Quantum Inflation: A General Approach to Quantum Causal Compatibility

Abstract

Causality is a seminal concept in science: any research discipline, from sociology and medicine to physics and chemistry, aims at understanding the causes that could explain the correlations observed among some measured variables. While several methods exist to characterize classical causal models, no general construction is known for the quantum case. In this work we present quantum inflation, a systematic technique to falsify if a given quantum causal model is compatible with some observed correlations. We demonstrate the power of the technique by reproducing known results and solving open problems for some paradigmatic examples of causal networks. Our results may find an application in many fields: from the characterization of correlations in quantum networks to the study of quantum effects in thermodynamic and biological processes.

Full text

Quantum Inflation: A General Approach to Quantum Causal Compatibility
Elie Wolfe,
1
Alejandro Pozas-Kerstjens,
2
Matan Grinberg,
3
Denis Rosset,
4
Antonio Ac´ın,
2, 5
and Miguel Navascu´es
6
1
Perimeter Institute for Theoretical Physics, N2L 2Y5 Waterloo, Canada
2
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
3
Princeton University, Princeton, NJ USA 08544
4
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5
5
ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
6
Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria
Causality is a seminal concept in science: any research discipline, from sociology and medicine to
physics and chemistry, aims at understanding the causes that could explain the correlations observed
among some measured variables. While several methods exist to characterize classical causal models,
no general construction is known for the quantum case. In this work we present quantum inflation,
a systematic technique to falsify if a given quantum causal model is compatible with some observed
correlations. We demonstrate the power of the technique by reproducing known results and solving
open problems for some paradigmatic examples of causal networks. Our results may find an application
in many fields: from the characterization of correlations in quantum networks to the study of quantum
effects in thermodynamic and biological processes.
I. INTRODUCTION
Causality is an ubiquitous concept in science. It can be
argued that one of the main challenges in any scientific
discipline is to identify which causes are behind the
correlations observed among some measured variables,
encapsulated by their joint probability distribution.
Understanding this problem is crucial in many situa-
tions, such as, for example, the development of medical
treatments, taking data-based social policy decisions, the
design of new materials or the theoretical modeling of
experiments. More precise characterizations of causal
correlations enable better decision among competing
explanation for given statistics, a task known as
causal
discovery
. Advances in causal understanding also enables
quantification of causal effects from purely observational
data, thus extracting counterfactual conclusions even in
instances where randomized or controlled trials are not
feasible, a task known as causal inference [1–3].
Bayesian causal networks, in the form of directed
acyclic graphs (DAGs), provide the tools to formalise
such problems. These graphs, examples of which are
shown in Fig. 1, encode the causal relations between
the various variables in the problem, which could be
either observed or non-observed. The latter, also known
as latent, are required in many relevant situations in
order to explain correlations among the observed. The
fundamental task addressed in this work underlies both
causal discovery and causal inference, and is known as the
causal compatibility
problem. It consists of deciding
whether a given joint probability distribution over some
observed variables can be explained by a given candidate
Bayesian causal network. Equivalently, the objective of
causal compatibility can be viewed as characterizing the set
of distributions compatible with a given Bayesian network.
In all cases, the measured variables in a causal network
are, by definition, classical. However, causal networks may
be classical or quantum depending on whether correlations
are established by means of classical or quantum informa-
tion. Because of its importance and broad range of appli-
cations, there is a vast literature devoted to understanding
the problem of causal compatibility for classical causal net-
works, see for instance Ref. [
1
]. On the contrary, very little
is known for the quantum case despite the factthat nature is
ultimately quantum and quantum effects are expected to be
crucial for the understanding of many relevant phenomena
in many scientific disciplines. Moreover the two problems
are known to be different, as one of the consequences of
Bell’s theorem [4,5] is that quantum causal networks can
explain correlations for which the analogous classical net-
work fails [
6
–
10
]. Our work addresses these issues and
provides a systematic construction to tackle the problem
of causal compatibility for quantum causal networks.
As mentioned, several results already exist in the classi-
cal case. Whenever the network does not contain any latent
variable the solution is rather simple and it suffices to check
whether all the conditional independences associated to the
network topology are satisfied [
1
]. The problem, however,
becomes much more difficult as soon as the network also
includes latent variables, as their presence generally implies
non-trivial inequalities on the observed probabilities. A
general method to tackle the causal compatibility problem,
known as the
inflation technique
, was obtained in [
11
]. It
consists of a hierarchy of conditions, organised according to
their computational cost, that are necessary for a Bayesian
network to be be able to explain the observed correlations.
Moreover, the hierarchy is asymptotically sufficient, in the
sense that the candidate Bayesian network is compatible if,
and only if, all conditions in the hierarchy are satisfied [
12
].
When moving to quantum causal scenarios, the problem
of causal compatibility presents several new features. In the
classical case, the cardinality of the latent variables can be
upper bounded and, therefore, the problem is decidable. In
the quantum case, however, a similar upper bound cannot
exist because the problem of quantum causal compatibility
is undecidable, as implied by recent results on quantum
correlations [
13
]. Yet, this does not preclude the existence
of a method similar to inflation to tackle the question.
arXiv:1909.10519v1 [quant-ph] 23 Sep 2019

2
B
U
A
X Y
(a)
A
BC ρBC
ρAC ρAB
(b)
UBC
B C
UAB
A
X Y Z
(c)
A
B C
X
Y
UAS
UBC
S
(d)
FIG. 1. DAG representation of different causal scenarios. The red, dashed circles are latent variables, and the yellow, single-lined
circles denote visible variables. (a) The Bell scenario is one of the simplest causal structures exhibiting a classical-quantum gap,
that is, where there exist distributions that can be realized with quantum latent variables (denoted as
U=ρ
), but not with classical
ones (denoted as
U=λ
). (b) The triangle scenario also presents a classical-quantum gap. Alternatively, the triangle scenario can
also be defined with one visible variable (called measurement choice) influencing each of
A
,
B
and
C
. (c) The tripartite-line causal
scenario, where two causally independent parties
A
and
C
share each some resource with a central party
B
. (d) Arbitrary causal
structures contain directed edges beyond the traditional network connections of latent-to-terminal edges and root-to-terminal
edges. A method for analyzing correlations in general structures is given in Section IV.
Unfortunately, the inflation technique cannot be straight-
forwardly adapted to the quantum case because it relies on
information broadcasting, a primitive that is not plausible
with quantum information [14,15]. Other causal analysis
techniques which are fundamentally quantum have been
proposed. Notably among these is the quantum entropy
vector approach of [
16
], which is applicable to all causal
structures but uses only those constraints on entropies im-
posed by the causal structure, or the scalar extension of [
17
],
which imposes stronger constraints but cannot be applied
to causal structures in which all observed nodes are causally
connected, such as the triangle scenario of Fig. 1(b).
The main result of our work is the construction of
quantum inflation
, a systematic technique to study
causal compatibility in any quantum Bayesian network. It
can be seen as a quantum analogue of the classical inflation
technique which avoids the latter’s reliance on information
broadcast. We first explain in Sec. II how quantum
inflation works by means of an example, and provide the
details of the construction in Sec. III. Then, we describe
how to address quantum causal compatibility in complex
scenarios in Sec. IV. Also, quantum inflation can serve as a
tool for assessing causal compatiblity with classical models.
We detail this construction in Sec. V. Finally, in Sec. VI
we apply quantum inflation to characterize correlations
achievable in various tripartite quantum causal networks,
and after that we conclude in Sec. VII.
II. QUANTUM INFLATION BY EXAMPLE
Consider the quantum causal network depicted in
Fig. 2(a), whereby three random variables
a
,
b
,
c
are
generated by conducting bipartite measurements over the
ends of three bipartite quantum states
ρAB
,
ρBC
,
ρAC
.
We are handed the distribution
Pobs
(
a,b,c
) of observed
variables and asked if it is compatible with this model.
How to proceed?
Suppose that there existed indeed bipartite states
ρAB , ρBC , ρAC
of systems
A0B0, B00 C0, A00C00
, and com-
muting measurement operators
Ea, Fb, Gc
, acting on
systems
A0A00,B0B00,C0C00
, respectively, which were able
to reproduce the correlations
Pobs
(
a,b, c
). Now imagine
how the scenario would change if
n
independent copies
ρi
AB ,ρi
BC ,ρi
AC
,
i=1,...,n
of each of the original states were
distributed instead, as depicted in Fig. 2(b). Call
ρ
the over-
all quantum state before any measurement is carried out.
For any
i,j
=1
,...,n
we can, in principle, implement measure-
ment
{Ea}a
on the
ith
copy of
ρAC
and the
jth
copy of
ρAB
:
we denote by {Ei,j
a}athe corresponding measurement op-
erators. Similarly, call
{Fi,j
b}b
, (
{Gi,j
c}c
) the measurement
{Fb}b({Gc}c) over the states ρi
AB ,ρj
BC (ρi
BC ,ρj
AC ).
The newly defined operators and their averages under
state
ρ
satisfy non-trivial relations. For example, for
H
=
E,F,G
and
i6
=
k,j 6
=
l
the operators
Hi,j
a
and
Hk,l
b
act on dif-
ferent Hilbert spaces, and hence
hHi,j
a,Hk,l
bi=0
. Similarly,
expressions such as
DE1,1
aE1,2
a0F2,2
bEρ
and
DE1,2
aE1,1
a0F1,2
bEρ
can be shown identical, since one can arrive at the second
one from the first one just by exchanging
ρ1
AB
with
ρ2
AB
.
More generally, for any function
Q
(
{Ei,j
a,F k,l
b,Gm,n
c}
) of
the measurement operators and any three permutations
π,π0,π00 of the indices 1,...,n, one should have
Q({Ei,j
a,F k,l
b,Gm,n
c})ρ
=DQ({Eπ(i),π0(j)
a,F π0(k),π00(l)
b,Gπ00(m),π(n)
c})Eρ.(1)
Finally, note that, if we conducted the measurements
{Ei,i, F i,i, Gi,i }n
i=1
at the same time (something we can
do, as they all commute with each other), then the
measurement outcomes
a1,...,an,b1,...,bn,c1,...,cn
would
be distributed according to
*n
Y
i=1
Ei,i
aiFi,i
biGi,i
ci+ρ
=
n
Y
i=1
Pobs(ai,bi,ci).(2)

3
A
BC ρBC
ρAC ρAB
=
A0
A00
Ea
C00
A00 A0
B0
B00
C0
ú
(a)
A
BC
ρBC
2
ρAC
2ρAB
2
ρBC
1
ρAC
1ρAB
1
=E1,1
a
(b)
FIG. 2. Quantum inflation in the triangle scenario. (a) In the
original scenario, by probing systems
A0,A00
with the quantum
measurement
{Ea}
, a value
a
for the random variable
A
is
generated. The values
b,c
for the random variables
B
and
C
are
produced similarly. (b) In quantum inflation, we distribute
n
(in
the case shown,
n=
2) independent copies of the same states to
the parties, which now use the original measurement operators
on different pairs of copies of the states they receive. For
instance, the measurement operators
{E1,1
a}a
act on the states
corresponding to copies
ρ1
AB
and
ρ1
AC
, and the measurements
with other superindices are defined in an analogous way.
If the original distribution
Pobs
(
a,b,c
) is compatible with
the network in Fig. 2(a), then there should exist a Hilbert
space
H
, a state
ρ
:
H → H
and operators
{Ei,j
a}i,j,a
,
{Fk,l
b}k,l,b
,
{Gm,n
c}m,n,c
satisfying the above relations.
If such is the case, we say that
Pobs
(
a, b, c
) admits an
nth-order quantum inflation
. By increasing the index
of
n
, we arrive at a hierarchy of conditions, each of
which must be satisfied by any compatible distribution
Pobs(a,b,c).
At first glance, disproving the existence of a quantum
inflation looks as difficult as the original feasibility prob-
lem. However, the former task can be tackled via non-
commutative polynomial optimization (NPO) theory [
18
].
Originally developed to characterize quantum nonlocal-
ity [
19
,
20
], the general goal of NPO theory is to optimize the
expectation value of a polynomial over operators subject
to a number of polynomial operator and statistical con-
straints. This is achieved by means of a hierarchy of semidef-
inite programming tests [
21
], see also Appendix A. In our
particular case, we are dealing with a feasibility problem.
The polynomial operator constraints we wish to enforce on
Ei,j
a,F k,l
b,Gm,n
c
are that they define complete families of pro-
jectors, which commute when acting on different quantum
systems. The statistical constraints are given by Eqs. (1-2).
If for some
n
we were able to certify, via NPO theory, that
Pobs
(
a,b,c
) does not admit an
nth
-order quantum inflation,
then we would have proven that
Pobs
(
a,b,c
) does not admit
a realization in the quantum network of Fig. 2(a).
The method just described can be easily adapted to
bound the statistics of any network in which the observed
variables are defined by measurements on the quantum
latent variables, such as the triangle scenario. To test the
incompatibility of a distribution
Pobs
, we would consider
a modified network with
n
copies of each of the latent vari-
ables, extend the original measurement operators to act on
all possible copies of each system and work out how operator
averages relate to
Pobs
and to each other. Finally, we would
use NPO theory to disprove the existence of a state and op-
erators satisfying the inferred constraints. In Section IV we
further show how to extend the notion of quantum inflation
to prove infeasibility in general quantum causal structures,
where there might be causal connections among observed
variables, as well as from observed to latent variables.
III. DETAILED DESCRIPTION
To illustrate the details of the construction, we first
consider a subset of causal scenarios in which single
measurements are applied to different quantum states.
They correspond to two-layer DAG’s in which arrows
coming from a first layer, consisting in both observed and
latent variables, go to a second layer of observed variables.
Each of the variables in the second layer is regarded as
an outcome variable, since it is the result of conducting
a measurement on a quantum state. The set of all classical
observed parents of such a variable can be understood as
the measurement setting used to produce this outcome.
The essential premise of quantum inflation is to ask what
would happen if multiple copies of the original (unspecified)
quantum states were simultaneously available to each party.
In this gedankenexperiment the parties use copies of their
original measurement apparatus to perform
n
simultaneous
measurements on the
n
copies of the original quantum states
now available to them. There are different ways in which a
party can align her measurements to act on the states now
available, thus we must explicitlyspecify upon which unique
set of Hilbert spaces a given measurement operator acts non-
trivially. Let us therefore denote measurement operators by
ˆ
Os|k
i|m≡ˆ
OOutcome variable=k,Spaces=s,
Setting=m,Outcome=i,
where the four indices specify
1. k
, the index or name of the outcome variable in the
original causal graph,
2. s, the Hilbert spaces the given operator acts on,
3. i, the measurement setting being used,
4. m, the outcome associated with the operator.
For example, using an
n=
2 quantum inflation of the
triangle scenario (see Fig. 2) one would find that
s
for
outcome variable
k=A
may be sampled from precisely
four possibilities, each value being a different tuple:
s∈{A0
1,A00
1},{A0
1,A00
2},{A0
2,A00
1},{A0
2,A00
2},.
where
A0
i
(
A00
i
) denotes the factor
A
of the Hilbert space
where ρi
AB (ρi
AC ) acts.
These operators will be regarded as the non-commuting
variables of an NPO problem where the polynomial con-
straints are derived according to rules pertaining to the op-

4
erators’ projective nature and as well as a number of commu-
tation rules. The statistical constraints are then imposed
from symmetry under permutations of the state indices
and enforcing consistency with the observed probabilities.
Projection rules
For fixed
s,k,m
, the non-commuting variables
{ˆ
Os|k
i|m}i
must correspond to a complete set of measurement
operators. Since we do not restrict the dimensionality of
the Hilbert space where they act, we can take them to be
a complete set of projectors. That is, they must obey the
relations
ˆ
Os|k
i|m=( ˆ
Os|k
i|m)†,ˆ
Os|k
i|mˆ
Os|k
i0|m=δii0ˆ
Os|k
i|m,∀s,k,i,i0,m (3a)
Xi
ˆ
Os|k
i|m=1,for all s,k,m. (3b)
These relations imply, in turn, that each of the non-
commuting variables is a bounded operator. Hence, by
[
18
], the hierarchy of SDP programs provided by NPO is
complete, i.e., if the said distribution does not admit an
nth
-order inflation, then one of the NPO SDP relaxations
will detect its infeasibility.
Commutation rules
Operators acting on different Hilbert spaces must
commute. More formally,
hˆ
Os1|k1
i1|m1,ˆ
Os2|k2
i2|m2i=0 if s1∩s2=∅.(4)
Later on we consider modifying these commutation rules
so as to construct an alternative SDP for constraining the
correlations of classical causal structures.
Symmetry under permutations of the indices
The critical ingredient that relates the inflated network
structure to the original network is that all averages of
products of the noncommuting variables must be invariant
under any permutation
π
of the source indices. Call
ρ
the
overall quantum state of the inflated network (since we do
not cap the Hilbert space dimension, we can assume that
all state preparations in the original network are pure).
Then we have that
ˆ
Os1|k1
i1|m1·ˆ
Os2|k2
i2|m2·...·ˆ
Osn|kn
in|mnρ
=Dˆ
Oπ(s1)|k1
i1|m1·ˆ
Oπ(s2)|k2
i2|m2·...·ˆ
Oπ(sn)|kn
in|mnEρ.(5)
An example of such statistical constraints imposed in the
triangle scenario was given in Eq.
(1)
. Another example,
for an inflation level n=3, is the following:
DE{A0
1,A00
1}
0E{A0
2,A00
2}
1F{B00
3,B0
1}
0G{C00
1,C0
2}
0Eρ
apply ρ1
AB↔ρ2
AB
=DE{A0
2,A00
1}
0E{A0
1,A00
2}
1F{B00
3,B0
2}
0G{C00
1,C0
2}
0Eρ(6a)
apply ρ1
AB↔ρ3
AB
=DE{A0
2,A00
1}
0E{A0
3,A00
2}
1F{B00
3,B0
2}
0G{C00
1,C0
2}
0Eρ(6b)
apply ρ1
BC ↔ρ3
BC
=DE{A0
2,A00
1}
0E{A0
3,A00
2}
1F{B00
1,B0
2}
0G{C00
1,C0
2}
0Eρ,(6c)
where, for readability, we identified
ˆ
Os|A
a|∅
(
ˆ
Os|B
b|∅
) [
ˆ
Os|C
c|∅
]
with Es
a(Fs
b) [Gs
c].
Consistency with the observed probabilities
Finally, as described by Eq.
(2)
in Section II, certain
product averages can be related to products of the probabil-
ities of the distribution
Pobs
that one wishes to test. More
specifically, let
n
be the order of the considered inflation,
and denote by
~
jk
the set of Hilbert spaces where variable
k
acts, with the copy labels of all Hilbert spaces equal to
j∈{
1
,...,n}
(e.g.:
~
1A
=
{A0
1,A00
1}
,
~
2A
=
{A0
2,A00
2}
in Fig. 2).
Then, for any set of indices {ij,k,mj,k }j,k, we have that
*n
Y
j=1Y
k
ˆ
O~
jk|k
ij,k|mj,k +ρ
=
n
Y
j=1
Pobs \
k
(ij,k|mj,k ,k)!,(7)
where (i|m,k) denotes the event of probing kwith setting
mand obtaining the result i.
IV. ARBITRARY CAUSAL SCENARIOS
In the previous section we provided a systematic method
to characterize the correlations achievable in a subset of
quantum causal networks, namely two-layer DAG’s where
no node has both parents and children. In this section we
extend those ideas to characterize correlations in arbitrary
causal structures.
A. Classical Exogenous Variables
The first important case that must be dealt with is that
of the so-called exogenized causal structures, where all
unobserved nodes are root nodes but otherwise classical
variables can have both parents and children. We address
the case of arbitrary, non-exogenized causal structures in
the next section.

5
X
A B
ρ
(a)
X
A B
ρA
(b)
FIG. 3. Example of interruption of generic causal struc-
tures. (a) The instrumental causal structure, and (b) its
interruption. Note that (a) the random variable
A
has both
parents and children. The constraint to impose in order
for the interruption to behave as the original scenario is
P(a)(A=a,B=b|X=x)= P(b)(A=a,B =b|X=x,A#=a).
There exists a procedure that can be used to map the
correlations of any exogenized causal structure to the cor-
relations of a unique two-layer DAG associated with the
original structure. We call such procedure
interruption
.
Graphically, interruption modifies a graph as follows: For
every observed variable
Ai
which is neither a root node nor
a terminal node, introduce a new variable
A#
i
and replace
all edges formerly originating from
Ai
by edges originating
from
A#
i
. In the interruption graph,
Ai
becomes a terminal
node and
A#
i
is a root node. Proceeding in this fashion,
any causal structure can be converted into a two-layer sce-
nario. Graphical examples of interruption are shown in
Fig. 3. Conceptually, interruption has extensive precedent
in literature regarding classical causal analysis. It is closely
related to the Single-World Intervention Graphs (SWIGS)
pioneered by [
22
], as well as the
e
-separation technique
introduced in [
23
]. Interruption previously has been used
to show Tsirelson inequalities constraining the set of quan-
tum correlations compatible with the Bell scenario can be
ported to quantum constraints pertaining to the Instrumen-
tal scenario [
24
], see also the proof of Theorem 25 in Ref. [
7
].
B. Quantum Exogenous Variables
Thanks to Evan’s exogenization procedure [
25
], classical
non-exogenous structures can be transformed into
exogenous causal structures with the same predictive
power. The procedure consists in replacing all arrows from
a parent node to a latent node with arrows from the parent
node to the children of the latent node. This operation is
repeated for all parents of all latent nodes such that finally
all latent variables become parentless.
Unfortunately, when applied to quantum latent
variables, exogenization results in a new quantum network
that, in general, does not predict the same distributions of
observed events as its predecessor. The example in Fig. 4,
evidencing this compatibility mismatch, is wholly due to
Stefano Pironio.
To make this explicit, in Fig. 4(a) the variable
S
can
serve as a setting, which adjusts the state
ρBC
before it is
sent to B and C. Thus, it is possible for
P
(
A,B|X,Y ,S=
0)
A
B C
X
Y Z
ρAS
ρBC
S
(a)
A B C
X Y ZρS
(b)
FIG. 4. In (a) there is a causal structure with
ρBC
being a non-
exogenous unobserved quantum node. In (b) there is a different
causal structure, corresponding to the classical latent reduction
of the former. While these two graphs would be equivalent if
the unobserved nodes were classical, they are demonstrably
inequivalent when the unobserved nodes are quantum.
to maximally violate a Bell inequality for A and B, and
P
(
A,C|X,Z,S=
1) to maximally violate a Bell inequality
for A and C. No quantum state prepared independently
of
S
can do so, due to the monogamy of quantum correla-
tions [
26
,
27
]. Consequently, it is not possible to reproduce
such correlations within the causal network of Fig. 4(b).
One way to deal with this predicament is to regard
observable variables with unobserved children as random
variables indicating the classical control of a quantum
channel. Thus, in Fig. 4(a) we treat the root variable
S
as the classical control for a quantum channel acting
on the
BC
subsystems. That is, one understands
ρABC |S=s
=
ˆ
UsρAS ˆ
U†
s
, where, for all values of
s
,
ˆ
Us
is a
unitary operator that commutes with any operator acting
solely on
A
’s subsystem. As such, the joint distribution
of the values of the visible variables
A,B,C
conditioned on
the root visible nodes can be understood as generated by
P(A=a,B=b,C=c|X=x,Y =y,Z =z,S=s)
=Dˆ
U†
sˆ
Aa
xˆ
Bb
yˆ
Cc
zˆ
UsEρAS
=Dˆ
Aa
xˆ
U†
sˆ
Bb
yˆ
Cc
zˆ
UsEρAS
.
This interpretation can be made without loss of generality,
since the subspace
S
of the complete Hilbert space
AS
can
be understood as containing the subspaces corresponding
to Band C.
As in the exogenous case, an
nth
-order inflation of a
causal structure with non-exogenous quantum variables
requires taking
n
copies of the unobserved root nodes.
Each unitary operator
ˆ
Us
in the original causal structure
gives rise to operators of the form
ˆ
O{Sj}|U
s
in the inflated
graph, where
j
denotes the copy of the Hilbert space where
ˆ
Us
acts. The unitary or outcome operators associated
to the descendants of any such “unitary node” (
B,C
, in
Fig. 4) inherit the copy label jof the Hilbert space Sj.
With this last prescription, the symmetry relabelling
rule, Eq.
(5)
, still holds. However, the projection rules
(3a-3b) only hold if the non-commuting variable
k
in
question corresponds to an outcome variable in the original
graph. If
k
corresponds to a unitary variable, then the

6
operator ˆ
Os|k
mmust be subject to the constraints
ˆ
Os|k
m(ˆ
Os|k
m)†=( ˆ
Os|k
m)†ˆ
Os|k
m=1.(8)
The commutation rule
(4)
remains valid upon qualifying
that the Hilbert spaces listed in
s1∪s2
must be simulta-
neously co-existing in the original graph. For example, in
Fig. 4(a), the operators corresponding to the Hilbert spaces
associated to the outcome variables
B
and
C
co-exist after
the transformation
Us
is applied over system
S
. It follows
that the corresponding measurement operators
Os|B
b|y
,
Os
0|C
c|z
commute. Finally, rule
(7)
expressing consistency with the
observed probabilities must also be amended, to take into
account that descendants of a unitary variable must be
bracketed by the corresponding unitary and its adjoint.
Note that the aforementioned operator and statistical
constraints are all polynomial, and thus they can all be
enforced in the framework of NPO theory.
Interruption, classical exogenization, and the treatment
for quantum exogenous variables hereby presented cover
all possible nontrivial causal influences in arbitrary
quantum causal structures. Quantum inflation is therefore
a technique of full applicability to bound the quantum
correlations achievable in any causal scenario.
V. SDP FOR CLASSICAL COMPATIBILITY
The quantum inflation technique can be easily adapted
for solving the problem of causal compatibility with an
arbitrary classical causal structure. It is known that any
correlation achievable with only classical sources can be
realized in terms of commuting measurements acting
on a quantum state [
28
]. Therefore, in order to detect
correlations incompatible with classical networks, one
must generalize the commutation relations in the original
quantum inflation method to the constraint that any pair
of measurement operators commute. That is,
ˆ
Os1|k1
i1|m1·ˆ
Os2|k2
i2|m2=ˆ
Os2|k2
i2|m2·ˆ
Os1|k1
i1|m1,
for all
s1,s2,k1,k2,m1,m2,i1,i2
. This defines a hierarchy
of constraints that classical correlations compatible with
a given causal structure must satisfy.
The associated NPO hierarchy is guaranteed to converge
at a finite level. In fact, for hierarchy levels higher
than
N·m·
(
d−
1)—where
N
is the number of parties,
m
is the number of settings per party and
d
is the
number of outcomes per measurement—application of the
commutation relations allows one to reduce any product
of the operators involved into one of shorter length. For a
fixed inflation level, the problem solved at the highest level
of the NPO hierarchy is analogous to the linear program
solved in classical inflation [
11
] at the same inflation level.
In contrast with the original classical inflation technique,
the classical variant of the quantum inflation technique
described in this article uses semidefinite programming,
and exhibits far more efficient scaling with the inflation hi-
erarchy than the original linearprogramming approach [
12
].
One must note that this gain in efficiency comes at the
expense of introducing further relaxations in the problem.
Nevertheless, this classical variant of quantum inflation
is capable of recovering a variety of seminal results of
classical causal compatibility, such as the incompatibility
of the W and GHZ distributions with classical realizations
in the triangle scenario. It also identifies the distribution
described by Fritz [
6
], known to have a quantum realization
in the triangle scenario, as incompatible with classical
realizations. For all these results, the relaxed SDP
formulation is far less memory-demanding than the raw
linear programming formulation. Furthermore, the SDP
approach is the only method that can be used when using
inflation to assess causal compatibility in the presence of
terminal nodes which can take continuous values.
In conclusion, not only can quantum inflation be
leveraged to obtain results for networks with classical
sources, but we argue that it is the most suitable technique
to be used for addressing causal compatibility with
classical realizations in large networks.
VI. RESULTS
In the following we demonstrate the power of quantum
inflation by reproducing known results and solving open
problems in different tripartite causal networks.
A. The Triangle Scenario
The triangle scenario consists of three parties that
are influenced in pairs by bipartite latent variables, as
depicted in Fig. 1(b). Additionally, each of the visible
variables may be influenced by another visible variable,
that represent a measurement choice.
The first example we study in this scenario is the
so-called W distribution. This distribution is defined by
the task of all parties outputting the outcome 0 except
one, which should output 1. Explicitly, it is
PW(a,b,c):=(1
3a+b+c=1
0 otherwise .(9)
The W distribution was proven in [
11
] not to be realizable
in the triangle scenario where the latent variables are
classical. Additionally, it is easy to see that it is realizable
with tripartite classical randomness. It can be shown
that
PW
does not admit a second-order quantum inflation.
Therefore, a quantum realization of the W distribution
in the triangle scenario is impossible.
Quantum inflation is robust, and certifies that when
mixing the W distribution with white noise, the resulting
distribution
PW,v
(
a,b,c
)
:
=
vPW
(
a,b,c
)+(1
−v
)
/
8 does not
have a quantum realization in the triangle for all visibilities
v
higher than 3(2
−√3
)
≈
0
.
8039. This result is obtained by
solving the NPO program associated to a second-order in-

7
A
X
B
Y
C
ZρBC
ρAC ρAB
FIG. 5. The quantum triangle scenario with settings. Each
of the visible variables
A
,
B
and
C
is now influenced not only
by the latent variables, but from additional visible variables
X,Y,Zthat represent measurement choices.
flation and the set of monomials
L2
(see Appendix Bfor the
definition of this set), restricted to operators of length
≤
3.
B. The Triangle Scenario with Settings
As mentioned before, one can also consider more
complicated networks that include additional observable
variables to encode for choices of discrete measurement
settings. Fig. 5shows this type of network for the case of the
triangle scenario. In this setup we study the Mermin-GHZ
distribution, defined by PMermin(a,b,c|x,y,z):=





1/8x+y+z=0 mod2,
1+(−1)a+b+c/8x+y+z= 1,
1−(−1)a+b+c/8x+y+z=3.
(10)
Quantum inflation also allows one to prove that the
Mermin-GHZ distribution is not compatible with a
quantum realization in the triangle scenario with inputs,
by showing that
PMermin
does not admit a second-order
quantum inflation. Additionally, its noisy version
PMermin,v
can be proven not to have a quantum realization
for any visibility vhigher than p2/3≈0.8165.
C. The Tripartite-Line Scenario
Quantum inflation is organised as an infinite hierarchy
of necessary conditions. There are however situations in
which it recovers the quantum boundary at a finite step.
An example of these situations is provided by the tripartite-
line scenario of Fig. 1(c), which underlies phenomena such
as entanglement swapping. The main characteristic of this
structure is that there is no causal connection between the
extreme variables
A
and
C
. As a consequence of this, all
correlations realizable in the tripartite-line scenario satisfy
the following factorization relation
X
b
Pobs(a,b,c|x,y,z) = Pobs(a|x)Pobs (c|z).(11)
∈ BQ
∈ BL
v
01
40.328 1
21
FIG. 6. Summary of results recoverable with quantum inflation
in the tripartite-line scenario. Quantum inflation correctly
recovers that all
P2PR,v
with visibility
v>
1
/
2 are incompatible
with the quantum tripartite line scenario already at the NPO
hierarchy finite level
S2
when assessing compatibility with a
second-order inflation. When imposing that all measurements
commute, it witnesses that all
P2PR,v
with visibility
v>
0
.
328
cannot be realized in terms of classical hidden variables.
This scenario has been thoroughly studied in the
literature [
29
,
30
]. In fact, it is known that the probability
distribution
P2PR(a,b,c|x,y,z):=1+(−1)a+b+c+xy+yz /8,(12)
despite satisfying the constraint of Eq.
(11)
, cannot be
realized in the tripartite-line scenario in terms of classical
or quantum latent variables. However, it is known that
its mixture with white noise,
P2PR,v :=vP2PR+ (1−v)/8
,
can be realized if the visibility parameter
v
is sufficiently
small [
30
].
P2PR,v
admits a realization in terms of quantum
latent variables for any 0
≤v≤
1
/
2, and in terms of classical
latent variables for any 0≤v≤1/4.
Quantum inflation correctly recovers that all
P2PR,v
with visibility
v>
1
/
2 are incompatible with the quantum
tripartite line scenario. It does so by certifying that for
any
v >
1
/
2, the corresponding
P2PR,v
does not admit
a second-order inflation, and this infeasibility is found
already at the NPO hierarchy level corresponding to the set
S2
(see Appendix Bfor a definition of this monomial set).
Furthermore, we can also contrast against realizations in
terms of classical latent variables by imposing that all mea-
surements in the problem commute. While not being tight,
this classical version of quantum inflation witnesses that all
P2PR,v
with visibility
v>
0
.
328 cannot be realized in terms
of classical hidden variables. This result is obtained when
analyzing compatibility with a third-order inflation, solv-
ing the NPO problem associated to the corresponding set of
monomials
L1
(its definition can be found in Appendix B).
VII. CONCLUSIONS
We introduced the quantum inflation technique, a
systematic method to discern whether an observable distri-
bution is compatible with a causal explanation involving
quantum degrees of freedom. The technique is of general
applicability, in that it can be employed to analyze corre-
lations achievable by any quantum causal structure with,
potentially, visible-to-visible, latent-to-visible, visible-to-
latent or latent-to-latent connections. Furthermore, we

8
discussed how a slight modification allows also to study
causal realizations in terms of classical latent variables.
We used quantum inflation to study realizations of
famous tripartite distributions in different causal struc-
tures, proving that the W and Mermin-GHZ distributions
cannot be generated in the triangle scenario with quantum
latent variables and bounding their noise resistance. We
also showed, in the entanglement swapping scenario, how
quantum inflation is capable of recovering known results.
The implementation of quantum inflation comprises
two different hierarchies: the one of inflations, and for
each inflation, the NPO hierarchy used to determine
whether a distribution admits such an inflation. While
asymptotic convergence has been proven for the latter,
that of the former is an open question. Nevertheless, we
have identified situations in which tight results can be
obtained at finite steps of the hierarchies.
Quantum inflation can find an application in many
fields. A clear first application is found in multi-party
quantum information protocols [
31
]. From a more general
perspective, and due to the central role that causality
has in science, we expect quantum inflation to become a
fundamental tool for analyzing causality in any situation
where a quantum behavior is presumed.
ACKNOWLEDGMENTS
We acknowledge useful discussions with Stefano Pironio.
This work is supported by Fundaci´o Obra Social “la Caixa”
(LCF/BQ/ES15/10360001), the ERC CoG QITBOX,
the AXA Chair in Quantum Information Science, the
Spanish MINECO (QIBEQI FIS2016-80773-P and Severo
Ochoa SEV-2015-0522), Fundaci´o Cellex, Generalitat de
Catalunya (SGR 1381 and CERCA Programme) and the
Austrian Science fund (FWF) stand-alone project P 30947.
This research was supported by Perimeter Institute for
Theoretical Physics. Research at Perimeter Institute is
supported in part 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 Economic Development, Job
Creation and Trade. This publication was made possible
through the support of a grant from the John Templeton
Foundation. The opinions expressed in this publication
are those of the authors and do not necessarily reflect the
views of the John Templeton Foundation.
[1]
J. Pearl, Causality: Models, Reasoning, and Inference
(Cambridge University Press, 2009).
[2]
S. Morgan and C. Winship, Counterfactuals and Causal
Inference: Methods and Principles for Social Research
(Cambridge University Press, 2007).
[3]
I. Shpitser and J. Pearl, “Complete Identification Methods
for the Causal Hierarchy,” J. Mach. Learn. Res.
9
, 1941
(2008).
[4]
J. S. Bell, “On the Problem of Hidden Variables in
Quantum Mechanics,” Rev. Mod. Phys. 38, 447 (1966).
[5]
N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and
S. Wehner, “Bell nonlocality,” Rev. Mod. Phys.
86
, 419
(2014).
[6]
T. Fritz, “Beyond Bell’s theorem: correlation scenarios,”
New J. Phys. 14, 103001 (2012).
[7]
J. Henson, R. Lal, and M. F. Pusey, “Theory-independent
limits on correlations from generalized Bayesian networks,”
New J. Phys. 16, 113043 (2014).
[8]
C. J. Wood and R. W. Spekkens, “The lesson of
causal discovery algorithms for quantum correlations:
causal explanations of Bell-inequality violations require
fine-tuning,” New J. Phys. 17, 033002 (2015).
[9]
R. Chaves, R. Kueng, J. Bohr Brask, and D. Gross, “Unify-
ing Framework for Relaxations of the Causal Assumptions
in Bell’s Theorem,” Phys. Rev. Lett.
114
, 140403 (2015).
[10]
E. Wolfe, D. Schmid, A. Bel´en Sainz, R. Kunjwal, and
R. W. Spekkens, “Bell Quantified: The Resource Theory
of Nonclassicality of Common-Cause Boxes,” (2019),
arXiv:1903.06311.
[11]
E. Wolfe, R. W. Spekkens, and T. Fritz, “The Inflation
Technique for Causal Inference with Latent Variables,” J.
Causal Inference 7(2019),1609.00672.
[12]
M. Navascu´es and E. Wolfe, “The Inflation Technique
Completely Solves the Causal Compatibility Problem,”
(2017), arXiv:1707.06476.
[13]
W. Slofstra, “The set of quantum correlations is not
closed,” For. Math., Pi 7, e1 (2019).
[14]
H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and
B. Schumacher, “Noncommuting Mixed States Cannot
Be Broadcast,” Phys. Rev. Lett. 76, 2818 (1996).
[15]
H. Barnum, J. Barrett, M. Leifer, and A. Wilce, “Cloning
and Broadcasting in Generic Probabilistic Theories,”
(2006), quant-ph/0611295.
[16]
R. Chaves, C. Majenz, and D. Gross, “Informa-
tion–theoretic implications of quantum causal structures,”
Nat. Comm. 6(2015).
[17]
A. Pozas-Kerstjens, R. Rabelo,
L
. Rudnicki, R. Chaves,
D. Cavalcanti, M. Navascu´es, and A. Ac´ın, “Bounding
the sets of classical and quantum correlations in networks,”
(2019), arXiv:1904.08943.
[18]
S. Pironio, M. Navascu´es, and A. Ac´ın, “Convergent
relaxations of polynomial optimization problems with non-
commuting variables,” SIAM J. Optim. 20, 2157 (2010).
[19]
M. Navascu´es, S. Pironio, and A. Ac´ın, “Bounding the
set of quantum correlations,” Phys. Rev. Lett.
98
, 010401
(2007).
[20]
M. Navascu´es, S. Pironio, and A. Ac´ın, “A convergent
hierarchy of semidefinite programs characterizing the set of
quantum correlations,” New J. Phys. 10, 073013 (2008).
[21]
L. Vandenberghe and S. Boyd, “Semidefinite programming,”
SIAM Rev. 38, 49 (1996).
[22]
T. S. Richardson and J. M. Robins, Single World
Intervention Graphs (SWIGs) : A Unification of the
Counterfactual and Graphical Approaches to Causality
(Now Publishers Inc, 2013) Working Paper #128, Center

9
for Stat. & Soc. Sci., U. Washington.
[23]
R. J. Evans, “Graphical methods for inequality constraints
in marginalized DAGs,” in IEEE International Workshop
on Machine Learning for Signal Processing (2012).
[24]
T. Van Himbeeck, J. Bohr Brask, S. Pironio, R. Ra-
manathan, A. B. Sainz, and E. Wolfe, “Quantum
violations in the Instrumental scenario and their relations
to the Bell scenario,” Quantum 3, 186 (2019).
[25]
R. J. Evans, “Margins of discrete Bayesian networks,”
Annals Stat. 46, 2623 (2018).
[26]
B. Toner, “Monogamy of non-local quantum correlations,”
Proc. Roy. Soc. A 465, 59 (2009).
[27]
M. P. Seevinck, “Monogamy of correlations versus
monogamy of entanglement,” Quant. Info. Proc.
9
, 273
(2010).
[28]
F. Baccari, D. Cavalcanti, P. Wittek, and A. Ac´ın,
“Efficient Device-Independent Entanglement Detection for
Multipartite Systems,” Phys. Rev. X 7, 021042 (2017).
[29]
C. Branciard, N. Gisin, and S. Pironio, “Characterizing
the Nonlocal Correlations Created via Entanglement
Swapping,” Phys. Rev. Lett. 104, 170401 (2010).
[30]
C. Branciard, D. Rosset, N. Gisin, and S. Pironio, “Bilocal
versus nonbilocal correlations in entanglement-swapping
experiments,” Phys. Rev. A 85, 1 (2012).
[31]
E. Wolfe
et al.
, “Classifying and Bounding Non-Classicality
in Quantum Networks,” (in preparation).
[32]
T. Moroder, J.-D. Bancal, Y.-C. Liang, M. Hofmann,
and O. G¨uhne, “Device-Independent Entanglement
Quantification and Related Applications,” Phys. Rev.
Lett. 111, 030501 (2013).
Appendix A: Non-Commutative Polynomial
Optimization
A generic NPO problem can be cast as
p∗= min
(H,X,ρ)hp(X)iρ
such that qi(X)0∀i=1...mq,
that is, finding a Hilb ert space
H
, a positive-semidefinite op-
erator
ρ
:
H→H
with trace one, and a list of bounded oper-
ators
X
=(
X1...Xn
) in
H
(where
XiXj6
=
XjXi
in general)
that minimize the expectation value
hp(X)iρ
=
Tr[ρ·p(X)]
of the polynomial operator
p
(
X
) given some polynomial
constraints
qi
(
X
)

0, where
qi
(
X
)

0 means that the op-
erator
qi
(
X
) should be positive semidefinite. One can also
add to the optimization statistical constraints of the form
hrj(X)iρ≥
0, for
j
=1
,...,mr
. Note that the NPO formal-
ism can also accommodate equality constraints of the form
q
(
X
)= 0 or
hrj(X)iρ
=0, since they are equivalent to the
constraints
q
(
X
)
,−q
(
X
)

0 and
hrj(X)iρ,h−rj(X)iρ≥
0,
respectively. The procedure for solving these problems is
described in Ref. [
18
], and uses a hierarchy of relaxations
where each of the steps is a semidefinite program. The so-
lutions to these problems form a monotonically-increasing
sequence of lower bounds on the global minimum p∗:
p1≤p2≤···≤p∞≤p∗.
If the constraints
{qi
(
X
)

0
}i
imply (explicitly or
implicitly, see Ref. [
18
]) that all non-commuting variables
X1,...,Xn
are bounded (and they do so in all the NPO prob-
lems considered in this work), then the sequence of lower
bounds is asymptotically convergent. That is, p∞=p∗.
In our case, given some observed correlations, we deal
with a feasibility problem about the existence of a quantum
state and measurements subject to polynomial operator
and statistical constraints arising from the causal networks
and the observed correlations. This feasibility problem can
be mapped into an optimization problem in different ways.
For instance, while not being the most practical procedure,
the easiest way of doing it is by considering a constant poly-
nomial
p
(
X
)= 1 as the function to be optimized. This prob-
lem has solution equal to 1 provided that the polynomial
and statistical constraints are simultaneously satisfiable.
Any step in the hierarchy is therefore a necessary SDP
test to be satisfied for the causal model to be compatible
with the observed correlations. Note that the same formal-
ism can be used to optimize polynomials of the operators,
such as, for instance, Bell-like inequalities, over quantum
correlations compatible with a given causal structure [
31
].
Appendix B: Monomial Sets for NPO Problems
The hierarchies of semidefinite programs that bound the
solutions of NPO problems can be described in terms of
sets of products of the non-commutative operators in the
problem. In this article we use two different hierarchies,
that are both asymptotically complete. The levels in the
first hierarchy are known as
NPA levels
[
19
,
20
]. The
NPA level
n
,
Sn
, is associated to the set of all products
of operators in the problem, of length no larger than
n
.
For example, the set
S2
associated to the inflations of the
quantum triangle scenario discussed in the main text is
S2={1}∪{Hi,j
p}p,i,j ∪{Hi,j
pHk,l
q}p,q,i,j,k,l ,
where
H
=
E, F,G
. On the other hand, some problems
may achieve tighter results at lower levels if one instead
considers
local levels
[
32
]. The local level
n
,
Ln
, is built
from the products of operators that contain at most
n
operators of a same party. For instance, in the quantum
triangle scenario, the set L1associated to its inflations is
L1={1}∪{Hi,j
p}p,i,j ∪{Hi,j
p(H0)k,l
q}p,q,i,j,k,l
∪{Ei,j
aFk,l
bGm,n
c}a,b,c,i,j,k,l,m,n,
where
H6
=
H0
. While both hierarchies are asymptotically
complete, they satisfy the relation
Sn⊂Ln6⊂Sn+1
(in fact,
the smallest set of the NPA hierarchy that contains
Ln
is
Spn
, where
p
is the number of parties), and thus the use
of finite levels of one or the other hierarchy may vary with
the specific problem to solve.