Quantum violations in the Instrumental scenario and their relations to the Bell scenario

Abstract

The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and postquantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalites, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario are more efficient than their Bell scenario counterpart as regards the number of random input choices involved, which may have potential implications for device-independent protocols.

Full text

Quantum violations in the Instrumental scenario
and their relations to the Bell scenario
Thomas Van Himbeeck,
1, 2
Jonatan Bohr Brask,
3
Stefano Pironio,
1
Ravishankar Ramanathan,
1
Ana Bel´en Sainz,
4
and Elie Wolfe
4
1
Laboratoire d’Information Quantique, Universit´e Libre de Bruxelles, 1050 Bruxelles, Belgium
2
Centre for Quantum Information & Communication, Universit´e Libre de Bruxelles, Belgium
3
Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
4
Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada, N2L 2Y5
(Dated: April 12, 2018)
The causal structure of any experiment implies restrictions on the observable correlations between
measurement outcomes, which are different for experiments exploiting classical, quantum, or post-
quantum resources. In the study of Bell nonlocality, these differences have been explored in great
detail for more and more involved causal structures. Here, we go in the opposite direction and
identify the simplest causal structure which exhibits a separation between classical, quantum, and
postquantum correlations. It arises in the so-called Instrumental scenario, known from classical
causal models. We derive inequalities for this scenario and show that they are closely related to
well-known Bell inequalites, such as the Clauser-Horne-Shimony-Holt inequality, which enables us
to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating
the first two. The relations that we uncover imply that the quantum or post-quantum advantages
witnessed by the violation of our Instrumental inequalities are not fundamentally different from
those witnessed by the violations of standard inequalities in the usual Bell scenario. However,
non-classical tests in the Instrumental scenario are more efficient than their Bell scenario counterpart
as regards the number of random input choices involved, which may have potential implications for
device-independent protocols.
I. INTRODUCTION
Classical and quantum physics provide fundamentally
different predictions about the correlation which can be
observed in experiments with multiple parties. Under-
standing the exact nature of this difference is a central
problem in the foundations of quantum physics and is
also important for applications in information processing.
In any experiment, the causal structure of the setup
imposes restrictions on the observable correlations. De-
pending on whether the experiment is modeled using
classical random variables, quantum states and measure-
ments, or postquantum resources, these limitations may
be different, leading to observable differences between
classical models, quantum mechanics, and general prob-
abilistic theories. This was first pointed out by Bell [
1
],
who found that models which attempt to describe an
experiment in terms of causal relations between classical
random variables, and where the actions of one party
cannot influence the local observations of separate parties,
imply restrictions on the observable correlations, known
as Bell inequalities. Measurements on entangled quantum
states shared between the observers, on the other hand,
can lead to violation of these inequalities.
This discovery sparked the study of Bell nonlocality
which by now is an active field of research and a cor-
nerstone of quantum theory [
2
]. Bell’s original setting
involves two non-communicating parties each selecting
a measurement to perform, and each obtaining a mea-
surement outcome. Later studies have considered many
variations of this causal structure. For example, the use
of sequential measurements [
3
], multiple sources [
4
,
5
],
FIG. 1. DAGs for the standard Bell scenario, and the in-
strumental scenario. Circles and squares denote observable
and unobservable variables respectively. Arrows denote causal
influence.
(a)
Bipartite Bell scenario. Each party has an input
(
X
and
Y
) and an output (
A
and
B
), which are observable.
An unobservable shared source Λ may influence the outputs.
(b)
Instrumental scenario. The second party has no input, but
the output of the first party is communicated to the second.
multiple parties [
6
], and communication between the par-
ties [
7
,
8
]. In general, the causal structures of all these
variants are more complicated than Bell’s original setting.
Here, we go in the opposite direction and identify the sim-
plest causal structure that exhibits a separation between
classical, quantum, and postquantum correlations.
To be a little more specific about what we mean by
‘simple’, let us first note how a causal structure can be
represented. In a given experiment there is a number
of observable variables. For instance, on a measurement
apparatus, one variable (the ’input’) may correspond to
the setting of a knob determining the measurement to
arXiv:1804.04119v1 [quant-ph] 11 Apr 2018

2
be made, and another variable (the ‘output’) to the mea-
surement outcome. In addition, there may be hidden or
latent variables, which are not observed, but which me-
diate correlations between the observable variables. For
instance, setting the knob on an apparatus in one way
may determine the reading of a distant apparatus through
an unobserved electromagnetic field. We can represent
the causal relationships between all these variables on
a directed acyclic graph (DAG), where the nodes corre-
spond to variables, and the edges between them signify
causal influence. For classical models, all the variables are
classical random variables. For quantum or postquantum
models, the hidden variables may be replaced by quantum
states or more generally by the resources of generalized
probabilistic theories (GPT), such as Popescu-Rohrlich
boxes [
9
]. Fig. 1(a) shows the DAG for the standard,
bipartite Bell scenario. We consider a causal structure to
be simpler the simpler the corresponding DAG is, i.e. the
fewer nodes and edges it has.
The causal structure represented by a DAG constrains
the possible correlations between the observed variables,
depending on whether the hidden variables are classi-
cal, quantum or GPT. For instance in the Bell DAG of
Fig. 1(a) the correlations between the observed random
variables
A, B, X, Y
are characterized by the conditional
probabilities
p
=
{pAB|X Y
(
ab|xy
)
}
, which in the classical
case take the form:
p∈ CBell iff p(ab|xy) = X
λ
p(λ)p(a|xλ)p(b|yλ).(1)
This is simply the usual Bell locality condition, and it
leads to linear constraints on
p
(
ab|xy
), which are Bell
inequalities1.
In the quantum case,
p∈ QBell iff p(ab|xy) = tr ρ Ea|x⊗Fb|y,(2)
where
ρ
denotes a quantum state produced by Λ and
distributed to the quantum devices in
A
and
B
; for each
x
,
{Ea|x}a
is a POVM defining a valid measurement with
outcomes
a
; and for each
y
,
{Fb|y}b
is a POVM defining
a valid measurement with outcomes b.
For the GPT case,
p∈ GBell iff p(ab|xy) = (ea|x|(eb|y|◦|Ψ) ,
where, using the notation of [
10
],
|
Ψ) denotes a GPT
generalization of the quantum state
ρ
in (2), and (
ea|x|
,
(
eb|y|
GPT generalizations of the quantum measurement
operators
Ea|x
,
Eb|y
. An example of a GPT beyond
quantum theory is the one known as boxworld [
11
], and
the set of such GPT correlations for the Bell scenario
coincides with the set of no-signalling correlations.
1
Note that in (1), we omitted the subscripts in the conditional
probabilties, writing
p
(
ab|xy
) instead of
pAB|XY
(
ab|xy
). We will
follow this use throughout the paper when there is no risk of
misunderstanding.
These definitions can be generalized to arbitrary DAGs
beyond the Bell scenario. The classical case has been
studied extensively in the classical causality literature [
12
].
Definitions of quantum and GPT correlations for arbitary
DAGs were introduced by Henson, Lal, and Pusey (HLP)
[
13
]. We will not present the HLP formalism in detail,
as we will not need it, and refer the interested reader to
their paper. It suffices to say that when thinking of a
set of correlations, be it classical, quantum, or any other
GPT, we can think of it as arising from ‘measurements’
being performed on a ‘state’, where the measurements
and state are dubbed classical, quantum or GPT. Since
classical states are a subset of quantum states which are
in turn a subset of GPT states, it follows that the sets of
correlations associated with the various generalization of
a causal structure form a hierarchy,
CDAG ⊆ QDAG ⊆ GDAG .
While the classical, quantum, and GPT sets are strictly
distinct in the Bell scenario, this is not always the case
for an arbitrary DAG. HLP have introduced a necessary
condition for these three sets to be distinct. Given a DAG,
one can thus evaluate the HLP condition. If this condition
is not satisfied, then the sets of classical, quantum, and
GPT correlations are equal, i.e., the causal structure
represented by the DAG is uninteresting as it does not
lead to observable differences between these theories. If
the HLP condition is satisfied, then one cannot conclude
anything yet: classical, quantum, and GPT models might
lead to observable differences, or might not – some further
analysis is required.
In their paper, HLP have applied their criterion to
all possible DAGs with 7 nodes or less [
13
], identifying
all DAGs that possibly admit a separation between clas-
sical, quantum, and postquantum correlations. They
have found a single DAG that is simpler than the Bell
DAG, where ‘simple’ means that it involves fewer nodes
and edges. This DAG is represented in Fig. 1(b). It
has been studied previously in the classical causality lit-
erature, where it is known as the ‘Instrumental DAG’
[
14
,
15
], a nomenclature we will follow. We show here
that the Instrumental scenario does indeed provide a sep-
aration between the sets of classical, quantum, and GPT
correlations. We derive an inequality which must hold
for classical correlations and relate it to the well known
Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [
16
]
for the scenario of Fig. 1(a). In so doing, we identify
its maximal quantum and GPT violations. We start by
describing in more detail the instrumental scenario and
relating it to the Bell scenario.
II. THE INSTRUMENTAL SCENARIO AND ITS
RELATION TO THE BELL SCENARIO
Imagine possessing some quantum implementation of
the Bell scenario: this consists of three devices. The first
device (Alice’s apparatus) accepts as input one classical

3
system (Alice’s choice of setting) and one quantum system
(Alice’s share of
ρ
), and outputs a classical system (Alice’s
measurements outcome). The second device (Bob’s appa-
ratus) again accepts one classical input and one quantum
input, and returns one classical output. The third device
is Λ itself, which has no inputs, and outputs a bipartite
quantum system.
Now, consider taking the quantum Bell scenario imple-
mentation, and modifying it as follows: Instead of letting
Bob choose his setting
y
freely, copy Alice’s classical out-
put
a
and wire the copy into Bob’s classical input. This
creates a new scenario, characterized by the conditional
probabilities
p
(
ab|x
), since
y
is no longer freely chosen.
Of course,
p
(
ab|x
) =
p
(
ab|x, y
=
a
) so we could describe
the probabilities
p
(
ab|x
) that would characterize the new
scenario without ever needing to actually perform the
hypothetical modification, so long as we have a priori
knowledge of p(ab|xy).
But this hypothetical scenario which we have imag-
ined constructing by modifying the quantum Bell sce-
nario is identically the quantum Instrumental scenario
per Fig. 1(b). Furthermore, the same is true for the clas-
sical and GPT variants of the scenarios, as we can simply
take the third device Λ to be a source of either classical or
GPT states
ρ
. This leads us to the following fundamental
statement
p∈ TInstr iff p(ab|x) = p0(ab|x, y=a) where p0∈ TBell
(3)
where
T
is a placeholder for a correlation set, such as
classical C, quantum Q, or GPT G.
In this sense, correlations in the Instrumental scenario
are essentially postselections of Bell scenario correlations:
An experimenter might perform many runs of the Bell
scenario experiment, but then postselect to examine only
those experimental runs when
y=a
. This postselected
data will exhibit Instrumental scenario statistics, and
moreover, every
TInstr
correlation can arise via this sort
of postselection on Bell scenario statistics2.
To make this relationship between DAGs more concrete,
note that compatibility with the Instrumental scenario
is defined nearly identically to compatibility with the
Bell scenario, except all references to the variable
Y
get
overwritten with references to the variable A:
p∈ CInstr iff p(ab|x) = Pλp(λ)p(a|xλ)p(b|aλ),
p∈ QInstr iff p(ab|x) = tr ρ Ea|x⊗Fb|a,
p∈ GInstr iff p(ab|x) = (ea|x|(eb|a|Ψ) .
Even though the Quantum- and GPT-Bell-Scenario sets
are in general distinct from the Classical-Bell-Scenario
2
The recognition that every
TInstr
correlation must be compatible
with
TBell
is an example of a device-independent causal infer-
ence technique which we call Interruption. Relating pairs of
DAGs in this manner is explored more generally in a forthcoming
manuscript, currently in preparation [17].
FIG. 2. The Signalling-Between-Outputs scenario considered
in Refs. [
8
,
18
]. This scenario relaxes the no-communication
assumption between the variables
A
and
B
, thus modifying
the usual Bell scenario depicted in Fig. 1(a) by the addition
of an edge A→B.
set, it may be that their postselections (defining the cor-
responding Instrumental-Scenario correlation sets) all co-
incide. Indeed, it is well known that postselection of
genuinely quantum, nonlocal data may admit a purely
classical explanation. This effect is at the basis, for in-
stance, of the infamous detection loophole in Bell ex-
periments [
2
]. It is thus not obvious a priori that the
Instrumental scenario should admit a separation between
classical, quantum and GPT correlations; it might be
another example of an uninteresting DAG, which simply
happens not to be identified by the HLP criterion.
The Instrumental scenario can also be understood as
a Bell scenario with relaxed causality constraints. Such
relaxations of the Bell scenario have been considered pre-
viously. For instance, the no-communication assumption
between the variables
A
and
B
in the Bell scenario has
been relaxed in Refs. [
8
,
18
], leading to the Signalling-
Between-Outputs scenario represented in Fig. 2. Other
works have considered modified Bell scenarios wherein
one no longer assumes that the measurement inputs
X
and
Y
could have been set-up freely [
19
,
20
]. The In-
strumental scenario represents simultaneous relaxation of
both the measurement-freedom and no-communication as-
sumptions: not only may the outcome of the measurement
performed at
B
depend directly on the distant outcome
A
, but furthermore the measurement setting
Y
is not
chosen freely but is instead fixed entirely by the value
A
output by the distant measurement device.
We remark that testing for membership in
TInstr
, by
admitting an extension to
TBell
per Eq.
(3)
, generalizes to
any well defined correlation set in the Bell scenario. For
example, one might consider relaxations of the quantum
set corresponding to levels of the Navascu´es-Pironio-Ac´ın
(NPA) hierarchy [
21
,
22
], including the set known as Al-
most Quantum Correlations [
23
], or tests for compatibil-
ity with a restricted Hilbert space dimension [
24
–
26
]. All
those correlation membership tests for the Bell scenario
can be applied to the Instrumental scenario by simply
introducing existential quantifiers: Does an extension of
p
(
ab|x
)
≡p
(
ab|x, y=a
) exist to
p
(
ab|x, y6=a
) such that the
relevant condition for membership in
TBell
is satisfied?
This modification is especially easy for testing NPA-level

4
compatibility, as those semidefinite tests already natively
support unspecified probabilities.
III. GEOMETRY OF THE
INSTRUMENTAL-SCENARIO CORRELATIONS
Before attempting to find a gap between classical and
quantum correlations in the instrumental scenario, let us
take a general geometric perspective to enhance our un-
derstanding. Every correlation in the Bell scenario can be
thought of as a high-dimensional vector,
d=|A||B||X||Y|
,
where the coordinates are given by the many different
probabilities
p
(
ab|xy
). Every correlation in the Instrumen-
tal scenario can be though as a somewhat lower dimension
vector,
d=|A||B||X|
, where the coordinates are given by
the probabilities
p
(
ab|x
). The set of all correlations in
TInstr
are formed by axial projection of those coordinates
p(ab|x, y6=a) of TBell .
In both the Bell and Instrumental scenarios, all sets
of correlations are convex. The sets
GBell
and
CBell
are
the no-signalling polytope and the local polytope respec-
tively, whereas the set
QBell
is a convex set but not a
polytope [
2
,
27
–
29
]. The projections of polytopes are also
polytopes, so we know that
GInstr
and
CInstr
will also form
polytopes. To obtain the Instrumental scenario polytopes
from the Bell scenario polytopes, we can use Fourier-
Motzkin elimination or any other polytope projection
technique [
30
–
34
]. Alternatively, we can directly compute
CInstr
by taking the convex hull of all deterministic strate-
gies in the Instrumental scenario. We have performed
these operations for small cardinalities of the observed
variables X, A, B using the polytope software PORTA.
In the simplest case where
X, A, B ∈ {
0
,
1
}
are all
binary, we find that
CInstr
=
GInstr
and that these sets are
fully characterized by the trivial normalization
X
ab
p(ab|x)=1
and positivity
p(ab|x)≥0
conditions, together with the additional set of constraints
pAB|X(a0|x) + pAB|X(a1|x0)≤1 for all a, x 6=x0,(4)
which can be expressed compactly as
max
aX
b
max
xp(ab|x)≤1.(5)
As
CInstr ⊆ QInstr ⊆ GInstr
, the above constraints also
fully characterize the quantum set QInstr.
Since the normalization condition is the only generic
equality constraint satisfied by correlations in the Instru-
mental scenario, the sets
CInstr
,
QInstr
, and
GInstr
are
full-dimensional in the space of normalized probability
distributions. This should be contrasted with the Bell sce-
nario where
CBell
,
QBell
, and
Gell
are not full-dimensional
in the space of normalized probability distributions, since
they also satisfy the no-signaling equality constraints,
expressing that the marginal distribution of
b
cannot de-
pend on
x
and the marginal distribution of
a
on
y
. This
full-dimensional property of the Instrumental scenario is
not limited to the
|X|=|A|=|B|=2
case, but is valid for
any cardinalities of the inputs and outputs. Indeed, a
method to determine the complete set of equality con-
straints satisfied by classical correlations compatible with
an arbitrary DAG has been given in [
35
]. Applying it to
the Instrumental DAG yields no other equalities than the
normalization conditions in the classical case – and hence
in the quantum and GPT case as well, since they contain
classical correlations as a subset.
Even though the Instrumental scenario does not contain
no-signaling constraints – indeed the input
b
can depend
on
x
through
a
– some remnant of the no-signaling condi-
tions are preserved when projecting the Bell scenario to
the Instrumental scenario, as expressed by the inequalities
(4), which can be interpreted as limiting the magnitude
by which bcan depend on xwhen ais kept constant.
We can understand that such inequalities are GPT-
inviolable from the definition of
GInstr
as as a projec-
tion of the no-signalling polytope. As an example,
let us derive one of the inequalities (4) from the fol-
lowing two positivity inequalities for the Bell scenario:
pAB|X Y
(11
|
10)
≥
0 and
pAB|X Y
(10
|
00)
≥
0. Sum-
ming those two inequalities together and then using no-
signalling to express the probabilities as linear combina-
tions of those where Alice’s output matches Bob’s input,
i.e.,
pAB|X Y
(11
|
10)
→pB|Y
(1
|
0)
−pAB|X Y
(01
|
10) and
pAB|X Y
(10
|
00)
→pB|Y
(0
|
0)
−pAB|X Y
(00
|
00), we obtain
pB|Y(0|0) + pB|Y(1|0)
−pAB|X Y (00|00) −pAB|X Y (01|10) ≥0,
or, equivalently,
pAB|X Y (00|00) + pAB|X Y (01|10) ≤1.
Having eliminated the non-Instrumental probabilities
p
(
ab|x, y 6
=
a
), the final inequality (as translated for the
Instrumental scenario) reads
pAB|X(00|0) + pAB|X(01|1) ≤1,(6)
This proves that eq.
(6)
– an instance of (4) – is an
Instrumental scenario inequality which is GPT inviolable.
Expressed in the general form (5), these inequalities
are valid for
CInstr
,
QInstr
, and
GInstr
for arbitrary number
of inputs and outputs
|X|
,
|A|
,
|B|
. They were originally
derived by Pearl [
14
] for the classical Instrumental sce-
nario and have come to be known as the instrumental
inequalities. Henson, Lal, and Pusey then showed that
Pearl’s instrumental inequalities (5) are satisfied by all
GPTs for arbitrary inputs and outputs [13].

5
To summarize, we have found that in the case
|X|=|A|=|B|=2
the instrumental inequalities (5) are the
unique facets, besides the trivial positivity facets, of the
GPT polytope. We have verified that this is also the
case for
|X|=
2 and
|A|=|B| ≤ 4
. We also know that the
instrumental inequalities are satisfied by the GPT poly-
tope for arbitrary number of inputs and outputs, but we
leave it as on open question whether they are the unique
non-trivial facets in this general case.
In the simplest possible Bell scenario where
X,Y,A,B ∈ {0,1}
well-known bounds on the violation
of the CHSH inequality imply that
CBell (QBell (GBell
.
We have found, however, that
CInstr =QInstr =GInstr
for
the corresponding Instrumental sets, i.e. all non-classical
features of Bell correlations are washed out when post-
selecting them to obtain the Instrumental correlations of
the
X,A,B ∈ {0,1}
set-up. Though, we have established
this fact by fully characterizing the Instrumental poly-
topes using the software PORTA, it is also instructive
to see more explicitly how all non-local correlations of
the
X,Y,A,B ∈ {0,1}
Bell scenario admit a classical ex-
planation when projected to the Instrumental scenario.
Consider for instance the Popescu-Rorhlich (PR) correla-
tions
p(ab|xy) = 1/2 if b=a+xy
0 otherwise ,
which reach the algebraic value 4 for the CHSH expression.
Post-selecting the case where
p
(
ab|x, y
=
a
), we get the
following Instrumental scenario correlations
p(ab|x) = 1/2 if b=a(1 + x)
0 otherwise .
In other words,
p
(
a|x
)=1
/
2 and
b=a
if
x=0
, while
b= 0
if
x=1
. But now it is easy to see how these cor-
relations can be simulated classically. Consider a bi-
nary hidden variable
λ∈ {
0
,
1
}
that is unbiased, i.e.
p(λ=0) = p(λ=1) = 1/2
, and define
a:
=
λ
+
x
,
b:=λ a
.
We obviously have that
p
(
a|x
)=1
/
2 and
b=λ2=λ=a
if
x=0, while b=λ(λ+ 1)=0 if x=1, as required.
Since any GPT correlations in the Bell scenario can
be written as a mixture of classical correlations and PR
correlations, any GPT correlations can be written as a
mixture of classical correlations and post-selected PR
correlations. But since we have just seen that the later
ones are classical, this establishes that
GInstr
=
CInstr
.
More generally, it was shown in [
8
,
18
] by a similar ar-
gument that classical models can reproduce any GPT
correlations in the Signalling-Between-Outputs scenario
whenever
|X|=|Y|=|A|=2
and
|B|
is arbitrary. These
results translate to our case since the Instrumental DAG
is a subgraph of the Signalling-Between-Outputs DAG
in which the node
Y
is dropped. They imply that there
cannot be any separation between classical, quantum and
GPT correlations in the Instrumental scenario whenever
|X|=|A|=2 and |B|is arbitrary.
Using PORTA, we have also determined the facets of
the classical polytope in the case
|X|=
2,
|A|≤
4,
|B|≤
4
by taking the convex hull of all deterministic strategies.
We find again that Pearl’s instrumental inequalities are
the only non-trivial facets, implying that
Cinstr
=
Ginstr
in this case as well.
IV. A CLASSICAL INSTRUMENTAL
SCENARIO INEQUALITY WHICH ADMITS
QUANTUM VIOLATION
The case
X∈ {
0
,
1
,
2
}
,
A, B ∈ {
0
,
1
}
is more interesting
as we found that the facets of the classical polytope
CInstr
comprise, in addition to Pearl’s instrumental inequalities,
a new family of inequalities, one representative of which
is
IBonet := p(a=b|0) + p(b=0|1) + p(a=0, b=1|2) ≤2.(7)
Inequality
(7)
was found previously by Bonet [
15
] (also
using PORTA) who was looking for stronger classical con-
straints to complement Pearl’s instrumental inequalities
(5).
This inequality admits quantum violation, and more-
over also provides quantum/GPT separation, as we now
show. A quantum strategy violating (7) is as follows.
Let the source Λ distribute the two-qubit maximally
entangled state
|φ+i
= (
|
00
i
+
|
11
i
)
/√2
and let
A
perform the measurements
σx, σz,−
(
σx
+
σz
)
/√2
when
she receives the input
X
= 0
,
1
,
2, respectively, while
Bob measures (
σx
+
σz
)
/√2,
(
σx−σz
)
/√2
when he re-
ceives
a
= 0
,
1. A straightforward computation gives
IBonet = (3 + √2)/2'2.207 >2.
An example of GPT correlations violating Bonet’s in-
equality is given by
p(ab|x) = 1/2 if b=a+f(x, a)
0 otherwise ,
where
f
(0
, a
) =
a
,
f
(1
, a
) = 0, and
f
(2
, a
) =
a
+ 1.
Inserting these probabilities in (7) yields
IBonet
= 5
/
2
>
2.
It can be verified that these correlations are GPT valid as
they satisfy Pearl’s instrumental inequalities, which are
the unique (non-trivial) facets of the GPT polytope in
the
|X|
= 3,
|A|
=
|B|
= 2 case. Alternatively, they can
be seen as postselection of the GPT (i.e., no-signalling)
Bell correlations
p
(
ab|xy
) defined by
p
(
ab|xy
) = 1
/
2 if
b=a+f(x, y) and 0 otherwise.
V. RELATION TO THE CHSH INEQUALITY
AND DUMMY INPUTS
The fact that post-selections of the
|X|=|Y|=|A|=|B|=2
Bell scenario, where non-locality is
entirely detected by the violation of the CHSH inequality,
do not lead to non-classical Instrumental correlations
might suggest, naively, that violation of Bonet’s inequality
(7) uncover a stronger form of non-locality, requiring
violating beyond the CHSH inequality. We show that

6
this is not the case by relating Bonet’s inequality to
the CHSH inequality. That such a link must exist also
follows directly from the fact that all (non-trivial) facets
of the
|X|=3
,
|Y|=|A|=|B|=2
classical Bell polytopes
are liftings of the CHSH inequality [36].
Although we found inequality
(7)
by taking the convex
hull of deterministic strategies and without regard to the
relationship between the Bell and Instrumental scenarios,
it is enlightening to retrodictively explain
IBonet
as a
projection of the classical Bell scenario polytope.
Let us rewrite the expression
IBonet
per
(7)
in terms
of
p
(
ab|xy
); that is, let us interpret the facet of the clas-
sical Instrumental polytope as a valid inequality for the
Bell polytope. This operation is a trivial lifting of the
inequality via the mapping
p
(
ab|x
)
→p
(
ab|x, y=a
). We
find that
Lifting [IBonet]
=p(00|00) + p(11|01) + p(00|10) + p(10|11) + p(01|20) .
Using the normalization and no-signalling constraints
satisfied by the Bell scenario probabilities
pAB|X Y
, we
can rewrite this last expression as
Lifting [IBonet] = 1
4hCH S H i − p(11|20) + 3
2,
where
hCH S H i≡hA0B0i+hA0B1i+hA1B0i−hA1B1i,
with
hAxByi
the expectation value of
AB
given that
X
and
Y
take values
x
and
y
. From this retrodiction it
becomes immediately clear that the classical, quantum,
and GPT bounds of bounds IBonet are
IBonet ≤




2 Classical
(3 + √2)/2 Quantum
5/2 GPT
,
as this follows from
hCH S H i ≤ 




2 Classical
2√2 Quantum
4 GPT
,
as well as the fact that
−p
(11
|
20)
≤
0 in all physical
theories.
A perhaps surprising consequence of the retrodictive
interpretation of
IBonet
is that any nonclassical correla-
tion in the CHSH Bell scenario can be used as a resource
to generate nonclassical correlations in the Instrumen-
tal scenario, despite the fact that the Instrumental sce-
nario has coinciding GPT and classical polytopes for
|X|=|A|=2
. The trick which allows us to map arbitrary
non-classical No-Signalling correlations in the Bell sce-
nario where
|X|=|Y|=|A|=|B|=2
into non-classical corre-
lations in the Instrumental scenario is as follows: Starting
from a
pAB|X Y
in the standard CHSH scenario where
x∈ {
0
,
1
}
, trivially map it to
p0
AB|X Y
in an extended sce-
nario where
x∈ {
0
,
1
,
2
}
by setting
p0
(
ab|xy
) =
p
(
ab|xy
)
when
x=
0
,
1 and
p0
(
ab|x=
2
, y
) =
δa,0p
(
b|y
) when
x=
2.
That is, in the case
x
= 2, the output
a
is deterministi-
cally equal to 0. Then we have
p0
(11
|
20) = 0 and thus we
may substitute
1
4hCH S H ip0+p0(11|20) →1
4hCH S H ip
to recast
Lifting [IBonet]p0
for
|X|
= 3 as an explicit func-
tion of
p
for
|X|
= 2, with the trivial intermediate map
p→p0taken for granted:
Lifting [IBonet]p0=1
4hCH S H ip+3
2,(8)
In particular, this trivial map allows us to relate the extent
of the violation of
IBonet ≤
2 in the Instrumental scenario
entirely as a function of the extent of the violation of
hCH S H i ≤
2 in the Bell scenario. A direct consequence
is that any non-classical correlations in the
|X|
=
|A|
=
|Y|
=
|B|
= 2 Bell scenario, which necessarily violate the
CHSH inequality, give rise to non-classical correlations
violating Bonet’s inequality in the Instrumental scenario.
Another way to express this connection is as follows.
Writing
p0
in term of
p
, we can rewrite the relation (8)
explicitly as the identity
1
4hCH S H ip+3
2=p(00|00) + p(11|01) (9)
+p(00|10) + p(10|11) + pB|Y(1|0) .
Thus, instead of testing CHSH in the regular way, which
involves estimating the correlations for 4 choices of input
pairs (
x, y
)
∈ {
(0
,
0)
,
(0
,
1)
,
(1
,
0)
,
(1
,
1)
}
, one can alterna-
tively test it using 3 choices of an input
z
. (
i
) If
z=
0
,
1,
one uses
x=z
on Alice’s side and uses Alice’s outputs as
an input for Bob. This allows to evaluate the first four
terms on the righ-hand side of (9). (
ii
) If
z=
2, one uses
y=
0 as an input for Bob and registers his output (without
testing Alice). This allows to evaluate the last term of
(9).
VI. TILTED INSTRUMENTAL INEQUALITIES
The results of the last section show that, at least in
the specific input-output configuration we considered,
the Instrumental scenario is essentially equivalent to
the Bell scenario for the mere purpose of detecting non-
classicality, i.e., correlations in the CHSH Bell scenario are
non-classical if and only if they give rise to non-classical
correlations in the Instrumental scenario.
However, it is well known that many interesting proper-
ties of non-classical correlations do not merely reduce to
testing their non-classicality. For example, in the CHSH
Bell scenario, deciding if given correlations
p
are non-
classical can entirely be decided by testing the CHSH
inequality, and therefore other types of inequalities, such

7
as the tilted CHSH inequalities introduced in [
37
] are irrel-
evant from this perspective. However, such tilted CHSH
inequalities are useful for other purposes. For instance
they can be used to certify in a device-independent setting
more randomness that would be possible using the CHSH
inequality [37,38].
Similarly, the Instrumental scenario may provide new in-
teresting features not directly present in the Bell scenario.
As an example, notice that the smallest Bell scenario relies
on the violation of the CHSH inequality and hence need
two random bits at the inputs
(|X|=|Y|=2)
. Interestingly,
we have seen that the same non-classical resources (using
the same quantum state and measurements) can be tested
in the Instrumental scenario using only a single random
trit
(|X|=3)
. Indeed, we have seen that we can simply
interpret the Instrumental Bonet’s inequality as a new
way to test the violation of the CHSH inequality.
This could potentially have implications for device-
independent (DI) randomness certification [
39
,
40
] as it
could lead to schemes generating the same outputs ran-
domness as a standard CHSH protocol but consuming
less input randomness. This requires further examination,
however, as many parameters, such as the amount by
which the random inputs can be biased (resulting in a
consumption of randomness per round inferior to 2 bits
even using the CHSH inequality) or the sensitivity to sta-
tistical fluctuations, determine the randomness expansion
rate and such parameters might be more favorable in the
standard Bell scenario.
In the Bell scenario, the tilted CHSH inequalities have
useful properties from the perspective of randomness certi-
fication. Interestingly, we now show that such inequalities
admit an Instrumental version through a link analogous
to the one relating the CHSH and Bonet’s inequalities.
Let us start with the Instrumental version of the tilted
CHSH inequalities. For α≥1, define
Iα=1−α
2p(a= 0|0) + p(a= 0|1)
+α p(a= 0, b = 0|0) + p(a= 1, b = 1|0)
+α p(a= 0, b = 0|1) + p(a= 1, b = 0|1)
+α p(a= 0, b = 1|2).
Whenever
α
= 1, we recover Bonet’s expression
(7)
.
Iα
defines valid (though not necessarily facet-defining) in-
equalities for the Instrumental scenario, and the tech-
niques presented above can be used to relate its lifting to
tilted CHSH inequalities and derive its classical, quantum,
and GPT bounds. In particular, one can show that
Lifting [Iα] = 1
4hCH S Hαi − α p(11|20) + 1 + α
2,
where
hCH S Hαi ≡ αhA0B0i+hA0B1i+αhA1B0i−hA1B1i.
The tilted CHSH inequalities were introduced in Ref. [
37
],
and have the following properties
hCH S Hαi ≤ 




2αClassical
2√α2+ 1 Quantum
2(1 + α) GPT
.(10)
Hence, the corresponding bounds for the tilted instrumen-
tal inequalities read
Iα≤




1 + αClassical
(1 + α+√α2+ 1)/2 Quantum
3/2 + αGPT
,
and they can be achieved by using the same trivial map-
ping from the Bell scenario to the Instrumental scenario.
We see then that these tilted inequalities admit not just
quantum violations, but also a GPT violation beyond the
quantum bound, for all α≥1.
It was shown in Ref. [
37
] that correlations maximally
violating the tilted CHSH inequality of Eq.
(10)
can be
used to certify 2
−ln
(2)
/α
bits of randomness, which can
be made arbitrarily close to 2 by increasing
α
. By exploit-
ing the link with
Iα
, the same amount of randomness can
be certified by violating a tilted Instrumental inequality
instead. As here, at most a single trit of randomness
is used at the input, while 2
−
bits can be certified at
the ouput (for

arbitrarily small), we can generate two
random bits for the price of (at most) one trit.
VII. DISCUSSION
The original motivation of Bell was to provide a testable
criterion for whether Nature is compatible with a classical
local causal description. In such an experiment, ideally
one does not wish to make any assumptions about the non-
existence of spurious communication channels between the
parties, which could be mediated via as-yet-undiscovered
physics. To rule out communication, which could explain
the observed correlations, one can instead arrange to have
space-like separation of the different parties’ measure-
ment events. Any communication would then need to
be superluminal, in violation of special relativity. The
minimal causal structure in which such an experiment
can be implemented is that of Fig. 1(a), and the minimal
scenario is that of CHSH (binary inputs and outputs for
each party). Several conclusive tests imposing space-like
separation have recently been realised [41–43].
However, one of the consequences of Bell nonlocality
is to enable device-independent (DI) information process-
ing. Conditioned on the violation of a Bell inequality, it
becomes possible to certify the security or correct func-
tioning of an information processing protocol, without
any detailed knowledge of its implementation [
39
,
40
,
44
–
50
]. Prominent examples are quantum key distribution
and random number expansion and amplification. In DI
settings, it is typically assumed that devices are shielded,
i.e. that the experimenters control the inputs which enter

8
into the devices, and that the devices do not leak infor-
mation on spurious side channels. For DI information
processing therefore, the minimal non-trivial setting is
the Instrumental scenario Fig. 1(b) considered here.
We have shown here that the Bell and Instrumental
scenarios are closely related. Though correlations in the
simplest Bell scenario, the CHSH scenario, always admit a
classical model if they are directly projected on the Instru-
mental scenario, we have shown that their non-classical
nature is entirely preserved in the Instrumental scenario
provided some purely classical local processing is first
applied on Alice’s side. This finding has important impli-
cations: given some non-classical resource
p
(
ab|xy
) in an
arbitrary Bell scenario, determining whether this resource
gives rise to a non-classical behavior in the Instrumen-
tal scenario cannot simply be answered by considering
the Instrumental probabilities
p
(
ab|x
) =
p
(
ab|x, y
=
a
)
(and determining if they are in the classical Instrumen-
tal polytope, e.g., using linear programming). Instead,
one should also take into account all possible local clas-
sical transformations that can be applied to the given
non-classical correlation
p
. By failing to consider such
trivial, free transformations of a correlation one obtains
false negatives from the standard causal inference tools –
correlations appear to be compatible with the classical In-
strumental DAG, but actually are not
3
. This observation
applies to other DAG derived from Bell-type scenarios,
such as the Signalling-Between-Outputs scenario of Fig. 2.
Another outcome of our results is that the Instrumental
versions of the CHSH and tilted CHSH inequalities require
less input choices than their standard Bell versions. We
have briefly discussed the potential of such Instrumental
inequalities for DI randomness certification, but this is
an issue that deserves further investigation.
From a fundamental point of view, we have identified
a fully device-independent scenario (in particular, that
does not rely on several independent hidden sources [
51
])
that require three random input choices only, whereas the
CHSH scenario requires in total four (2
×
2) random input
choices. We leave it as an open question whether it is
possible to find a DI scenario where a random choice be-
tween two values only is sufficient to observe non-classical
correlations.
NOTE ADDED
The results presented here partly overlap with those
obtained independently in [
52
], where Bonet’s inequal-
ity and the violating quantum and GPT correlations of
Section IV were also introduced, but where the one-to-
one relation between Bonet’s inequality and the CHSH
inequality presented in Section Vwas not noticed. All
our results up to Section Vhave been orally presented by
S.P. at the Quantum Networks 2017 Workshop, Oxford
(UK) in August 2017.
ACKNOWLEDGMENTS
This research was supported by the Fondation Wiener-
Anspach, the Interuniversity Attraction Poles program of
the Belgian Science Policy Office under the grant IAP P7-
35 photonics@be, and the Perimeter Institute for Theoret-
ical Physics. 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. S.P. is Research Asso-
ciates of the Fonds de la Recherche Scientifique (F.R.S.-
FNRS).
[1]
J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,”
Physics 1, 195 (1964).
[2]
Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Va-
lerio Scarani, and Stephanie Wehner, “Bell nonlocality,”
Rev. Mod. Phys. 86, 419–478 (2014).
[3]
Sandu Popescu, “Bell’s Inequalities and Density Matrices:
Revealing “Hidden” Nonlocality,” Phys. Rev. Lett.
74
,
2619–2622 (1995).
[4]
C. Branciard, N. Gisin, and S. Pironio, “Characteriz-
ing the Nonlocal Correlations Created via Entanglement
Swapping,” Phys. Rev. Lett. 104, 170401 (2010).
[5]
Tobias Fritz, “Beyond Bell’s theorem: correlation scenar-
ios,” New J. Phys. 14, 103001 (2012).
3Such free transformations are analogous to the concept of inter-
ventions in the classical causality literature and are known to
improve the reliability of causal inferences [12].
[6]
J.-D. Bancal, S. Pironio, A. Acin, Y.-C. Liang, V. Scarani,
and N. Gisin, “Quantum non-locality based on finite-speed
causal influences leads to superluminal signalling,” Nat.
Phys. 8, 867–870 (2012).
[7]
B. F. Toner and D. Bacon, “Communication Cost of
Simulating Bell Correlations,” Phys. Rev. Lett.
91
, 187904
(2003).
[8]
J. B. Brask and R. Chaves, “Bell scenarios with commu-
nication,” J. Phys. A 50, 094001 (2017).
[9]
S. Popescu and D. Rohrlich, “Quantum nonlocality as an
axiom,” Found. Phys. 24, 379–385 (1994).
[10]
Giulio Chiribella, Giacomo Mauro DAriano, and Paolo
Perinotti, “Probabilistic theories with purification,” Phys.
Rev. A 81, 062348 (2010).
[11]
Peter Janotta, “Generalizations of Boxworld,” Proc. Theo.
Comp. Sci. 95, 183192 (2012).
[12] J. Pearl, Causality (Cambridge University Press, 2009).

9
[13]
Joe Henson, Raymond Lal, and Matthew F. Pusey,
“Theory-independent limits on correlations from general-
ized Bayesian networks,” New J. Phys.
16
, 113043 (2014).
[14]
J. Pearl, “On the Testability of Causal Models with La-
tent and Instrumental Variables,” in Proc. 11th Conf.
Uncertainty in Artificial Intelligence (1995) pp. 435–443.
[15]
Blai Bonet, “Instrumentality Tests Revisited,” in Proc.
17th Conf. Uncertainty in Artificial Intelligence (2001)
pp. 48–55.
[16]
John F. Clauser, Michael A. Horne, Abner Shimony, and
Richard A. Holt, “Proposed Experiment to Test Local
Hidden-Variable Theories,” Phys. Rev. Lett.
23
, 880–884
(1969).
[17]
“Causal Inference for Generalized Bayesian Networks,” In
preparation.
[18]
R. Chaves, R. Kueng, J. B. Brask, and D. Gross, “Unify-
ing Framework for Relaxations of the Causal Assumptions
in Bell’s Theorem,” Phys. Rev. Lett.
114
, 140403 (2015).
[19]
Jonathan Barrett and Nicolas Gisin, “How Much Mea-
surement Independence Is Needed to Demonstrate Nonlo-
cality?” Phys. Rev. Lett. 106, 100406 (2011).
[20]
Gilles P¨utz, Denis Rosset, Tomer Jack Barnea, Yeong-
Cherng Liang, and Nicolas Gisin, “Arbitrarily Small
Amount of Measurement Independence is Sufficient to
Manifest Quantum Nonlocality,” Phys. Rev. Lett.
113
,
190402 (2014).
[21]
Miguel Navascu´es, Stefano Pironio, and Antonio Acn,
“A convergent hierarchy of semidefinite programs charac-
terizing the set of quantum correlations,” New J. Phys.
10, 073013 (2008).
[22]
Miguel Navascu´es, Stefano Pironio, and Antonio Ac´ın,
“Bounding the Set of Quantum Correlations,” Phys. Rev.
Lett. 98, 010401 (2007).
[23]
Miguel Navascu´es, Yelena Guryanova, Matty J. Hoban,
and Antonio Acn, “Almost quantum correlations,” Nat.
Comm. 6, 6288 (2015).
[24]
Miguel Navascu´es, Gonzalo de la Torre, and Tam´as
V´ertesi, “Characterization of Quantum Correlations with
Local Dimension Constraints and Its Device-Independent
Applications,” Phys. Rev. X 4, 011011 (2014).
[25]
Miguel Navascu´es and Tam´as V´ertesi, “Bounding the Set
of Finite Dimensional Quantum Correlations,” Phys. Rev.
Lett. 115, 020501 (2015).
[26]
Jamie Sikora, Antonios Varvitsiotis, and Zhaohui Wei,
“Minimum Dimension of a Hilbert Space Needed to Gen-
erate a Quantum Correlation,” Phys. Rev. Lett.
117
,
060401 (2016).
[27]
David Avis and Tsuyoshi Ito, “Comparison of two bounds
of the quantum correlation set,” in 1st Inter. Conf. on
Quant. Nano & Micro Tech. (2007).
[28]
Jean-Daniel Bancal, On the Device-Independent Approach
to Quantum Physics (Springer International Publishing,
2014).
[29]
Valerio Scarani, “The Device-Independent Outlook on
Quantum Physics,” Acta Physica Slovaca
62
, 347 (2012).
[30]
George B Dantzig and B Curtis Eaves, “Fourier-Motzkin
elimination and its dual,” J. Combin. Th. A
14
, 288 – 297
(1973).
[31]
Egon Balas, “Projection with a Minimal System of In-
equalities,” Comp. Optimiz. Applic. 10, 189–193 (1998).
[32]
Colin Jones, Eric C Kerrigan, and Jan Maciejowski,
Equality Set Projection: A new algorithm for the projec-
tion of polytopes in halfspace representation , Tech. Rep.
(Cambridge University Engineering Dept, 2004).
[33]
Demetrius V. Shapot and Alexander M. Lukatskii, “Solu-
tion Building for Arbitrary System of Linear Inequalities
in an Explicit Form,” Am. J. Comp. Math.
02
, 1–11
(2012).
[34]
S. I. Bastrakov and N. Yu. Zolotykh, “Fast method for
verifying Chernikov rules in Fourier-Motzkin elimination,”
Comp. Mat. & Math. Phys. 55, 160–167 (2015).
[35]
Robin J. Evans, “Margins of discrete Bayesian networks,”
arXiv:1501.02103 (2015).
[36]
Stefano Pironio, “All Clauser-Horne-Shimony-Holt poly-
topes,” J. Phys. A 47, 424020 (2014).
[37]
Antonio Ac´ın, Serge Massar, and Stefano Pironio, “Ran-
domness versus nonlocality and entanglement,” Phys. Rev.
Lett. 108, 100402 (2012).
[38]
C´edric Bamps, Serge Massar, and Stefano Pironio,
“Device-independent randomness generation with sublinear
shared quantum resources,” arXiv:1704.02130 (2017).
[39]
Roger Colbeck and Adrian Kent, “Private randomness
expansion with untrusted devices,” J. Phys. A
44
, 095305
(2011), earlier version published in R. Colbeck, “Quantum
and relativistic protocols for secure multi-party computa-
tion”, PhD thesis, Cambridge, 2006.
[40]
S. Pironio et al., “Random numbers certified by Bells
theorem,” Nature 464, 1021–1024 (2010).
[41]
B. Hensen et al., “Loophole-free Bell inequality violation
using electron spins separated by 1.3 kilometres,” Nature
526, 682–686 (2015).
[42]
Lynden K. Shalm et al., “Strong Loophole-Free Test of
Local Realism,” Phys. Rev. Lett. 115, 250402 (2015).
[43]
Marissa Giustina et al., “Significant-Loophole-Free Test
of Bell’s Theorem with Entangled Photons,” Phys. Rev.
Lett. 115, 250401 (2015).
[44]
D. Mayers and A. Yao, “Quantum cryptography with
imperfect apparatus,” in Proc. 39th Symposium on Foun-
dations of Computer Science (1998) pp. 503–509.
[45]
Jonathan Barrett, Lucien Hardy, and Adrian Kent, “No
Signaling and Quantum Key Distribution,” Phys. Rev.
Lett. 95, 010503 (2005).
[46]
A. Acin, N. Gisin, and L. Masanes, “From Bell’s Theorem
to Secure Quantum Key Distribution,” Phys. Rev. Lett.
97, 120405 (2006).
[47]
Antonio Ac´ın, Nicolas Brunner, Nicolas Gisin, Serge
Massar, Stefano Pironio, and Valerio Scarani, “Device-
Independent Security of Quantum Cryptography against
Collective Attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[48]
R. Colbeck and R. Renner, “Free randomness can be
amplified,” Nat. Phys. 8, 450–453 (2012).
[49]
C. A. Miller and Y. Shi, “Robust protocols for securely
expanding randomness and distributing keys using un-
trusted quantum devices,” in Proc. 46th Symposium on
Theory of Computing (2014) p. 417.

10
[50]
Thomas Vidick Rotem Arnon-Friedman, Renato Ren-
ner, “Simple and tight device-independent security proofs,”
arXiv:1607.01797 (2016).
[51]
Cyril Branciard, Denis Rosset, Nicolas Gisin, and Ste-
fano Pironio, “Bilocal versus nonbilocal correlations in
entanglement-swapping experiments,” Phys. Rev. A
85
,
032119 (2012).
[52]
Rafael Chaves, Gonzalo Carvacho, Iris Agresti, Valerio Di
Giulio, Leandro Aolita, Sandro Giacomini, and Fabio
Sciarrino, “Quantum violation of an instrumental test,”
Nat. Phys. 47, 291296 (2018).