spacing=nonfrench

# Causal Data Fusion with Quantum Confounders

Pedro Lauand<EMAIL_ADDRESS>Instituto de Física “Gleb Wataghin”,
Universidade Estadual de Campinas, 130830-859, Campinas, Brazil  Perimeter
Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5  Bereket
Ngussie Bekele  Perimeter Institute for Theoretical Physics, Waterloo,
Ontario, Canada, N2L 2Y5  Addis Ababa University, Addis Ababa, Ethiopia  Elie
Wolfe  Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada,
N2L 2Y5

###### Abstract

From the modern perspective of causal inference, Bell’s theorem — a
fundamental signature of quantum theory — is a particular case where quantum
correlations are incompatible with the classical theory of causality, and the
generalization of Bell’s theorem to quantum networks has led to several
breakthrough results and novel applications. Here, we consider the problem of
causal data fusion, where we piece together multiple datasets collected under
heterogeneous conditions. In particular, we show quantum experiments can
generate observational and interventional data with a non-classical signature
when pieced together that cannot be reproduced classically. We prove this
quantum non-classicality emerges from the fusion of the datasets and is
present in a plethora of scenarios, even where standard Bell non-classicality
is impossible. Furthermore, we show that non-classicality genuine to the
fusion of multiple data tables is achievable with quantum resources. Our work
shows incorporating interventions–a central tool in causal inference– can be a
powerful tool to detect non-classicality beyond the violation of a standard
Bell inequality. In a companion article [1], we extend our investigation
considering all latent exogenous causal structures with 3 observable
variables.

Introduction—Estimating cause-and-effect relations behind the correlations
observed among some measured variables is a central goal of science and one of
the most challenging aspects of causality theory [2, 3, 4]. The difficulty
stems from the fact that one cannot conclusively infer a cause-and-effect
relationship between two events or variables based only on the correlation
observed between them, i.e. "correlation does not imply causation". The reason
is that any correlation observed between two or more random variables can be
explained, in the classical regime, by a potentially unobserved common cause.
Comprehending the conditions under which such confounding factors can be
controlled, allowing for the derivation of a causal hypothesis from empirical
data, has paved the way for the development of several causal analysis tools.
Interventions [5] are the go-to tool in the field of causal analysis enabling
better decisions among competing explanations for given statistics, a task
known as causal discovery, and also to estimate causal relationships or
predict counterfactual quantities which allow us to analyze how relative
strengths of different causal pathways could play a role in establishing some
observed correlations, a task known as causal inference. However, there are
situations where intervening in the system may not be feasible, such as
ethical constraints or when the focus is on assessing causal effects in
previous experiments.

Instrumental variables [6] allow for the quantification of cause-and-effect
relationships even in the absence of interventions. To achieve this, some
causal assumptions must be met, namely, the instrument must influence the
cause of the cause-and-effect relationships we are interested in and it must
be independent of any confounding factors of the original variables.
Instrumental inequalities, constraints that any experiment in compliance with
the instrumental assumptions should respect, have been introduced [7] and
their violation explicitly proves that one does not have a proper instrument.
Concepts such as interventions and instrumental variables have become
fundamental tools for assessing causal effects across a range of disciplines
[8, 9, 10]. Despite its success, all such tools and applications rely on the
classical notion of causality that, since Bell’s theorem [11], we know cannot
account for all quantum correlations.

Bell’s theorem is a cornerstone of quantum physics and it provides a device-
independent [12] proof of the incompatibility of classical and quantum
predictions, that is, solely based on the causal assumptions of an experiment
and agnostic of any internal mechanisms or physical details of the measurement
and state preparation devices. Historically, the mismatch between quantum and
classical correlations was shown for a particular causal structure and, in the
last decade, has been generalized to causal structures of increasing
complexity and with different topologies [13]. This has led to a formalization
of a quantum common cause and, more generally, quantum causal models [14, 15,
16]. Typically, this incompatibility is proven by violating a Bell inequality
which requires only passive observations of the experiment under scrutiny.

The derivation of a Bell inequality can be seen as a particular case of a
causal inference task. Recently, it has been shown that the violation of a
Bell inequality is not the only signature of the incompatibility between
quantum correlations and causality theory [17, 18, 19, 20]. Specifically, they
show how quantum correlations can change the cause-and-effect relations that
can be inferred considering the estimation of counterfactual interventions.
Remarkably, this new phenomenon is present even when quantum correlations
cannot violate any standard Bell-type inequality in the simplest instrumental
scenario. This opens new possibilities to explore the role of causality within
quantum theory. Here, we consider _data fusion_ problem, which consists of
piecing together multiple datasets collected under heterogeneous conditions
[21]. We show the data fusion problem exhibits an opportunity for detecting
the non-classicality of quantum experiments in this new data regime,
considering joint probabilities for passive observations and _do-conditional_
probabilities for observations under intervention. Remarkably, these
experiments have the feature that, even though individually each data table
admits a classical model explanation, these datasets admit no single classical
model when considered jointly. We call this new type of non-classicality "non-
classicality from data fusion".

a1

a2

a3

Figure 2: Here we have the three scenarios considered. In Fig. 2a1 we have the
triangle scenario with communication between $A$ and $B$, represented by a
direct influence $A\rightarrow B$. In Fig. 2a2 we can see the relaxation of
the Unrelated Confounders (UC) scenario, with extra communication between $A$
and $C$. Finally, in Fig. 2a3 we can see a chain-like structure, $A\rightarrow
B\rightarrow C$ with a tripartite common cause.

We explore quantum non-classicality from data fusion for all latent exogenous
causal structures with 3 observable variables. We introduce a robust framework
able to asses the classical compatibility of the fusion between observational
data and multiple interventions in a generic causal structure. Then, we show
that the bounds constraining the possible data tables classically compatible
with a given causal structure admit quantum violations, i.e. the analogous
experiment with quantum confounders generates data (observational and
interventional) that cannot be explained with classical latent variable
models. In particular, we show that this non-classicality is present even in
causal structures where standard Bell non-classicality, involving only passive
observations, is impossible and that non-classicality genuine to multiple
interventions can also be achieved with quantum resources. This work focuses
on three main cases but we expand our investigation in a companion article [1]
which also provides all the required technical proofs of the results in this
work.

Causal Modeling—Causal assumptions can be captured by directed acyclic graphs
(DAGs), comprising a finite collection of nodes and a set of directed edges.
Each node denotes a random variable, categorized into observable nodes,
depicted by triangles, represented by classical variables, and latent nodes,
depicted by circles, signifying variables inaccessible to us. For passive
observations, we obtain the joint probability distribution of the observable
variables. In this work, we focus on the case of three observable variables,
namely $A$, $B$, and $C$. Passive observations of the observed variables are
captured by a joint probability distribution $P_{ABC}$. Causal models aim to
specify the mechanisms that play a role in establishing observable data. To
achieve this, it accounts for the dependencies of every variable with its
causal parents, denoted $\operatorname{Pa}{(X)}$, which can be defined as the
variables that share incoming edges with $X$. Classical compatibility is
defined in terms of the conditional probability distribution
$p_{X}(x|\operatorname{Pa}(x))$, where $\operatorname{Pa}(x)$ denotes that the
random variables in the set $\operatorname{Pa}(X)$ take their respective
values, for each node in the graph. From these response functions, we can
define the _Markov_ decomposition of the DAG. We will specify this
decomposition for each case throughout the text. For a general construction,
see our companion article [1]. In quantum causal models, each latent variable,
$\Lambda_{i}$, is represented by a density matrix $\psi_{\Lambda_{i}}$, and
observable variables are associated with positive semi-definite operator-
valued measurement effects (POVMs) properly normalized.

Interventions play a crucial role in causal inference, allowing us to
ascertain the causal connections among variables within a specified process
[2, 6]. Unlike passive observations, interventions involve locally changing
the underlying causal structure of an experiment, eliminating all external
influences that a given variable might have, and putting it under the
exclusive control of an observer. The probability of observing the variables
$B$ and $C$ when the variable $A$ undergoes an intervention and is
artificially set to $A=a$ is represented by do-conditional probabilities
$P_{BC}(b,c|\text{do}(A=a))$, eliminating the response function
$p_{A}(a|\operatorname{Pa}(a))$ from the model decomposition. Analogously, for
quantum causal models, we can substitute the operator
$E_{a|\operatorname{Pa}(a)}$ by $\mathds{1}$. Similarly, we can define the do-
conditionals $P_{AC}(a,c|\text{do}(B=b))$ and $P_{AB}(a,b|\text{do}(C=c))$.

The fusion of observational and interventional data can be understood as a
particular instance of the _compatibility problem_ , asking whether some
statistics of the experiment (observational and interventional) are jointly
compatible with a given causal structure. To achieve this, we use the
_interruption technique_ [22]. Truly, starting from a given causal structure
an interruption involves creating a modified scenario by splitting the node
upon intervention into two distinct variables. Moreover, we can express
multiple interventions by iteratively interrupting the nodes upon
intervention. This approach has precedent in the causal inference literature
in the "single-world intervention graphs" (SWIGs) [23] and also the node-
splitting procedure of Ref. [24].

Numerical Methods— In this work, we explore three qualitative distinct
numerical modeling approaches. The first numerical approach involves
characterizing a polytope [25], which can be naturally cast as a Linear
Program (LP). Primarily, this is used in cases with a single latent source and
the classical compatibility problem amounts to deciding whether a given point
is inside or outside some polytope. The second numerical approach is based on
quadratic optimization [26], a natural generalization of an LP problem.
Adopting the so-called _branch and bound methods_ [27], which involves
systematically and iteratively narrowing down the range of variables, dividing
them into subproblems that can each be approximated by a corresponding LP
problem. This branching subroutine enables the attainment of tighter
relaxations, leading to upper and lower bounds that gradually converge towards
the global optimal solution. The third numerical approach is that of the
_inflation technique_ [28]. The inflation approach considers when one has
access to many independent copies of the scenario’s variables and can put them
in different configurations. The use of the inflation technique with convex
optimization allows us to derive analytical causal compatibility inequalities
via convex duality. For details on our numerical methods, we refer the reader
to our companion article [1].

Quantum non-classicality from data-fusion— Herein we show how quantum
correlations can generate a non-classical signature violating bounds involving
the fusion of observational and interventional data. To guarantee that this
non-classicality is exclusive to the interplay between the observations and
the interventions under scrutiny, we further require that the individual data
tables admit a classical model explanation. In this regard, all quantum
violations we show go beyond the paradigmatic violation of a standard Bell-
type inequality and can only be attributed to the fusion of the data tables.
We focus on the cases shown in Fig. 2.

Consider the DAG shown in Fig. 2a1. We have $A$, $B$ and $C$ with bipartite
latent variables $\alpha$, $\beta$, and $\gamma$ between $B$ and $C$, $A$ and
$C$, and $A$ and $B$ respectively with an additional edge $A\rightarrow B$.
The classical model decomposition is given by

$\displaystyle P_{ABC}$ $\displaystyle(a,b,c)=$ (1)
$\displaystyle\sum_{\alpha,\gamma,\beta}p(\alpha)p(\gamma)p(\beta)p_{A}(a|\gamma,\beta)p_{B}(b|\alpha,\gamma,a)p_{C}(c|\alpha,\beta)$

and

$\displaystyle P_{BC}(b,c|do(A=a$ $\displaystyle))=$ (2)
$\displaystyle\sum_{\alpha,\gamma,\beta}p(\alpha)p(\gamma)p(\beta)p_{B}(b|\alpha,\gamma,a)p_{C}(c|\alpha,\beta).$

The data tables that can arise in quantum models are given by

$\displaystyle
P_{ABC}(a,b,c)=\Tr\left(\psi_{AB}\otimes\psi_{BC}\otimes\psi_{AC}\left(E_{a|b}\otimes
E_{b}\otimes E_{c|b,a}\right)\right)$ (3) $\displaystyle
P_{BC}(b,c|do(A=a))=\Tr\left(\psi_{AB}\otimes\psi_{BC}\otimes\psi_{AC}\left(\mathds{1}\otimes
E_{b}\otimes E_{c|b,a}\right)\right).$

This scenario can be seen as a generalization of the triangle network,
explored at length by the community [29, 30, 31]. For this causal structure,
we prove in Lemma 1 in our companion article that, when $|A|=2$, non-
classicality from data fusion of $\\{P_{ABC},P_{BC|do(A)}\\}$ is equivalent to
standard network non-classicality of $Q_{ABC|A^{\\#}}$ over the corresponding
SWIG, i.e. the graph resulting from interruption technique, which correspond
to the standard triangle scenario where $B$ has a binary input $A^{\\#}$ and
$A$ is binary. Then, we prove the existence of quantum compatible models that
exhibit non-classicality from data fusion by using a protocol of disguised
Bell non-classicality as network non-classicality, due to Fritz [32].
Furthermore, we use Lemma 1 to show the validity of inequality constraints
over $\\{P_{ABC},P_{BC|do(A)}\\}$ originated from Bell-like inequalities
classically valid for the triangle scenario that admit quantum violation. The
details of our proof of non-classicality for the specific protocol used can be
found in Section IVD and the corresponding causal inequality can be found in
Appendix E of our companion article.

Now, consider the DAG in Fig. 2a2. This scenario features $A$, $B$, and $C$
with two bipartite lantent variables $\gamma$ between $A$ and $B$, and
$\alpha$ between $B$ and $C$, with $B$ directly influencing $A$ and $C$, i.e.
$A\leftarrow B\rightarrow C$, and an arrow $A\rightarrow C$. We consider the
data tables $\\{P_{ABC},P_{BC|do(A)},P_{AC|do(B)}\\}$, a classical model
generates data tables given by

$\displaystyle
P_{ABC}(a,b,c)=\sum_{\alpha,\gamma}p(\alpha)p(\gamma)p_{A}(a|\gamma,b)p_{B}(b|\alpha,\gamma)p_{C}(c|\alpha,b,a)$
(4) $\displaystyle
P_{BC}(b,c|do(A=a))=\sum_{\alpha,\gamma}p(\alpha)p(\gamma)p_{B}(b|\alpha,\gamma)p_{C}(c|\alpha,b,a)$
$\displaystyle
P_{AC}(a,c|do(B=b))=\sum_{\alpha,\gamma}p(\alpha)p(\gamma)p_{A}(a|\gamma,b)p_{C}(c|\alpha,b,a),$

and quantum models respect

$\displaystyle
P_{ABC}(a,b,c)=\Tr\left(\psi_{AB}\otimes\psi_{BC}\left(E_{a|b}\otimes
E_{b}\otimes E_{c|b,a}\right)\right)$ (5) $\displaystyle
P_{BC}(b,c|do(A=a))=\Tr\left(\psi_{AB}\otimes\psi_{BC}\left(\mathds{1}\otimes
E_{b}\otimes E_{c|b,a}\right)\right)$ $\displaystyle
P_{AC}(a,c|do(B=b))=\Tr\left(\psi_{AB}\otimes\psi_{BC}\left(E_{a|b}\otimes\mathds{1}\otimes
E_{c|b,a}\right)\right).$

This causal structure can be regarded as a relaxation of the _Unrelated
Confounders_ (UC) scenario where $A$ can influence $C$ directly. The UC
scenario has only recently been considered from a quantum information
perspective [33, 34, 35] and is akin to the causal structure underlying the
entanglement-swapping experiment [36]. Section IVB and Appendix A of our
companion article show that the UC scenario exhibits robust quantum non-
classicality from data fusion when all variables are dichotomic. Furthermore,
we can show that from non-classicality from data fusion in the UC scenario we
can achieve non-classicality in its relaxation shown in Fig. 2a2. Truly, using
the arguments outlined in Section IVB we find that the inequality

$W:=P_{A}(0|do(B=0))^{2}-P_{A}(0|do(B=0))I+E+J\leq 0,$ (6)

where $I$ and $J$ are linear expressions given by

$\displaystyle I:=$ $\displaystyle 2P_{AB}(0,0)+P_{B}(1)+P_{BC}(1,0|do(A=0))$
(7) $\displaystyle+P_{ABC}(0,1,1)-2P_{ABC}(0,1,0)$ $\displaystyle J:=$
$\displaystyle P_{AB}(0,0)-2P_{B}(0)+2P_{AB}(0,0)-2P_{ABC}(0,1,0).$

and $E$ carries quadratic terms

$\displaystyle E:=2P_{AB}(0,1)P_{BC}(1,0|do(A=0))+P_{AB}(0,0)P_{B}(0)+$ (8)
$\displaystyle\left(2P_{B}(0)-P_{AB}(0,0)\right)\left(P_{BC}(1,0|do(A=0))+P_{ABC}(0,1,0)\right)$
$\displaystyle-P_{AB}(0,0)^{2}-P_{ABC}(0,1,0)^{2},$

must hold for all classically compatible data tables in the scenario of Fig.
2a2. Using the same protocol outlined in Theorem 1 of Section IVB, where $A$
and $B$ share a maximally entangled state $\phi^{+}$ with isotropic noise, and
$B$ and $C$ share a classical source $\alpha$. $A$ is assigned the
measurements $\sigma_{x}$ if $b=0$ and $\sigma_{z}$ if $b=1$ while $B$ is
associated with the measurements
$\frac{1}{\sqrt{2}}(\sigma_{x}+(-1)^{\alpha}\sigma_{z})$ with
$\alpha\in\\{0,1\\}$. $C$ deterministically outputs $\alpha$, regardless of
the value $b$ and $a$, i.e. $p_{C}(c|\alpha,b,a)=\delta_{c,\alpha}$. We can
achieve quantum violations of this inequality given by
$I^{Q}=\frac{1}{16}\left(18+7\sqrt{2}\right)$,
$J^{Q}=\frac{1}{2}\left(\sqrt{2}-1\right)$,
$E^{Q}=\frac{1}{128}\left(51+2\sqrt{2}\right)$, and
$P^{Q}_{A}(0|do(B=0))=\dfrac{1}{2}$. Reaching
$W^{Q}=\frac{1}{128}\left(38\sqrt{2}-53\right)\approx 0.005782$. Introducing
critical visibility $v$ to the quantum source, we can try to quantify this
non-classicality by mixing it with the maximally mixed state. We can show with
inflation technique critical visibility of $v\approx 0.96$, then we use
quadratic optimization to numerically certify the non-classicality emerging
from data fusion of the whole range of $\dfrac{1}{\sqrt{2}}<v\leq 1$.

Finally, we consider the causal structure depicted in Fig. 2a3 where we have a
chain-like structure for the observable variables, i.e. $A\rightarrow
B\rightarrow C$ and a single tripartite common cause $\Lambda$ between them.
Classically, the data tables $P_{ABC}(a,b,c)$, $P_{BC}(b,c|do(A=a))$ and
$P_{AC}(a,c|do(B=b))$ are given by

$\displaystyle
P_{ABC}(a,b,c)=\sum_{\lambda}p(\lambda)p_{A}(a|\lambda)p_{B}(b|a,\lambda)p_{C}(c|b,\lambda)$
(9) $\displaystyle
P_{BC}(b,c|do(A=a))=\sum_{\lambda}p(\lambda)p_{B}(b|a,\lambda)p_{C}(c|b,\lambda)$
$\displaystyle
P_{AC}(a,c|do(B=b))=\sum_{\lambda}p(\lambda)p_{A}(a|\lambda)p_{C}(c|b,\lambda).$

and quantum models yields

$\displaystyle P_{ABC}(a,b,c)=\Tr\left(\psi_{ABC}\left(E_{a}\otimes
E_{b|a}\otimes E_{c|b}\right)\right)$ (10) $\displaystyle
P_{BC}(b,c|do(A=a))=\Tr\left(\psi_{ABC}\left(\mathds{1}\otimes E_{b|a}\otimes
E_{c|b}\right)\right)$ $\displaystyle
P_{AC}(a,c|do(B=b))=\Tr\left(\psi_{ABC}\left(E_{a}\otimes\mathds{1}\otimes
E_{c|b}\right)\right).$

Initially, we only require each data table to be classically attainable for
certifying non-classicality from data fusion. However, any combination of data
tables could potentially result in a violation. Now, we introduce an
additional criterion: each pair of data tables must also be classically
attainable within our model. This added constraint ensures that the non-
classicality observed cannot be attributed to specific interactions between
observations and interventions or between different interventions. Instead, it
can only be ascribed to the fusion of all data tables. This novel form of
violation is termed non-classicality from data fusion in the three-way
synthesis. We give a precise definition in Section IIC of [1].

We show there exist quantumly compatible data tables when all variables are
dichotomic, that are non-classical in the three-way synthesis, we use a pure
non-maximally entangled three-qubit state

$|\psi_{ABC}\rangle:=\dfrac{e^{-i\frac{\pi}{8}}}{\sqrt{2}}|0_{A}\rangle|\psi^{-}_{BC}\rangle+\dfrac{e^{i\frac{\pi}{8}}}{\sqrt{2}}|1_{A}\rangle|\theta^{+}_{BC}\rangle$
(11)

where
$|\psi^{-}_{BC}\rangle=\left(|01_{BC}\rangle-|10_{BC}\rangle\right)/\sqrt{2}$
and
$|\theta^{+}_{BC}\rangle=\left(|01_{BC}\rangle+i|10_{BC}\rangle\right)/\sqrt{2}$.
$A$ is associated the measurement $\sigma_{x}$, $B$ is associated a
measurement dependending on the outcome $a$,
$\frac{\sigma_{x}+(-1)^{a}\sigma_{y}}{\sqrt{2}}$, and $C$ depending on $b$,
$\frac{\sigma_{x}+(-1)^{b}\sigma_{y}}{\sqrt{2}}$. We prove that the inequality

$\displaystyle D:=$ $\displaystyle P_{ABC}(1,0,0)+P_{ABC}(1,1,1)-P_{AC}(0,0)+$
(12) $\displaystyle P_{AC}(0,0|do(B=0))+P_{BC}(b\neq c|do(A=1))\leq 1$

is valid for all classically compatible data tables, and using the outlined
protocol we find $D^{Q}=1+\frac{2-\sqrt{2}}{16\sqrt{2}}$ reaching a violation
of $\beta_{Q}=\frac{2-\sqrt{2}}{16\sqrt{2}}\approx 0.025888$. Furthermore, to
prove that this protocol is pairwise compatible we show that it respects
specific inequality constraints shown in Appendix B of [1].

Discussion—The generalization of Bell’s theorem to other causal structures has
sparked new forms and new applications of non-classical behavior. However, a
central concept in causality theory - that of an intervention - remains
significantly unexplored. Unlike observations, an intervention locally changes
the underlying causal relations, erasing all causes acting on the variable we
intervene upon. We show that non-classicality considering this new data regime
of passive observations and interventions jointly is a generic feature. To
exemplify this, we prove here the existence of quantum non-classicality from
data fusion in three completely saturated scenarios, i.e. when all probability
distributions over three observable variables $P_{ABC}$ admit a classical
explanation for any cardinality of the variables. However, when the
observations $P_{ABC}$ are considered jointly with interventions we can prove
the existence of quantum non-classicality. Furthermore, we show that more
refined notions of non-classicality from data fusion are achievable with
quantum resources, where the violations are genuine to multiple interventions.

Finally, we expand our investigation in a companion article [1] for all latent
exogenous causal structures with three observed variables. We show
interventions represent a potent new tool for comprehending and observing non-
classical behavior and leave their potential practical applications to
information processing, like e.g. randomness extraction [37] and communication
complexity [38], for future work. Furthermore, we have focused on the scenario
where all the observed variables are classical but generalizations where
observed variables are made quantum, moving to the paradigm of quantum
steering [39] open an interesting venue for future work. For instance, the
teleportation protocol [40], remote state preparation [41], and dense coding
[42], which have an underlying instrumental causal structure, could be
generalized to new network scenarios with quantum communication. We hope our
work might trigger further developments in these directions.

## Acknowledgements

This work was supported by the São Paulo Research Foundation FAPESP (Grant No.
2022/03792-4). Research at Perimeter Institute is supported by the Government
of Canada through the Department of Innovation, Science and Economic
Development Canada and by the Province of Ontario through the Ministry of
Research, Innovation and Science.

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