s get a bit closer to the physics of the quantum
world.

##  5 From rock-paper-scissors to Bell experiments

As we noted at the beginning, the recent Nobel Prizes were awarded for Bell
experiments. These confirmed the strange correlations, predicted by quantum
theory, which Bell took to show that the quantum world is unavoidably non-
local. Given that these so-called Bell correlations were important enough to
win Nobel Prizes, readers may be surprised to learn that they can easily be
reproduced in a version of our rock-paper-scissors game. The only change we
need is to have Alice and Bob each flip a coin before they make their choice.

In this variant – call it _quantum_ rock-paper-scissors (QRPS) – Alice and Bob
each send two pieces of information to Charlie: their choice of rock-paper-
scissors, and the result of their coin flip. So Charlie gets four values, two
choices and two coin outcomes. This is precisely the same amount of
information generated in each run of a Bell experiment. In that context, these
values are called the two _measurement settings_ and the two _measurement
outcomes._ The Bell correlations are particular relationships between these
four values, in long lists of experimental results. (In one kind of Bell
experiment, for example, they specify among other things that whenever the two
settings are the same, the two outcomes must be different.)

In QRPS it is very easy for Charlie to set up a filter, keeping some results
and throwing away others, to make sure that the set of results he keeps
satisfies the Bell correlations. By using the right filter, Charlie can ensure
that the selected results look _exactly_ like the data generated in the
familiar kind of Bell experiment of Clauser, Aspect and Zeilinger.222See
[Bacciagaluppi & Hermens 2021] for the same point in the real QM context.

Of course, this doesn’t mean that there is any strange non-locality in QRPS.
As in the earlier version, the correlations are a selection artefact, a result
of collider bias. But since the results are now a perfect copy of real Bell
experiments, it is worth paying careful attention to differences between the
two.

The most obvious difference is that quantum rock-paper-scissors and real Bell
experiments look like time-reversed versions of each other. In QRPS Alice and
Bob send their choices _to_ Charlie, later in time. In a spacetime diagram
with time running up the vertical axis, the structure looks like an upside-
down ‘V’ – see the left hand side of Figure 2. We’ll say that cases like this
are ‘$\wedge$-shaped’. In real Bell experiments, Alice and Bob receive their
particles _fro_ m the source, which is earlier in time. So the structure looks
like $\lor$, as in the right hand side of Figure 2 – we’ll say that they are
‘$\lor$-shaped’.

In a moment we’re going to introduce a $\lor$-shaped version of QRPS, to
eliminate this difference. But first let’s summarise what we learned from the
$\wedge$-shaped case. We saw that it is easy for Charlie to set up a filter,
to make sure that the results he keeps satisfy the Bell correlations. But we
don’t need any non-locality between Alice and Bob, to explain what’s going on.
The correlations are simply collider bias. As in the original rock-paper-
scissors case (without constraint at the collider) they are simply a selection
artefact.

AliceCharlieBobAliceCharlieBobTimeSpace Figure 2: $\wedge$-shaped and
$\lor$-shaped experiments.

We could reintroduce God or an AGI at this point, to add a constrained
collider to QRPS. There would be one interesting difference from the original
game. In that case, we saw that the effect of the constraint was to give Alice
and Bob control over each other’s choices, making it hard to maintain that
they both had freedom to choose. In QRPS, as in the analogous real Bell
experiments, that problem goes away: Alice and Bob each get some influence
over the result of the other’s coin toss, but we can still treat both of their
own choices as completely free.

That difference aside, a constrained version of QRPS would be as unrealistic
as for the original game, and wouldn’t tell us anything new. So let’s turn
instead to a $\lor$-shaped version of QRPS, where realistic constrained
colliders will be much easier to find.

##  6 Flipping the game

As we said, QRPS looks like a time-reversed version of a Bell experiment,
$\wedge$-shaped rather than $\lor$-shaped. Instead of two particles _leaving_
a common source and _going to_ separate observers, it’s the other way round.
The information travels _from_ Alice and Bob, _to_ Charlie at the common point
in the future.

Can we flip this quantum rock-paper-scissors, to make it $\lor$-shaped not
$\wedge$-shaped? It might look easy. We can have Charlie toss the two coins
and send them to Alice and Bob, so that the results (heads or tails) become
Alice and Bob’s measurement outcomes. But if that’s all we do, Charlie won’t
know what choices Alice and Bob are going to make when he sends out the coins.
That means there’s no way for him to put bias into the results, in the way
that he could in the $\wedge$-shaped case. There’s no way that Charlie can
produce the Bell correlations, in other words.

But suppose we let Charlie _know in advance_ what choices Alice and Bob are
going to make – we give him a crystal ball, say. Then it is very easy for him
to manage the coins so that the net results, gathered over many plays of the
game, satisfy the Bell correlations. The trick is for Charlie to toss one
coin, and then choose the result for the other coin based on a rule that takes
into account Alice and Bob’s future choices.

So this game, too, generates the same kind of Bell correlations as the famous
experiments of Clauser, Aspect, and Zeilinger. Let’s ask the same question we
did about $\wedge$-shaped QRPS. Does the new $\lor$-shaped version involve
some kind of spooky action at a distance from Alice to Bob, and vice versa?

Alice’s choiceCharlie’s selection of outcomesBob’s choice Figure 3: A past
collider.

We hope that readers will be inclined to say ‘No’ to this question. After all,
the basic causal structure of the new $\lor$-shaped version is something like
Figure 3. Thanks to Charlie’s crystal ball, and the rule he uses for selecting
outcomes, Alice’s and Bob’s choices both influence what outcomes Charlie
produces, in every case. This means that Charlie’s selection procedure is a
collider, and we have to be on our guard for collider bias.

For this reason, attentive readers might suspect that collider bias plays the
same role in explaining the results the new $\lor$-shaped QRPS as it did in
the $\wedge$-shaped case. But there’s one very big difference between these
two cases – which brings us to our third ingredient.

##  7 Initial control

In the $\wedge$-shaped version of QRPS, Charlie had to apply a filter, and
throw away results he didn’t want. But in the $\lor$-shaped case, he gets to
_choose_ the results, in the light of what he learns from the crystal ball and
his (single) coin toss. _He doesn’t have to throw anything away._ In this
case, then, Charlie himself can constrain the collider. All he needs is an
ordinary ability we take for granted, to control the initial conditions of an
experiment.

Perhaps we shouldn’t take this for granted. It is actually a remarkable
ability, one that depends on the fact that we live in a place in which
abundant low-entropy energy can be harnessed by creatures like us. But by
ordinary standards, there’s nothing surprising about it. We have much more
control over the _initial_ conditions of experiments than over their _final_
conditions. It’s easy to arrange the balls on a pool table into precise
positions _before_ the initial break, but virtually impossible to play the
game so that they all _end up_ in those positions.

Let’s call this familiar fact _initial control._ It is the third ingredient in
our recipe. Looking ahead a bit (so to speak), the final ingredient is going
to be something that does the job of the crystal balls, in giving Charlie
access to information about future choices by Alice and Bob. We’re going to
find that on the shelf already in the quantum world, under the name
_retrocausality._ But before we go there, let’s summarise what we learn from
the new $\lor$-shaped QRPS.

Thanks to the crystal balls, we have the causal structure shown in Figure 3,
with a collider in the past, where Charlie chooses the outputs. Thanks to
initial control, it is easy to make it a constrained collider, so that Charlie
doesn’t need to throw any results away. And this means that there can be
connection across the constrained collider (CCC), from Alice to Bob, and vice
versa.

In other words, the combination of the collider structure in Figure 3 and the
constraint provided by initial control gives us CCC. If we are happy to use
causal language, we can say that it gives us the kind of zigzag causal
connections shown in Figure 4, from Alice’s choice to Bob’s outcome and Bob’s
choice to Alice’s outcome.333Two notes. First, there is disagreement about
whether the term ‘causation’ is appropriate in real Bell experiments, for the
connection between setting choices on one side and outcomes on the other. For
present purposes, we set aside that issue, though reasons for declining to use
the term in real cases are likely to apply here, too, since the correlations
are the same by definition. Second, we also set aside the question as to
whether Alice and Bob should be said to influence their own outcomes; that is
possible in real Bell experiments, of course, but doesn’t provide a way to
evade the argument for influence from each side to the other. (Why do we call
it a Parisian Zigzag? More on that in a moment.)

Alice’s choiceAlice’s outcomeCharlie’s selection of outcomesBob’s choiceBob’s
outcome Figure 4: The Parisian Zigzag.

In the light of this, let’s go back to this question: does $\lor$-shaped QRPS
involve some kind of spooky action at a distance from Alice to Bob, and vice
versa? At this point we need to be careful about what we mean by action at a
distance. As we have just seen, there is indeed some influence, or connection,
from Alice to Bob, and vice versa. Since they are at a distance from each
other, and a direct connection might need to be faster than light, we might
still want to call it action at a distance, or nonlocality. (One of the recent
Nobel laureates once told us that he thought such a zigzag should still count
as a nonlocal effect.)

However, the connection between Alice and Bob is indirect, and depends
entirely on processes which don’t themselves require anything faster than
light. So whatever we call it, it doesn’t have the relativity-challenging
character normally associated with so-called spooky action at a distance in
QM. And it is not very mysterious: we know exactly what it is, namely,
connection across a constrained collider.

The crystal balls were mysterious, of course, but once we gave ourselves
those, the explanation of the connection between Alice and Bob is
straightforward. Imagine if something like this could explain the results of
real Bell experiments – that would be a nail in the coffin of the quantum
spooks!

##  8 Retrocausality

Well, let’s see. We need our final ingredient. In $\lor$-shaped QRPS, we gave
Charlie a crystal ball, to allow causation to work backwards – in other words,
to allow Alice and Bob’s choices to feed into the algorithm Charlie uses to
select the measurement outcomes. In the real world, of course, we don’t find
magical crystal balls on any actual shelf.

In the quantum world, however, retrocausality is a familiar hypothesis. In
that sense, it is certainly available off the shelf. It was first proposed in
the late 1940s by the Parisian physicist Olivier Costa de Beauregard [Costa de
Beauregard 1953], who suggested that in the quantum world causal influence
might follow a zigzag path, as in Figure 4 – that’s why we called it the
Parisian Zigzag.

Retrocausality remained a niche idea for many years, though it has long had
some distinguished proponents. In the 1950s one of them, at least briefly, was
the British physicist Dennis Sciama [Sciama 1958], who taught an astonishing
generation of physicists, including Stephen Hawking. Sir Roger Penrose,
himself a recent Nobel laureate, has long been sympathetic to the idea.
There’s a story from the 1990s of Penrose drawing a zigzag at a quantum
workshop at the Royal Society in London, and joking ‘I can get away with
proposing this kind of thing, because I’m already a Fellow here.’ (Now that he
has a Nobel Prize it is even easier, presumably!)444For a recent version of
Penrose’s views on this topic, see [Penrose 2023]. Another famous long-term
proponent is the Israeli physicist Yakir Aharonov, whose interest dates from a
well-known 1964 paper [Aharonov, Bergmann & Lebowitz 1964].

More recently, we ourselves have written in _Aeon_ and elsewhere about the
advantages of retrocausal approaches to QM, both in avoiding action at a
distance, and in respecting time-symmetry. We argued that quantum theory
provides new reasons to think that microscopic time-symmetry requires
retrocausality [Price & Wharton 2016, Wharton & Price 2016].

Recent popular discussions of the retrocausal approach to QM may be found
here, here and here, for example.555See [Merali 2022, Becker 2018, Zyga 2017].
For specialist audiences, recent discussions include the following: [Price &
Wharton 2015, Leifer & Pusey (2017), Leifer (2017), Friedrich & Evans 2019,
Wharton & Argaman 2020, Norsen & Price 2021, Adlam 2022]. It is now a
sufficiently familiar proposal that no adequate survey of the puzzles of
quantum theory can afford to ignore it. Some commentators on the recent Nobel
Prize announcement got themselves into trouble for doing so [Hance &
Hossenfelder 2022].

So retrocausality in QM is a well-known idea, and has well-known points in its
favour. However, an additional striking advantage seems to have been
overlooked. Retrocausality suggests a simple mechanism for ‘ _the_
characteristic trait of quantum mechanics’ (Schrödinger), ‘its weirdest
feature’ (Weinberg) – in other words, for the strange connections between
separated systems called quantum entanglement.

Starting with retrocausality, our proposal goes like this, in four easy steps.
We’ve highlighted the use of our four ingredients.

  1. 1.

Retrocausality automatically introduces colliders into Bell experiments, at
the point where the two particles are produced.

  2. 2.

That’s interesting because colliders produce collider bias and causal
artefacts – correlations that look like they involve causation, but really
don’t.

  3. 3.

But constraining a collider can turn a causal artefact into a real connection
across the collider.

  4. 4.

In the case of colliders in the past, as in Figure 3, constraint is easy. It
just follows from normal initial control of experiments.

Taken together, these steps suggest a simple explanation for the Parisian
Zigzag, and the strange kind of non-local connections in the quantum world,
revealed by Bell’s arguments. It is _connection across constrained colliders,_
where the colliders result from retrocausality and the constraints from
ordinary initial control of experimental setups.

Of course, more work is needed to show that this simple mechanism can actually
explain quantum entanglement. We don’t mean to claim that it is a trivial step
from $\lor$-shaped QRPS to real Bell experiments. What we do claim, and what
we take to be demonstrated by $\lor$-shaped QRPS, is that the combination of
retrocausality and initial control can give rise to a connection between
separated systems that looks very similar to entanglement. In our view, this
is such a striking fact – and entanglement is otherwise such a strange and
mysterious beast – that we propose the following hypothesis:

> Hypothesis (E=CCC): Quantum entanglement is retrocausal collider bias,
> constrained by initial control.

If this hypothesis turns out to be true, then in place of spooky action at a
distance, we’ll get Costa de Beauregard’s zigzag connections – which, as he
always emphasised, are much easier to reconcile with relativity – but now
explained as CCC. It will still be true that quantum theory gives us a new
kind of connection between the properties of distant systems. The experiments
of Clauser, Aspect and Zeilinger provide very convincing evidence that quantum
entanglement is a real phenomenon. But it would no longer look mysterious. On
the contrary, any world that combines retrocausality and initial control would
be expected to look like this.

In earlier work, as we noted above, we and others have argued that time-
symmetry requires retrocausality, once the world becomes quantised [Price &
Wharton 2016, Zyga 2017].666See also [Price 2012, Price & Wharton 2015, Leifer
& Pusey (2017), Leifer (2017)]. The same line of argument now seems to lead to
entanglement, once initial control is added to the picture. That’s what we
meant at the beginning, when we said that entanglement is what we might have
expected, if we’d taken seriously the time-symmetry of the microworld.777In
our most recent version of this proposal [Price & Wharton 2023b], we argue
that at least for an operational version of the argument, the retrocausality
does not need to be an independent postulate. In so far as it is needed, it
comes for free when we consider an idealised regime without initial control.

##  9 Avoiding causal loops and signalling

Finally, a note for readers who are worried that the cure is worse than the
disease – that retrocausality opens the door to a menagerie of paradoxes and
problems. The note says: Well spotted! As we described $\lor$-shaped QRPS,
with the crystal balls, you are absolutely right. For one thing, the crystal
balls give Charlie options much like those of the famous time-traveller,
encountering his own grandfather long before his parents met. What’s to stop
Charlie interfering with the course of history, say by bribing Bob to make a
different choice than the one shown in the crystal ball? (In the causal loop
literature, this is called ‘bilking’.)

Also – less dramatic, maybe, but especially interesting in comparison to QM –
the crystal balls allow Alice and Bob to send messages to Charlie, and hence
potentially, with his help, to signal to each other. This isn’t possible in
real Bell experiments, where Alice and Bob can’t signal to each other, despite
having some influence on each other’s measurement outcomes. So isn’t this bad
news for retrocausality?

These are good objections, but it is easy to modify the $\lor$-shaped QRPS
game to avoid them. What we need to do is to split Charlie’s functions into
two parts. Most of what he does gets replaced by a simple algorithm, inside a
black box, that takes in information about the two future measurement
settings, and spits out the two measurement outcomes. Charlie himself can’t
see inside the black box, and doesn’t have access to the future settings. But
he still has a vital job to do. The box has a knob on the front, with a small
number of options. Charlie controls that knob, and if he wants the device to
produce the Bell correlations, he needs to choose the right option. In the
terminology of QM, that’s called ‘preparing the initial state’. If that’s all
that Charlie does, and the quantum black box takes care of the rest, the door
to the menagerie is closed. Alice and Bob can no longer signal to Charlie, or
to each other. Everything works as in orthdox QM, except that we now have the
prospect of an explanation of entanglement.

This means that if nature wants retrocausality without retrosignalling (and
without the paradoxes that retrosignalling would lead to), it is going to need
black boxes – places in nature where observers like Charlie can’t see the
whole story. In normal circumstances, such black boxes would seem like another
kind of magic. Charlie is a clever guy, after all. What’s to stop him taking a
peek inside any kind of box?

But in the quantum case, many readers will already know the answer to this
question. What’s to stop Charlie taking a peek is Werner Heisenberg, or more
precisely his famous Uncertainty Principle. Ever since Heisenberg, quantum
theory has been built on the idea that there are new limits to what it is
possible to know about physical reality. One of the central questions is
whether this is just a restriction on our knowledge of reality, or whether
reality itself is somehow fuzzy. As Schrödinger put it in 1935, after
describing his famous Cat Experiment: ‘There is a difference between a shaky
or out-of-focus photograph and a snapshot of clouds and fog banks.’
[Schrödinger 1935b]

The Cat Experiment was supposed to support the out-of-focus photograph option,
the view that the Uncertainty Principle is just a restriction on our
_knowledge_ of reality. Schrödinger thought it was obvious that the cat
couldn’t actually be somehow neither alive nor dead. Like Einstein,
Schrödinger favoured the view that the quantum description is incomplete, and
that reality contains further details, hidden behind Heisenberg’s veil.

In the decades since 1935, most physicists who care about these issues have
concluded that Einstein and Schrödinger were wrong. Bell’s Theorem, together
with the quantum predictions being confirmed by Clauser, Aspect, Zeilinger and
many others, has often been interpreted as showing that the spooky action at a
distance which Einstein hoped to avoid with additional ‘hidden variables’, is
an inevitable part of the quantum world.

Retrocausality is already the most interesting challenge to that view. By
taking the first option on Schrödinger’s list – by treating QM as an
unavoidably fuzzy picture of a sharper reality – it can allow the kind of
quantum black boxes needed to avoid retrosignalling and paradoxes. How
satisfying, then, if it also explains the other thing that Schrödinger put his
finger on in 1935, when he invented the term ‘entanglement’, and called it ‘
_the_ characteristic trait of quantum mechanics.’888We are grateful to Emily
Adlam, Guido Bacciagaluppi and Nathan Argaman for comments on previous
versions. HP is grateful to audiences in Toronto, Bonn and Dortmund.

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