0.8pt

# On safe post-selection for Bell tests with ideal detectors:
Causal diagram approach

Pawel Blasiak  Institute of Nuclear Physics Polish Academy of Sciences,
PL-31342 Kraków, Poland<EMAIL_ADDRESS>Ewa Borsuk  Institute of
Nuclear Physics Polish Academy of Sciences, PL-31342 Kraków, Poland
<EMAIL_ADDRESS>Marcin Markiewicz  International Centre for Theory of
Quantum Technologies, University of Gdańsk, PL-80308 Gdańsk, Poland
<EMAIL_ADDRESS>
###### Abstract

Reasoning about Bell nonlocality from the correlations observed in post-
selected data is always a matter of concern. This is because conditioning on
the outcomes is a source of non-causal correlations, known as a selection
bias, rising doubts whether the conclusion concerns the actual causal process
or maybe it is just an effect of processing the data. Yet, even in the
idealised case without detection inefficiencies, post-selection is an integral
part of experimental designs, not least because it is a part of the
entanglement generation process itself. In this paper we discuss a broad class
of scenarios with post-selection on multiple spatially distributed outcomes. A
simple criterion is worked out, called the all-but-one principle, showing when
the conclusions about nonlocality from breaking Bell inequalities with post-
selected data remain in force. Generality of this result, attained by adopting
the high-level diagrammatic tools of causal inference, provides safe grounds
for systematic reasoning based on the standard form of multipartite Bell
inequalities in a wide array of entanglement generation schemes, without
worrying about the dangers of selection bias. In particular, it can be applied
to post-selection defined by single-particle events in each detection chanel
when the number of particles in the system is conserved.

##  1 Introduction

The study of experimental correlations provides a window into the underlying
causal mechanisms, even when their exact nature remains obscured. In his
seminal works [1], John Bell showed that seemingly innocuous assumptions about
the causal structure of realistic models leave a mark on the observed
statistics. The conclusion has been that the violation of certain inequalities
is incompatible with the assumption of locality and free choice (or
measurement independence). Surprisingly, such violations can systematically
occur in quantum theory, potentially undermining our dearly held assumptions
about how nature works. Given how troubling this conclusion might be, it is
hardly surprising how thoroughly Bell’s result has been scrutinised in the
last few decades, both theoretically [2, 3, 4] and experimentally [5]. Its
importance is acknowledged by the term ’Bell nonlocality’ which refers to
experimental situations demonstrating inconsistency of the observed
correlations with causal (or realist) explanations maintaining both
assumptions at the same time. It is believed to be a source of quantum
advantage in the communication [6] and information tasks [7].

A simplified picture of a Bell experiment consists of a series of measurements
made by space-time separated parties on systems prepared in some entangled
state. However, it comes with a challenge as to the straight conclusion
regarding Bell nonlocality when it comes to the analysis of real experimental
designs. An important issue concerns the presence of post-selection in the
data collection process. One source of the problem lies with the measurement
part of the experiment in which some of the events are missed out due to the
inefficiencies of real detectors. It is called the detection loophole [8, 9]
and will not be addressed in this paper. In the following we shall assume
ideal detectors and thus focus on post-selection due to the preparation part
of the experiment.

It is often the case that post-selection in the experiment is due to the
specifics of the entanglement generation process itself. Typically this boils
down to the occurrence of a certain pattern in the outcomes deemed to be
interesting for the purpose at hand, i.e., exhibiting entanglement [10, 11,
12]. Some popular techniques of this sort include: heralding by some other
event (cf., event-ready detection [13]), time-bin entanglement [14] or
selecting single-particle detections in each experimental channel (cf., recent
proposals in Refs. [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30] or a variety of integrated photonic implementations [12]). So, the
generic structure of events is richer than that required for the intended Bell
inequality, and post-selection aims at retaining only those experimental
trials, based on some well-defined criterion, which are potentially
interesting for the violation of the desired inequality. This poses an issue
regarding the legitimacy of the conclusion about Bell nonlocality in such
scenarios, since conditioning is often a source of non-causal correlations. In
the field of causal inference the problem is known as a selection bias or
Berkson’s paradox [31, 32, 33, 34]. The difficulty being that, in the presence
of post-selection, it might be conditioning that leads to correlations
breaking Bell inequalities without necessarily claiming Bell nonlocality. How
critical it is for the analysis of Bell experiments may attest the effort to
close the detection loophole [9], which exploits post-selection due to
detector inefficiencies. Here we will assume ideal detectors and focus only on
post-selection due to entanglement generation.

It is interesting to ask about general conditions

when post-selection, due to entanglement generation, does not compromise the
conclusion of nonlocality from the violation of some given Bell inequality.

So far, this problem has been discussed only for some particular scenarios for
two and three parties, and the analysis typically involved the entire pattern
of experimental outcomes present in a given experiment [35, 36, 37]. In some
cases, those issues can be overcome by certain modifications in the
experimental arrangements (like for time-bin entanglement in Refs. [38, 39,
40, 41]). However, apart from those particular cases no attempt has been made
at a general analysis of Bell nonlocality in the presence of post-selection
due to the specifics of entanglement generation process. We note that
comprehensiveness of such an analysis would require a discussion of both
assumptions, locality and free choice, underlying the derivation of Bell
inequalities.

In this paper we give a general criterion for any multipartite scenario,
called the all-but-one principle, that can be expressed by the following
simple intuition:

if post-selection can be resolved with one party excluded, then it is safe for
Bell nonlocality arguments.

Meaning that, in such a case, the reasoning based on the standard Bell
inequalities is justified despite the issues of post-selection. Crucially, the
generality of the result owes to the high-level diagrammatic tools of causal
inference honed by Judea Pearl [31, 32, 33, 34]. We give a full poof of this
criterion preceded with a brief discussion of the selection bias and Bell
nonlocality under post-selection.

##  2 Selection bias and d-separation rules

Post-selection is a procedure of rejecting some of the data from the analysis
of an experiment. Technically, it boils down to estimating experimental
probabilities subject to some additional condition which depends on the
outcomes. It is crucial to make a warning that post-selection is not a
harmless procedure, since it is often a source of additional correlations in
the retained data. This is potentially dangerous for the task of
identification of causal relationships between the variables from the observed
correlations. In the field of causal inference the problem is known as a
selection bias or Berkson’s paradox.

Let us illustrate the problem with a simple example due to Elwert & Winship
[34]. Consider three features of Hollywood actors who could be beautiful $B$,
talented $T$, and some of them make it to be celebrities $C$. We may
reasonably expect that beauty $B$ and talent $T$ contribute to an actor being
considered a celebrity $C$ (pushed to the extreme, imagine that one of these
features is enough to become a celebrity), but in general population beauty
$B$ and talent $T$ are completely unrelated to one another. Suppose that this
is the whole story and hence Fig. 1 (on the left) illustrates the causal
diagram behind the data. Now, if we focus on the subpopulation of those actors
who made it to the status of celebrities $C$, then correlation between beauty
$B$ and talent $T$ appears (despite the fact that they were independent to
begin with). See Fig. 1 (on the right). Clearly, seeing an unattractive
celebrity makes it more likely that the person is a talented actor. And vice
versa, celebrities who are bad actors are more often found to be good looking.
(Note that pushed to the extreme, this inverse relation may even become a
certain conclusion). These correlations are non-causal, i.e., they arise
merely due to conditioning or restricting the data generated by the causal
diagram in Fig. 1 (on the left). This example illustrates the warning against
careless attribution of causal origin to correlations in the post-selected
data.

Figure 1: Selection bias. Consider three variables $B$ (beauty), $T$ (talent)
and $C$ (celebrity) modelled by a causal diagram on the left. It follows that
the variables $B$ and $T$ are independent, since the only path joining the
variables is blocked by a collider $C$ (Rule 1). However, on the right,
conditioning on the collider $C$, depicted by a red box, unblocks the path
between $B$ and $T$ making the variables likely dependent (Rule 3).

Interestingly, the pattern of independencies between the variables can be
deduced from the structure of the causal diagram itself. It has been shown to
boil down to the so called d-separation criterion, see [31, 32, 34, 33]. In a
nutshell, the idea consists of inspecting all paths in the causal diagram
connecting two variables of interest:

if all those paths are blocked then the variables are necessarily independent
(otherwise the variables are likely dependent).

The concept of blocking a path is defined by the following three simple
$d$-separation rules (see Fig. 2 for illustration):

######  Rule 1.

A path is blocked if there is a collider along the way, that is a node with
pair of arrows on the path that collide head-to-head.

######  Rule 2.

Conditioning on a non-collider blocks the path (where non-collider is a node
along the way with pair of arrows meeting head-to-tail or tail-to-tail).

######  Rule 3.

Conditioning on a collider (or its descendant) removes the block from Rule 1.

In the following analysis, those rules will provide systematic insight into
the pattern of conditional independences arising from specific post-selection
procedures.

Figure 2: Illustration of the d-separation rules. In the causal diagram nodes
$X$ and $Y$ are connected via two paths $X\rightarrow B\leftarrow Y$ and
$X\rightarrow A\rightarrow Y$. The path $X\rightarrow B\leftarrow Y$ is
blocked since $B$ is a collider by Rule 1 and conditioning on $B$ or $C$
unblocks the path by Rule 3. Whereas path $X\rightarrow A\rightarrow Y$ is
unblocked since $A$ is a non-collider and conditioning on $A$ blocks the path
by Rule 2. If both paths are blocked then $X$ and $Y$ are said to be
d-separated and then the variables are independent. In this example it happens
only in the case of conditioning on $A$ which is depicted on the right by a
red box.

##  3 Bell nonlocality and three causal assumptions

In the following, we consider the usual Bell-type scenario with several
parties $A,B,C,...$ conducting experiments in space-time separated regions.
The whole experiment consists of a series of trials in which each party
chooses a setting $x,y,z,...$ and makes a measurement registering the outcome
$a,b,c,...$ . For further convenience, let us denote the set of possible
outcomes by $\mathcal{O}$, i.e., we have respectively
$\mathcal{O}_{\scriptscriptstyle A}$, $\mathcal{O}_{\scriptscriptstyle B}$,
$\mathcal{O}_{\scriptscriptstyle C}$, … . Then, after many repetitions, the
parties compare their results calculating the statistics given by the set of
distributions $P_{\scriptscriptstyle abc...|xyz...}$ which describe the
probability of observing outcomes $a,b,c,...$ given measurements $x,y,z,...$
were made. For conciseness, following the terminology in Ref. [4], we call
such obtained set of probabilities $\mathcal{P}\equiv\\{P_{\scriptscriptstyle
abc...|xyz...}\\}_{\scriptscriptstyle xyz...}$ a "behaviour". Note that all
probabilities in the behaviour $\mathcal{P}$ are supposed to be calculated
without rejecting any trial from the experiment (no post-selection is made).

Figure 3: Causal structure in a Bell experiment. One the left, the graph
$\mathcal{G}$ describes causal relations between variables $a,b,c,...$
(measurement outcomes), $x,y,z,...$ (choice of settings), and some hidden
variable $\lambda$. It generates the behaviour
$\mathcal{P}\equiv\\{P_{\scriptscriptstyle
abc...|xyz...}\\}_{\scriptscriptstyle xyz...}$. On the right, the graph
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ incorporates post-selection into
the experiment by conditioning (red box) on the additional outcome-dependent
variable $\mathcal{K}(a,b,c,...)$. The latter introduces potential selection
bias into such obtained behaviour
$\mathcal{P}_{\scriptscriptstyle\mathcal{K}}\equiv\\{P_{\scriptscriptstyle
abc...|xyz...\,\mathcal{K}}\\}_{\scriptscriptstyle xyz...}$.

Bell inequalities are algebraic constraints in the form

$\displaystyle\mathcal{I}(\mathcal{P})$ $\displaystyle\equiv$
$\displaystyle\sum_{\scriptscriptstyle abc...\atop
xyz...}\,s_{\scriptscriptstyle xyz...}^{\scriptscriptstyle
abc...}\,P_{\scriptscriptstyle abc...|xyz...}\ \leqslant\
I_{\scriptscriptstyle L}\,,$ (1)

where $s_{\scriptscriptstyle xyz...}^{\scriptscriptstyle abc...}$ and
$I_{\scriptscriptstyle L}$ are some numbers. These inequalities are derived
under three assumptions called realism, locality and free choice. The realism
assumption posits that the observed correlations can be explained by the
causal influence between the variables relevant for the experiment, that is
measurement outcomes $a,b,c,...$ and settings $x,y,z,...$, as well as some
unobserved (hidden) variables collectively denoted by $\lambda$. Thus, by
conditioning on a priori unknown $\lambda$, we can always write [1, 2, 3, 4]

$\displaystyle P_{\scriptscriptstyle abc...|xyz...}$ $\displaystyle=$
$\displaystyle\sum_{\scriptscriptstyle\lambda}\ P_{\scriptscriptstyle
abc...|xyz...\,\lambda}\cdot P_{\scriptscriptstyle\lambda|xyz...}\,.$ (2)

Then, by invoking spatio-temporal structure of the experiment certain
conditional independencies between the variables can be justified. First, the
variables in different space locations cannot affect each other and the causal
influences propagate respecting temporal order of events. Second, the hidden
variable $\lambda$ is identified to be in the common past of variables
representing the outcomes $a,b,c,...$ , but not the variables representing
choice of the settings $x,y,z,...$ and hence the latter cannot be affected by
$\lambda$. The ensuing causal structure of the variables modelling the
experiment is depicted in the causal graph $\mathcal{G}$ in Fig. 3 (on the
left). This readily translates into conditional independencies in the
statistics generated by those causal models.111Note that in this argument
causal relationships are considered as prior to the statistical relations,
with the latter derivable from an appropriate structural causal model
compatible with a given causal structure. This is the leitmotif of the causal
inference field [31, 32, 33, 34]. They are referred to as the locality
assumption

$\displaystyle P_{\scriptscriptstyle abc...|xyz...\,\lambda}$ $\displaystyle=$
$\displaystyle P_{\scriptscriptstyle a|x\lambda}\cdot P_{\scriptscriptstyle
b|y\lambda}\cdot P_{\scriptscriptstyle c|z\lambda}\cdot...\,,$ (3)

and the free choice assumption (also called the measurement independence
assumption)

$\displaystyle P_{\scriptscriptstyle\lambda|xyz...}$ $\displaystyle=$
$\displaystyle P_{\scriptscriptstyle\lambda}\,.$ (4)

Within the causal model framework, these relations are a straightforward
application of the $d$-separation rules to the diagram in Fig. 3 (on the
left). [Eq. (3) follows by iterative use of Rule 2 given conditioning on non-
collider node $\lambda$, and Eq. (4) is an application of Rule 1 to colliders
$a,b,c,...$ ; cf. proof of Theorem 1.]

To summarise, each Bell inequality Eq. (1) is a simple algebraic consequence
of the three assumptions in Eqs. (2)-(4). It means that the violation of some
Bell inequality entails the impossibility of explaining the observed behaviour
$\mathcal{P}$ in a causal model maintaining both locality and free choice at
the same time. The essence of Bell’s theorem is to point out situations in
which quantum theory predicts violation of those inequalities [1, 2, 3, 4].

##  4 Post-selection issues

Crucially, the statistics used for estimation of probabilities in the
behaviour $\mathcal{P}$ should include every experimental trial for a valid
conclusion from breaking Bell inequality in Eq. (1) to be drawn. In practice,
however, some sort of post-selection is always made. Let us formalise this
concept by assuming that the causal structure encoded in the causal graph
$\mathcal{G}$ remains the same, but admits a richer variety of outcomes
$\widetilde{\mathcal{O}}$, i.e., we have
$\widetilde{\mathcal{O}}_{\scriptscriptstyle
A}\supset\mathcal{O}_{\scriptscriptstyle A}$,
$\widetilde{\mathcal{O}}_{\scriptscriptstyle
B}\supset\mathcal{O}_{\scriptscriptstyle B}$,
$\widetilde{\mathcal{O}}_{\scriptscriptstyle
C}\supset{\mathcal{O}}_{\scriptscriptstyle C}$, … . It means that, in addition
to the outcomes of interest $\mathcal{O}$, the experiment predicts results
which will have to be rejected in the analysis. Then post-selection boils down
to conditioning on some outcome-dependent variable

$\displaystyle\mathcal{K}$ $\displaystyle\equiv$
$\displaystyle\mathcal{K}(a,b,c,...)\,.$ (5)

Say, for $\mathcal{K}=1$ we accept the result, otherwise for $\mathcal{K}=0$
the result is rejected. This procedure aims at recovering the proper structure
of outcomes for the intended Bell inequality Eq. (1). In other words, the
reduction $\widetilde{\mathcal{O}}\leadsto\mathcal{O}$ is achieved by making
sure that the unwanted results drop out under the conditioning, i.e.,
$P_{\scriptscriptstyle abc...|xyz...\,\lambda\mathcal{K}}=0$ if
$a\notin\mathcal{O}_{\scriptscriptstyle A}$ or
$b\notin\mathcal{O}_{\scriptscriptstyle B}$ or
$c\notin\mathcal{O}_{\scriptscriptstyle C}$, … . Note that the value of
$\mathcal{K}$ is decided only after the parties meet to compare their results.
Hence the causal graph takes the form
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ in Fig. 3 (on the right).

In this way we get a new behaviour
$\mathcal{P}_{\scriptscriptstyle\mathcal{K}}\equiv\\{P_{\scriptscriptstyle
abc...|xyz...\,\mathcal{K}}\\}_{\scriptscriptstyle xyz...}$ which looks like a
good candidate for a test of Bell inequalities Eq. (1). Indeed, all premises
seem to be satisfied, i.e., the ’right’ causal graph $\mathcal{G}$ with the
appropriate pattern of outcomes $\mathcal{O}$, except one detail: there is
conditioning in such obtained statistics. This raises worries as regards the
validity of the conclusions, reached by using
$\mathcal{P}_{\scriptscriptstyle\mathcal{K}}$ in Eq. (1), as a legitimate
proof of Bell nonlocality. The selection bias may serve here as a warning of
how easily post-selection can lead to false causal conclusions, cf. Fig. 1. In
the case of a Bell test it might happen that conditioning (post-selection)
bootstraps the correlations so that
$\mathcal{I}(\mathcal{P}_{\scriptscriptstyle\mathcal{K}})>I_{\scriptscriptstyle
L}$, while for the full statistics it remains
$\mathcal{I}(\mathcal{P})\leqslant I_{\scriptscriptstyle L}$ in agreement with
the causal graph $\mathcal{G}$ in Fig. 3 (i.e., with locality and free choice
maintained). This is possible because conditioning ruins the independence
structure of Eqs. (3) and (4) which is required to prove Eq. (1). To see this,
note that the outcomes $a,b,c,...$ play the role of colliders in the causal
graph $\mathcal{G}$ but conditioning on their descendent $\mathcal{K}$ in
graph $\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ in Fig. 3 opens paths that
were previously blocked, cf. Rule 1 and Rule 3, thereby introducing
correlations into the data which may fake Bell inequalities. How critical it
might be for the analysis of Bell nonlocality may attest the effort to close
the detection loophole [3, 4, 8, 9] (which is a case of post-selection too).

Having warned against jumping to hasty conclusions with post-selected data, it
is then natural to ask:

When is it possible to make a conclusive Bell argument in the post-selected
regime?

In order to make it precise, we assume that the considered causal structure
for the experiment is given by the diagram
$\mathcal{G}{\scriptscriptstyle\mathcal{K}}$ in Fig. 3 and make the following
definition:

######  Definition 1 (Safe post-selection).

Post-selection procedure specified by the variable $\mathcal{K}(a,b,c,...)$ is
considered to be safewhitesafe if the locality and free choice assumptions
still hold in the post-selected regime, i.e.,

$\displaystyle P_{\scriptscriptstyle abc...|xyz...\,\lambda\mathcal{K}}$
$\displaystyle=$ $\displaystyle P_{\scriptscriptstyle
a|x\lambda\mathcal{K}}\cdot P_{\scriptscriptstyle b|y\lambda\mathcal{K}}\cdot
P_{\scriptscriptstyle c|z\lambda\mathcal{K}}\cdot...\,,$ (6)

and

$\displaystyle P_{\scriptscriptstyle\lambda|xyz...\,\mathcal{K}}$
$\displaystyle=$ $\displaystyle P_{\scriptscriptstyle\lambda|\mathcal{K}}\,.$
(7)

An immediate consequence is the observation:

######  Corollary 1.

If post-selection $\mathcal{K}$ is safe, then in the post-selected regime the
same set of Bell inequalities holds, i.e.,

$\displaystyle\mathcal{I}(\mathcal{P})\ \leqslant\ I_{\scriptscriptstyle L}$
$\displaystyle\ \Rightarrow\ $
$\displaystyle\mathcal{I}(\mathcal{P}_{\scriptscriptstyle\mathcal{K}})\
\leqslant\ I_{\scriptscriptstyle L}\,.$ (8)

It follows from the fact that each Bell inequality is obtained by algebraic
manipulation of the expression on the l.h.s. of Eq. (1) assuming Eqs. (3) and
(4) hold. Clearly, the same must be true for
$\mathcal{P}_{\scriptscriptstyle\mathcal{K}}$ since the same algebra, now with
Eqs. (6) and (7), must give the same result.

In this way, the problem of validity of reasoning in the post-selected regime
with the same Bell inequalities is phrased in terms of conditional
independencies in a given causal structure. Given post-selection
$\mathcal{K}$, the latter can be efficiently scrutinised with the diagrammatic
tools of causal inference ($d$-separation rules) applied to the causal graph
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$.

##  5 Main result

Figure 4: Illustration of the all-but-one principle. Suppose that in a
repeated experiment two particles arrive to Alice (A) and Bob (B) in all
possible configurations depicted on the left. Let the meaningful measurements
are only those performed on a single particle by each party. If the number of
particles is conserved, then Alice alone will know which trials to discard
(and similarly for Bob). On the right, there is another party Charlie (C) and
three particles are distributed in all possible ways. Again, conservation of
particles allows to decide by any two of them $AB$, $AC$ or $BC$ which trials
need to be rejected (without knowing what is happening respectively in $C$,
$B$ or $A$).

Suppose that the behaviour $\mathcal{P}$ produced by causal graph
$\mathcal{G}$ features extra correlations due to the specifics of the
preparation procedure (e.g. the number of particles in the experiment is
conserved). This often serves as a means of simplification for post-selection
$\mathcal{K}$. Let us consider the following property:

######  Definition 2 (All-but-one).

Post-selection $\mathcal{K}$ conforms to the all-but-onewhiteall-but-one
principle, if it can be fully determined without knowing one of the outcomes.
Formally, this boils down to a condition on the form of the variable
$\mathcal{K}=\mathcal{K}(a_{\scriptscriptstyle 1},a_{\scriptscriptstyle
2},...,a_{\scriptscriptstyle N})$ which requires that it can be reduced to

$\displaystyle\mathcal{K}\,=\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\,,$ (9)

for each $k=1,...\,,N$. Here, the symbol $\not\\!{a}_{\scriptscriptstyle k}$
means that the outcome for $k$-th party is missing.

Informally, this means that in a Bell scenario all parties except one is
always enough to know whether post-selection ends with a success or not
($\mathcal{K}=1\ \text{or}\ 0$). For example, for three parties $A$, $B$ and
$C$, already two of them may decide if post-selection takes place or not,
i.e., we have

$\displaystyle\mathcal{K}(a,b,c)\,=\,\mathcal{K}(b,c)\,=\,\mathcal{K}(a,c)\,=\,\mathcal{K}(a,b)\,.\
\ \ \ $ (10)

######  Example.

A typical situation where all-but-one principle can be readily applied is when
the number of particles is conserved. Suppose that $N$ particles are
distributed among $N$ parties which receive the particles in different
configurations. Let the interesting measurement results are only those when
there is a single particle per party. This means that, on top of the valid
experimental runs there will be trials in which some parties will register no
or more than one particle. Hence the experiment must resort to post-selection
which consists of retaining only those trials when each party reports a single
particle on their side. Observe that since the total number of particles $N$
is conserved, such a post-selection conforms to the all-but-one principle.
This is because gathering the outcomes from $N-1$ parties is enough to infer
the number of particles received by the missing one (i.e., $N-1$ parties
registering a single particle may conclude that the remaining one registers a
single particle too, since the total number of particles is $N$) and hence to
resolve post-selection only by themselves. See Fig. 4 for an illustration and
Refs. [10, 16, 12, 15, 17, 11, 18, 19, 20, 21, 22, 23, 24, 25, 27, 26, 28, 29,
30, 26] for some experimental designs with post-selection of the all-but-one
type.

Let us observe that the all-but-one principle has non-trivial consequences for
the causal graph $\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ in Fig. 3.
Namely, if the statistics generated by graph $\mathcal{G}$ gives a promise of
Definition 2, then one of the arrows pointing to $\mathcal{K}$ in graph
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ can be always erased without in
any way affecting the generated statistics (in particular, this means that
$\mathcal{P}_{\scriptscriptstyle\mathcal{K}}$ will remain unchanged).

Now we can state our main result:

Figure 5: Graphical proof of Theorem 1 (two parties). Each graph illustrates
structure of conditioning in Eqs. (16) - (20) depicted by red boxes around the
variables. Marked in green are paths joining variables of interest for which
the respective independencies are inferred by the $d$-separation Rules 1 \- 3.
In the first three graphs (on the left) conditioning on the non-collider
$\lambda$ blocks the paths, while for the last two graphs (on the right) the
paths are blocked by the colliders $a$ and $b$ respectively. Note that without
erasing one of the arrows coming to $\mathcal{K}$, as allowed by condition Eq.
(11), the inference of conditional independencies would not be possible.

######  Theorem 1.

For arbitrary number of parties, post-selection which conforms to the all-but-
onewhiteall-but-one principle is always safewhitesafe.

###### Proof.

Here we sketch the proof for two parties $A$ and $B$ which serves to
illustrate the main ideas. For the full proof see Appendix. We need to justify
that Eqs. (6) and (7) hold under the following condition:

$\displaystyle\mathcal{K}\,=\,\mathcal{K}(b)\,=\,\mathcal{K}(a)\,,$ (11)

i.e., the all-but-one principle in Definition 2.

As for Eq. (6) the reasoning follows the usual route starting with the
standard chain rule, i.e., we have

$\displaystyle P_{\scriptscriptstyle ab|xy\lambda\mathcal{K}}$
$\displaystyle=$ $\displaystyle P_{\scriptscriptstyle
a|bxy\lambda\mathcal{K}}\cdot P_{\scriptscriptstyle
b|xy\lambda\mathcal{K}}\,.$

Then the proof boils down to justification of the following conditional
independencies:

$\displaystyle\begin{array}[]{llllll}P_{\scriptscriptstyle
a|bxy\lambda\mathcal{K}}&=&P_{\scriptscriptstyle
a|xy\lambda\mathcal{K}}&&\text{since}&a\perp\\!\\!\\!\perp
b\,|\,xy\lambda\mathcal{K}(b)\\\\[3.0pt] &=&P_{\scriptscriptstyle
a|x\lambda\mathcal{K}}&&\text{since}&a\perp\\!\\!\\!\perp
y\,|\,x\lambda\mathcal{K}(b)\,,\\\\[3.0pt] P_{\scriptscriptstyle
b|xy\lambda\mathcal{K}}&=&P_{\scriptscriptstyle
b|y\lambda\mathcal{K}}&&\text{since}&b\perp\\!\\!\\!\perp
x\,|\,y\lambda\mathcal{K}(a)\,.\end{array}$ (15) (16)

Each of them can be inferred from the causal graph
$\mathcal{G}_{\scriptscriptstyle{K}}$ and application of the $d$-separation
Rule 2 to the non-collider node $\lambda$. See Fig. 5 for illustration.

Similarly, we can justify Eq. (7) and get

$\displaystyle\begin{array}[]{llllll}P_{\scriptscriptstyle\lambda|xy\mathcal{K}}&=&P_{\scriptscriptstyle\lambda|y\mathcal{K}}&&\text{since}&\lambda\perp\\!\\!\\!\perp
x\,|\,y\mathcal{K}(b)\\\\[3.0pt]
&=&P_{\scriptscriptstyle\lambda|\mathcal{K}}&&\text{since}&\lambda\perp\\!\\!\\!\perp
y\,|\,\mathcal{K}(a)\,.\end{array}$ (19) (20)

This time it follows from the $d$-separation Rule 1, since $a$ and $b$ are
colliders respectively, and Rule 3 does not apply (note that in neither case
$\mathcal{K}$ is a descendent). See Fig. 5 for illustration.

Note that crucial for this line of reasoning is the flexibility of the
expression $\mathcal{K}$ in Eq. (11), due to the all-but-one principle, which
permits to erase one of the arrows coming to $\mathcal{K}$. This trick
prevents unblocking certain paths required for the inference of conditional
independences. [For example, in last graph on the right in Fig. 5 retaining
arrow $b\rightarrow\mathcal{K}$ would have opened the (green) path
$\lambda\rightarrow b\leftarrow x$, since $\mathcal{K}$ becomes then a
descendent of the collider $b$ and Rule 3 applies.]

In general, the complexity of paths that need to be considered in the causal
graph $\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ grows with the number of
parties and then the $d$-separation tools prove indispensable for this kind of
analysis, see Appendix. ∎

##  6 Discussion

Because entanglement is not a property generated on demand, every Bell
experiment must resort to post-selection. However, this opens the doors to the
selection bias introducing non-causal correlations into the data, and thus
threatening the conclusions expected to be drawn from the experiment.
Therefore it is important that the analysis of Bell nonlocality takes this
fact into account. In this paper we gave a simple criterion, called the all-
but-one principle, that allows for safe reasoning in the post-selected regime.
Technically, we prove a theorem showing that Bell inequalities derived from
the full causal graph which includes conditioning due to post-selection of the
all-but-one type remain unchanged. It means that the conclusions drawn from
breaking Bell inequalities with such a post-selected data remain in full
force. Beyond the foundational research and application in multipartite
entanglement generation schemes [10, 11, 12], this criterion should be
significant for quantum cryptography and device independent certification [4,
6].

Novelty of the result reported in this work is three-fold: (a) it concerns any
multipartite scenario with an arbitrary number of outcomes and settings, (b)
it pertains to any Bell inequality that can be derived in a given scenario,
and (c) both assumptions of locality and free choice are explicitly discussed
in our analysis. The generality of the all-but-one principle should be
compared with other treatments of post-selection problem due to entanglement
generation [35, 36, 37].

Let us emphasise that the all-but-one principle draws on a special kind of
correlations built into the data due to the specifics of the preparation
procedure. Typically, if events of interest consist of arrival of a single
particle in each detection channel and the number of particles is known and
conserved, then such a post-selection fulfils the all-but-one principle. We
remark that it is a common situation in quantum optical schemes for
entanglement generation, see Refs. [10, 11, 12]. Some recent proposals based
on coincidence counts for high-dimensional multi-particle entanglement that
fall within the all-but-one principle include entanglement by path identity
[16, 17], entanglement without touching [18, 19, 20, 21] or spatial overlap of
indistinguishable particles [23, 24]. See also Refs. [25, 26, 27, 28, 29, 30].
For completeness, we note that the principle is not applicable to time-bin
entanglement scheme [14] which requires specific treatment [38, 39, 40, 41].

Note also that the all-but-one principle has limitations. Although applicable
in many theoretical settings it does not hold in situations with detector
inefficiencies (when the number of particles is not predictable). This is a
serious matter of concern for experimental tests of Bell inequalities leading
to the so called detection loophole which has to be analysed by other means
[3, 4, 8, 9]. We also note that an important experimental technique based on
event-ready-detection [13] is beyond the scope of the principle in the present
form (however, it allows for a straightforward extension to include that
scenario too).

We remark that in the paper we take a conservative approach to the analysis of
Bell nonlocality with causes propagating forward-in-time. For a discussion of
retrocausality see e.g. [42, 43]. Note also that we consider a situation in
which both assumptions of locality and free choice are maintained at the same
time. For a discussion of partial relaxation of those assumptions see [44] and
references therein.

Finally, let us highlight the role of conceptual tools of causal inference
[31, 32, 33, 34] in the present analysis. Not only this is an inspiring and
rigorous framework for the discussion of correlations vs cause-and-effect
relations, but comes equipped with the high-level diagrammatic tools
($d$-separation rules) which prove indispensable for the treatment of
multipartite Bell scenarios with many observers and outcomes. Despite a fairly
recent development of the field of causal inference outside of physics, those
methods have already successfully influenced the research in quantum
foundations, see e.g. [45, 46, 47, 48, 49, 50, 51, 52, 44, 53].

## Acknowledgments

We thank M. Żukowski for bringing post-selection issues to our attention. We
appreciate Y.-S. Kim and R. Lo Franco for helpful comments. We acknowledge
partial support by the Foundation for Polish Science (IRAP project, ICTQT,
contract no. MAB/2018/5, co-financed by EU within Smart Growth Operational
Programme).

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## Appendix

Here we prove Theorem 1 from the main text. For sake of illustration, we start
with the case of three parties. Then we give the full proof for any number of
parties.

##  $\bullet$ Proof of Theorem 1 for three parties A, B and C

[The following proof for three parties aims to better illustrate some
additional aspects which do not arise in the two-party case. It also serves to
emphasise the significance of $d$-separation tools of causal inference [31,
32, 33, 34] for this kind of analysis.]

###### Proof.

Let us consider a Bell experiment with three parties described by the causal
graph $\mathcal{G}_{\mathcal{K}}$ in Fig. 6 and assume that post-selection
conforms to the all-but-one principle in Definition 2, i.e., we have

$\displaystyle\mathcal{K}(a,b,c)\,=\,\mathcal{K}(b,c)\,=\,\mathcal{K}(a,c)\,=\,\mathcal{K}(a,b)\,.$
(21)

Crucially, this property allows erasing one of the three arrows coming to
$\mathcal{K}$ without affecting the generated statistics. This trick will be
used to infer conditional independencies in the post-selected behaviour
$\mathcal{P}_{\scriptscriptstyle\mathcal{K}}$.

Figure 6: Causal structure in a Bell experiment for three parties with post-
selection. On the left, unfolded on a plain is the causal graph
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ describing causal relations
between variables in a Bell experiment ($a,b,c$ \- outcomes, $x,y,z$ \-
measurement settings, and $\lambda$ \- hidden variable). Variable
$\mathcal{K}$ represents post-selection (where the red box means
conditioning). Cf. Fig. 3. On the right, three causal graphs which are
equivalent to graph $\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$, if the all-
but-one principle in Eq. (21) holds (then one of the three arrows coming to
$\mathcal{K}$ can be always erased without affecting the generated
statistics).

In order to prove Theorem 1 we need to justify that both Eqs. (6) and (7) in
Definition 1 hold under the condition in Eq. (21).

As a straightforward application of the chain rule we get

$\displaystyle P_{abc|xyz\lambda\mathcal{K}}\ =\
P_{a|bcxyz\lambda\mathcal{K}}\,\cdot\,P_{bc|xyz\lambda\mathcal{K}}\ =\
P_{a|bcxyz\lambda\mathcal{K}}\,\cdot\,P_{b|cxyz\lambda\mathcal{K}}\,\cdot\,P_{c|xyz\lambda\mathcal{K}}\,.$
(22)

Now, the prove of Eq. (6) boils down to showing the following sequence of
conditional independencies:

$\displaystyle\begin{array}[]{llllll}P_{a|bcxyz\lambda\mathcal{K}}&=&P_{a|cxyz\lambda\mathcal{K}}&&\text{since}&a\perp\\!\\!\\!\perp
b\,|\,cxyz\lambda\mathcal{K}(b,c)\\\\[3.0pt]
&=&P_{a|xyz\lambda\mathcal{K}}&&\text{since}&a\perp\\!\\!\\!\perp
c\,|\,xyz\lambda\mathcal{K}(b,c)\\\\[3.0pt]
&=&P_{a|xz\lambda\mathcal{K}}&&\text{since}&a\perp\\!\\!\\!\perp
y\,|\,xz\lambda\mathcal{K}(b,c)\\\\[3.0pt]
&=&P_{a|x\lambda\mathcal{K}}&&\text{since}&a\perp\\!\\!\\!\perp
z\,|\,x\lambda\mathcal{K}(b,c)\,.\end{array}\ \ $ (27)
$\displaystyle\begin{array}[]{llllll}P_{b|cxyz\lambda\mathcal{K}}&=&P_{b|xyz\lambda\mathcal{K}}&&\text{since}&b\perp\\!\\!\\!\perp
c\,|\,xyz\lambda\mathcal{K}(a,c)\\\\[3.0pt]
&=&P_{b|yz\lambda\mathcal{K}}&&\text{since}&b\perp\\!\\!\\!\perp
x\,|\,yz\lambda\mathcal{K}(a,c)\\\\[3.0pt]
&=&P_{b|y\lambda\mathcal{K}}&&\text{since}&b\perp\\!\\!\\!\perp
z\,|\,y\lambda\mathcal{K}(a,c)\,.\end{array}\ \ \ \ $ (31)
$\displaystyle\begin{array}[]{llllll}P_{c|xyz\lambda\mathcal{K}}&=&P_{c|yz\lambda\mathcal{K}}&&\text{since}&c\perp\\!\\!\\!\perp
x\,|\,yz\lambda\mathcal{K}(a,b)\\\\[3.0pt]
&=&P_{c|z\lambda\mathcal{K}}&&\text{since}&c\perp\\!\\!\\!\perp
y\,|\,z\lambda\mathcal{K}(a,b)\,.\end{array}\ \ \ \ \ \ \ $ (34)

All of them can be inferred by inspecting paths joining variables in question
in the causal graph $\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ and the use
of $d$-separation rules. See Fig. 7. In each case there are two possible paths
which are blocked by conditioning on the non-collider node $\lambda$ (Rule 2).
Note that in order to get the required conditional independencies all
conditions in Eq. (21) need to be used, i.e., the lack of the respective arrow
coming to $\mathcal{K}$ in each of the graphs in Fig. 7 is essential.

In a similar manner we can prove Eq. (7), that is we can prove the following
conditional independencies:

$\displaystyle\begin{array}[]{llllll}P_{\lambda|xyz\lambda\mathcal{K}}&=&P_{\lambda|yz\mathcal{K}}&&\text{since}&\lambda\perp\\!\\!\\!\perp
x\,|\,yz\mathcal{K}(b,c)\\\\[3.0pt]
&=&P_{\lambda|z\mathcal{K}}&&\text{since}&\lambda\perp\\!\\!\\!\perp
y\,|\,z\mathcal{K}(a,c)\\\\[3.0pt]
&=&P_{\lambda|\mathcal{K}}&&\text{since}&\lambda\perp\\!\\!\\!\perp
z\,|\,\mathcal{K}(a,b)\,.\end{array}\ \ \ \ $ (38)

Now there is always one path joining the relevant variables, and in each case
it is blocked by the respective collider $a$, $b$ and $c$ (Rule 1). See Fig.
7. Notice that, here as well, having all three arrows coming to $\mathcal{K}$
would spoil the proof, since it would lift the block from the respective
collider by conditioning on its descendent $\mathcal{K}$ (Rule 3).
∎

Figure 7: Graphical proof of Theorem 1 (three parties). Each graph illustrates
structure of conditioning in Eqs. (27) - (38) depicted by red boxes around the
variables. Marked in purple and green are paths joining variables of interest
for which the respective independencies are inferred by the $d$-separation
Rules 1 \- 3. In case of Eqs. (27) - (34) there are always two paths, and all
of them are blocked by conditioning on the non-collider $\lambda$ (Rule 2). As
for Eq. (38), in each case the only path is blocked by the respective collider
$a$, $b$ and $c$ (Rule 1). Note that dropping one of the arrows in each of the
graphs, as allowed by condition in Eq. (21) and explained in Fig. 6, is not
accidental (since otherwise it would open additional paths and spoil the
independence pattern).

##  $\bullet$ Proof of Theorem 1 for any number of parties

[The proof follows the lines of reasoning for the case of two and three
parties.]

###### Proof.

We consider the general case of $N$ parties in a Bell experiment with post-
selection conforming to the all-but-one principle in Definition 2, i.e we have

$\displaystyle\mathcal{K}\,=\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\qquad\text{for each \ $k=1,...\,,N$}\,,\quad$ (39)

where $\\!\not\\!{a}_{\scriptscriptstyle k}$ means that $k$-th variable is
missing from the list of all outcomes. The causal structure is then given by
the graph $\mathcal{G}_{\mathcal{K}}$ in Fig. 8. Note that condition Eq. (39)
entails that one of the $N$ arrows coming to $\mathcal{K}$ can be always
erased without changing the generated statistics (cf. Fig. 6). In particular,
this means that conditional independencies inferred by dropping one of those
arrows remain the same.

Figure 8: Causal structure in a Bell experiment for any number parties with
post-selection. On the left, unfolded on a plain is the causal graph
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$ describing relations between
variables in a Bell experiment ($a_{\scriptscriptstyle
1},...\,,a_{\scriptscriptstyle N}$ \- outcomes, $x_{\scriptscriptstyle
1},...\,,x_{\scriptscriptstyle N}$ \- measurement settings, and $\lambda$ \-
hidden variable). Variable $\mathcal{K}$ represents post-selection (where the
red box means conditioning). Cf. Fig. 3. On the right, the causal graph with
one arrow $a_{\scriptscriptstyle k}\rightarrow\mathcal{K}$ erased which is
equivalent to graph $\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$, if the all-
but-one principle in Eq. (39) holds.

The proof of Theorem 1 consists of justifying that both Eqs. (6) and (7) in
Definition 1 hold under the condition in Eq. (39).

By the repeated use of the chain rule, we get

$\displaystyle P_{a_{1}...\,a_{N}|x_{1}...\,x_{N}\lambda\mathcal{K}}\ =\
\prod_{\scriptscriptstyle k\,=\,1}^{\scriptscriptstyle N}\
P_{a_{k}|a_{k+1}...\,a_{N}x_{1}...\,x_{N}\lambda\mathcal{K}}\,.$ (40)

In order to prove Eq. (6) it remains to check that for each $k=1,...\,,N$ the
following sequence of conditional independencies holds:

$\displaystyle\\!\\!\\!\begin{array}[]{llllll}P_{a_{k}|a_{k+1}...\,a_{N}\,x_{1}...\,x_{N}\lambda\mathcal{K}}&=&P_{a_{k}|a_{k+2}...\,a_{N}\,x_{1}...\,x_{N}\lambda\mathcal{K}}&&\vspace{0.3cm}\\\
&&\quad\quad\text{since}\quad a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp
a_{\scriptscriptstyle k+1}\,|\,a_{\scriptscriptstyle
k+2}...\,a_{\scriptscriptstyle N}\,x_{\scriptscriptstyle
1}...\,x_{\scriptscriptstyle N}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\\\ &=&\dots\vspace{0.2cm}\\\
&=&P_{a_{k}|a_{l+1}...\,a_{N}\,x_{1}...\,x_{N}\lambda\mathcal{K}}&&\vspace{0.3cm}\\\
&&\quad\quad\text{since}\quad a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp
a_{\scriptscriptstyle l}\,|\,a_{\scriptscriptstyle
l+1}...\,a_{\scriptscriptstyle N}\,x_{\scriptscriptstyle
1}...\,x_{\scriptscriptstyle N}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\\\ &=&\dots\vspace{0.2cm}\\\
&=&P_{a_{k}|x_{1}...\,x_{N}\lambda\mathcal{K}}&&\vspace{0.3cm}\\\
&&\quad\quad\text{since}\quad a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp
a_{\scriptscriptstyle N}\,|\,x_{\scriptscriptstyle
1}...\,x_{\scriptscriptstyle N}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\,,\end{array}$ (49)

where $l>k$, and

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\begin{array}[]{llllll}P_{a_{k}|x_{1}...\,x_{N}\lambda\mathcal{K}}&=&P_{a_{k}|x_{2}...\,x_{N}\lambda\mathcal{K}}&\quad\text{since}\quad
a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp x_{\scriptscriptstyle
1}\,|\,x_{\scriptscriptstyle 2}...\,x_{\scriptscriptstyle
N}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\vspace{0.2cm}\\\ &=&\dots\vspace{0.2cm}\\\
&=&P_{a_{k}|x_{m+1}...\,x_{N}\lambda\mathcal{K}}&\quad\text{since}\quad
a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp x_{\scriptscriptstyle
m}\,|\,x_{\scriptscriptstyle m+1}...\,x_{\scriptscriptstyle
N}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\vspace{0.2cm}\\\ &=&\dots\vspace{0.2cm}\\\
&=&P_{a_{k}|x_{k}\,x_{l+1}...\,x_{N}\lambda\mathcal{K}}&\quad\text{since}\quad
a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp x_{\scriptscriptstyle
l}\,|\,x_{\scriptscriptstyle k}\,x_{\scriptscriptstyle
l+1}...\,x_{\scriptscriptstyle N}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\vspace{0.2cm}\\\ &=&\dots\vspace{0.2cm}\\\
&=&P_{a_{k}|x_{\scriptscriptstyle k}\lambda\mathcal{K}}&\quad\text{since}\quad
a_{\scriptscriptstyle k}\perp\\!\\!\\!\perp x_{\scriptscriptstyle
N}\,|\,x_{\scriptscriptstyle k}\lambda\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\,,\end{array}$ (57)

where $l>k>m$. These independencies can be justified by the $d$-separation
tools applied to the causal graph
$\mathcal{G}_{\scriptscriptstyle\mathcal{K}}$. It boils down to the inspection
of all paths joining variables in question. See Fig. 9 (cf. Fig. 7). In each
case there are $N-1$ possible paths which are all blocked by conditioning on
the non-collider node $\lambda$ (Rule 2). Like before, we use all conditions
in Eq. (39) to get the results (as readily seen from Fig. 9, where in each
case a different arrow $a_{\scriptscriptstyle k}\rightarrow\mathcal{K}$ is
missing).

Similarly, the proof of Eq. (7) boils down to the following conditional
independencies:

$\displaystyle\begin{array}[]{llllll}P_{\lambda|x_{1}...\,x_{N}\mathcal{K}}&=&P_{\lambda|x_{2}...\,x_{N}\mathcal{K}}&&\quad\text{since}\quad\lambda\perp\\!\\!\\!\perp
x_{\scriptscriptstyle 1}\,|\,x_{\scriptscriptstyle
2}...\,x_{\scriptscriptstyle N}\mathcal{K}(\\!\not\\!{a}_{\scriptscriptstyle
1},a_{\scriptscriptstyle 2},...\,,a_{\scriptscriptstyle N})\vspace{0.2cm}\\\
&=&\dots\vspace{0.2cm}\\\
&=&P_{\lambda|x_{k+1}...\,x_{N}\mathcal{K}}&&\quad\text{since}\quad\lambda\perp\\!\\!\\!\perp
x_{\scriptscriptstyle k}\,|\,x_{\scriptscriptstyle
k+1}...\,x_{\scriptscriptstyle N}\mathcal{K}(a_{\scriptscriptstyle
1},...\,,\\!\not\\!{a}_{\scriptscriptstyle k},...\,,a_{\scriptscriptstyle
N})\vspace{0.2cm}\\\ &=&\dots\vspace{0.2cm}\\\
&=&P_{\lambda|\mathcal{K}}&&\quad\text{since}\quad\lambda\perp\\!\\!\\!\perp
x_{\scriptscriptstyle N}\,|\,\mathcal{K}(a_{\scriptscriptstyle
1},...\,,a_{\scriptscriptstyle N-1},\\!\not\\!{a}_{\scriptscriptstyle
N})\,.\end{array}$ (63)

Here there is only one path joining the relevant variables that is blocked by
the respective collider $a_{\scriptscriptstyle k}$ (Rule 1). See Fig. 9 (cf.
Fig. 7). Note that the lacking arrow $a_{\scriptscriptstyle
k}\rightarrow\mathcal{K}$ is crucial, since otherwise it would unblock the
respective collider $a_{\scriptscriptstyle k}$ by conditioning on its
descendent $\mathcal{K}$ (Rule 3).

∎

Figure 9: Graphical proof of Theorem 1 (any number of parties). We give three
generic cases used for the justification of conditional independencies in Eqs.
(49) - (63). Red boxes around the variables depict the structure of
conditioning. Marked in purple, green, orange and blue are paths joining
variables of interest, $a_{\scriptscriptstyle k}$ and $a_{\scriptscriptstyle
l}$ (resp. $x_{\scriptscriptstyle l}$), for which the respective
independencies are inferred by the $d$-separation Rules 1 \- 3. In case of
Eqs. (49) and (57) there are $N-1$ paths, and all of them are blocked by
conditioning on the non-collider $\lambda$ (Rule 2). As for Eq. (63), the only
path joining $\lambda$ and $x_{\scriptscriptstyle k}$ is blocked by the
collider $a_{\scriptscriptstyle k}$ (Rule 1). Let us note that the lack of the
respective arrow $a_{\scriptscriptstyle k}\rightarrow\mathcal{K}$ in each of
the graphs is essential (in order not to introduce unwanted paths in case of
Eqs. (49) and (57), or to prevent lifting the block from the collider by Rule
3 in case of Eq. (63)).