Quantifying Bell: the Resource Theory of Nonclassicality
of Common-Cause Boxes
Elie Wolfe
1
, David Schmid
1,2
, Ana Bel
´
en Sainz
3,1
, Ravi Kunjwal
1,4
, and Robert W. Spekkens
1
1
Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
2
Institute for Quantum Computing and Dept. of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
3
International Centre for Theory of Quantum Technologies, University of Gda
´
nsk, 80-308 Gda
´
nsk, Poland
4
Centre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, CP 165, Universit
´
e libre de Bruxelles, 1050
Brussels, Belgium
June 5, 2020
We take a resource-theoretic approach to
the problem of quantifying nonclassicality in
Bell scenarios. The resources are conceptual-
ized as probabilistic processes from the set-
ting variables to the outcome variables hav-
ing a particular causal structure, namely, one
wherein the wings are only connected by
a common cause. We term them “common-
cause boxes”. We define the distinction be-
tween classical and nonclassical resources in
terms of whether or not a classical causal
model can explain the correlations. One can
then quantify the relative nonclassicality of
resources by considering their interconvert-
ibility relative to the set of operations that
can be implemented using a classical com-
mon cause (which correspond to local oper-
ations and shared randomness). We prove
that the set of free operations forms a poly-
tope, which in turn allows us to derive an ef-
ficient algorithm for deciding whether one re-
source can be converted to another. We more-
over define two distinct monotones with sim-
ple closed-form expressions in the two-party
binary-setting binary-outcome scenario, and
use these to reveal various properties of
the pre-order of resources, including a lower
bound on the cardinality of any complete set
of monotones. In particular, we show that the
information contained in the degrees of viola-
tion of facet-defining Bell inequalities is not
sufficient for quantifying nonclassicality, even
though it is sufficient for witnessing nonclas-
sicality. Finally, we show that the continuous
set of convexly extremal quantumly realiz-
able correlations are all at the top of the pre-
order of quantumly realizable correlations. In
addition to providing new insights on Bell
nonclassicality, our work also sets the stage
for quantifying nonclassicality in more gen-
eral causal networks.
Accepted in Quantum 2020-05-28, click title to verify. Published under CC-BY 4.0.1 1
arXiv:1903.06311v4 [quant-ph] 4 Jun 2020
Contents
1 Introduction 3
1.1 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 How to read this article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Motivating our approach and contrasting it with alternatives 4
2.1 Three views on Bell’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The resource theory suggested by the causal modelling paradigm . . . . . . . . . . . . . . . . 6
2.3 Contrast to the strictly operational paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Contrast to the superluminal causation paradigm . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Details of the resource theory 12
3.1 Free and nonfree common-cause boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 The free operations on common-cause boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Convexity of the set of free operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Resource theory preliminaries 18
4.1 Global features of a pre-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Features of resource monotones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Monotone constructions for any resource theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 A linear program for determining the ordering of any pair of resources 21
6 Two useful monotones 22
6.1 Preliminary facts regarding CHSH inequalities and PR boxes . . . . . . . . . . . . . . . . . . 22
6.2 Defining the two useful monotones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3 Closed-form expressions for M
CHSH
and M
NPR
for
(
2 2
2 2
)
-type resources . . . . . . . . . . . . . 25
7 Properties of the pre-order of common-cause boxes 26
7.1 Inferring global properties of the pre-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.2 Incompleteness of the two monotones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.3 At least eight independent measures of nonclassicality . . . . . . . . . . . . . . . . . . . . . . 30
8 Properties of the pre-order of quantumly realizable common-cause boxes 31
9 Conclusions and outlook 34
Appendices 37
A Comparing our framework with prior work 37
A.1 WPICC versus LOSR as the set of free operations . . . . . . . . . . . . . . . . . . . . . . . . 37
A.2 An oversight in the literature concerning how to formalize LOSR . . . . . . . . . . . . . . . . 37
A.3 Generalizing from Bell scenarios to more general causal structures . . . . . . . . . . . . . . . 41
B Proofs 44
B.1 Proof of Proposition 19: closed-form expression for M
NPR
(R) . . . . . . . . . . . . . . . . . . 44
B.2 Proof of Proposition 21: when the two monotones are complete . . . . . . . . . . . . . . . . . 46
B.3 Proof of Proposition 23: all nonfree resources of type
(
2 2
2 2
)
are orbital . . . . . . . . . . . . . . 47
B.4 Proof of Proposition 25: lower bound on the number of monotones in any complete set . . . 49
References 50
2
1 Introduction
Bell’s theorem [1, 2] highlights a precise sense in
which quantum theory requires a departure from a
classical worldview. Furthermore, violations of Bell
inequalities provide a means for certifying the non-
classicality of nature, independently of the correct-
ness of quantum theory. This is because Bell inequal-
ities can be tested directly on experimental data.
Experimental tests under very weak assumptions
have confirmed this nonclassicality [35]. Correla-
tions that violate Bell inequalities have also found
applications in information theory. Specifically, they
constitute an information-theoretic resource inso-
far as they can be used to perform various cryp-
tographic tasks in a device-independent way [614].
Consequently, much previous effort has been made
to quantify resourcefulness of correlations within
Bell scenarios [1522].
In this paper, we take a resource-theoretic ap-
proach to quantifying the nonclassicality of a given
correlation in a Bell scenario, grounded in a new per-
spective on Bell’s theorem. This is the perspective
of causal modelling, which differs from the tradi-
tional operational approaches both conceptually and
in practice. Nevertheless, the natural choice of the
set of free operations for the Bell scenario in our
framework coincides with the one proposed in some
previous works [16, 17], namely, local operations
and shared randomness (LOSR)
1
. See also the
subsequent works of Refs. [2325].
Our causal perspective on quantifying Bell non-
classicality also generalizes naturally to a framework
for quantifying the nonclassicality of correlations in
more general causal scenarios. We discuss this gen-
eralization in Section A.2, but leave its development
to future work.
1.1 Summary of main results
We now summarize the content and main results of
our article.
In Section 2, we articulate the view on Bell’s
theorem that motivates our approach—the causal
1
There is widespread agreement that the free operations should
somehow consist of local operations supplemented with shared
randomness, however, different authors have been led to for-
malize this idea differently, that is, they have been led to
distinct proposals for the the set of free operations. Indeed,
the formalization provided in Refs. [
18
,
20
,
21
] is inconsistent
with the one given in Refs. [
16
,
17
] and therefore also with
the one presented here. A detailed discussion of this issue can
be found in Appendix A.2.
modelling paradigm—and contrast it with two other
views on Bell’s theorem, namely, the strictly oper-
ational and superluminal causation paradigms. In
particular, we explain how the differences between
these views impacts how one conceptualizes Bell in-
equality violations as a resource, and we highlight
some of the advantages of our approach relative to
the alternatives. We also introduce the notion of par-
titioned process theories [26] as the mathematical
framework for resource theories that we adopt in
this article.
In Section 3, we provide a formal definition of the
resource theory to be studied. For bipartite Bell sce-
narios, we argue that the set of processes which nat-
urally constitute the resources in our approach is the
set of all bipartite processes with classical inputs and
outputs that can arise within a causal model with
a (possibly nonclassical) common cause between the
wings. We also argue that the natural set of free oper-
ations on such processes are those that are achieved
by embedding the process in a circuit for which the
only connection between the wings is a classical com-
mon cause, and we demonstrate that this is equiv-
alent to the set of local operations and shared ran-
domness, as the latter is formalized in Refs. [16, 17].
In Section 4, we introduce some of the central
concepts of any resource theory, including the no-
tion of a pre-order and its features, the notion of
monotones and complete sets thereof, and the no-
tions of cost and yield monotones, which underlie
the explicit monotone constructions that follow.
In Section 5, we show how one can use two in-
stances of a linear program to determine the or-
dering relation which holds between any pair of re-
sources (see Proposition 15 and the discussion that
follows it).
In Section 6, we define two monotones of particu-
lar interest. The first (defined in Eq. (33)) is based
on a yield construction relative to all resources in the
Clauser-Horne-Shimony-Holt (CHSH) scenario [27]
(a bipartite Bell scenario where the settings and out-
comes all have cardinality two) and where the yield
is measured by the value of the canonical CHSH
functional. The second (defined in Eq. (36)) is based
on a cost construction relative to a one-parameter
family of resources in the CHSH scenario and where
the cost is measured again by the value of the canon-
ical CHSH functional. Although both of these mono-
tones are originally defined in terms of an optimiza-
tion problem, we derive closed-form expressions for
each of them for resources within the CHSH scenario
3
(see Propositions 17 and 19 respectively). We show
that within the CHSH scenario [27], a variety of
monotones which have been previously studied are
all equivalent (up to a monotonic function) to the
first of these monotones (see Corollary 18). Because
our two monotones are provably not equivalent, this
result implies that the second of our monotones pro-
vides information beyond that given by previously
studied monotones.
In Section 7 we leverage our two monotones to
derive various global properties of the pre-order
induced by single-copy deterministic conversions.
Specifically, we prove that the pre-order:
is not complete (i.e., there exist incomparable
resources),
is not weak (the incomparability relation is not
transitive),
has both infinite width and infinite height,
is locally infinite.
We also prove that the two monotones just men-
tioned do not completely characterize the pre-order
of resources, by showing that they fail to do so even
for the special case of the CHSH scenario. We further
show (in Theorem 26) that no fewer than eight con-
tinuous monotones can do the job. We also show (in
Proposition 23) that the equivalence classes among
nonfree resources in the CHSH scenario (though not
in general) are given exactly by the orbits of the
symmetry group of deterministic free operations.
Finally, in Section 8, we show that all of the global
features of the pre-order hold even for the strict sub-
set of resources which can be realized in quantum
theory. We also prove (in Lemma 27) that every ex-
tremal quantumly realizable resource is at the top of
the pre-order of quantumly realizable resources, and
(in Proposition 28) that there are a continuous set of
incomparable resources at the top of this pre-order.
1.2 How to read this article
We will demonstrate in Section 3 that in spite of
the difference in our attitude towards Bell’s theo-
rem, the definition of the set of resources and the
set of free operations that is natural for the Bell
scenario within the causal modelling paradigm coin-
cides with a definition that has been proposed within
the strictly operational paradigm, namely, the one
proposed in Refs. [16, 17]. Because Bell scenarios
are the focus of our article, any reader who would
rather take the strictly operationalist attitude to-
wards Bell’s theorem can reinterpret all of our re-
sults through that lens. In particular, readers who
are already sympathetic to the notion that LOSR,
as defined in Refs. [16, 17], is the right choice of free
operations may wish to skip Sections 2 and 3.
To understand our conviction that LOSR consti-
tutes the right choice of free operations for Bell sce-
narios, however, readers are advised to read Sec-
tions 2 and 3.2. In particular, to understand how
our approach differs (advantageously) from other
approaches, readers are encouraged to examine Sec-
tions 2.3 and 2.4 as well as Appendix A.
Because Section 4 reviews basic definitions and
terminology for concepts related to resource theories,
any reader who has expertise on resource theories
may wish to skip this section. We note, however, that
some of the material presented therein is not found
in standard treatments, such as our discussion of
global properties of a pre-order and our discussion
of a scheme for constructing useful cost and yield
monotones.
The presentation of our novel technical results be-
gins in Section 5.
2 Motivating our approach and con-
trasting it with alternatives
2.1 Three views on Bell’s theorem
The traditional commentary on Bell’s theorem [28,
29] takes a particular view on how to articulate the
assumptions that are necessary to derive Bell in-
equalities. Among these assumptions, two are typ-
ically highlighted as deserving of the most scrutiny,
namely, the assumptions that are usually termed re-
alism and locality
2
. Abandoning one or the other of
these two assumptions is the starting point of most
commentaries on what to do in the face of violations
of Bell inequalities.
3
Furthermore, a schism seems to
have developed between the camps that advocate for
each of these two views [30].
Among the researchers who take Bell’s theorem
to demonstrate the need to abandon realism, there
is a contingent which adopts a purely operational at-
titude towards quantum theory, that is, an attitude
wherein the scientist’s job is merely to predict the
statistical distribution of outcomes of measurements
performed on specific preparations in a specified ex-
perimental scenario. We shall refer to the members
2
Note, however, that different authors will formalize these
assumptions in different ways.
3
See, however, the discussion of superdeterminism in footnote 7.
4
of this camp as operationalists [31]. For such re-
searchers, a violation of a Bell inequality is simply a
litmus test for the inadequacy of a classical realist ac-
count of the experiment. One particular type of oper-
ationalist attitude, which we shall term the strictly
operational paradigm, advocates that physical
concepts ought to be defined in terms of operational
concepts, and consequently that any properties of
a Bell-type experiment, such as whether it is sig-
nalling or not and what sorts of causal connections
might hold between the wings, must be expressed in
the language of the classical input-output function-
ality of that experiment. In other words, they advo-
cate that the only concepts that are meaningful for
such an experiment are those that supervene
4
upon
its input-output functionality.
5
Most prior work on
quantifying the resource in Bell experiments has
been done within this paradigm, and the characteris-
tic of experimental correlations that is usually taken
to quantify the resource is simply some notion of dis-
tance from the set of correlations that satisfy all the
Bell inequalities.
Consider, on the other hand, the researchers who
take realism as sacrosanct, and in particular those
who take Bell’s theorem to demonstrate the failure
of locality—that is, the existence of superluminal
causal influences [33, 34].
6
Researchers in this camp,
whom we shall refer to as advocates of the super-
luminal causation paradigm, would presumably
find it natural to quantify the resource of Bell in-
equality violations in terms of the strength of the
superluminal causal influences required to account
for them (within the framework of a classical causal
model). An approach along these lines is described
4
A
-properties are said to supervene on
B
-properties if every
A-difference implies a B-difference.
5
Some might describe what we have here called the strictly op-
erational paradigm as the “device-independent” paradigm [
32
],
however, we avoid using the latter term here because its usage
is not restricted to describing a particular type of empiricist
philosophy of science: it also has a more technical meaning in
the context of quantum information theory, wherein it indi-
cates whether or not a given information-theoretic protocol
depends on a prior characterization of the devices used therein.
Indeed, Bell-inequality-violating correlations have been shown
to be a key resource in cryptography because they allow for
device-independent implementations of cryptographic tasks[
6
14].
6
Although such influences do not imply the possibility of su-
perluminal signalling, they do imply a certain tension with
relativity theory if one believes that the latter does not merely
concern anthropocentric concepts such as signalling, but also
physical concepts such as causation.
in Refs. [35, 36]. Earlier work on the communication
cost of simulating Bell-inequality violations [37, 38]
is also naturally understood in this way.
7
In recent years, a third attitude toward Bell’s
theorem—inspired by the framework of causal infer-
ence [42]—has been gaining in popularity. In this ap-
proach, the assumptions that go into the derivation
of Bell inequalities are [43]: Reichenbach’s princi-
ple (that correlations need to be explained causally),
the framework of classical causal modelling, and the
principle of no fine-tuning (that statistical indepen-
dences should not be explained by fine-tuning of the
values of parameters in the causal model). Here, a
violation of a Bell inequality does not lead to the tra-
ditional dilemma between realism and locality, but
rather attests to the impossibility of providing a non-
fine-tuned explanation of the experiment within the
framework of classical causal models. This attitude
implies the possibility of a new option for what as-
sumption to give up in the face of such a violation.
Specifically, the new possibility being contemplated
is that one can hold fast to Reichenbach’s principle
and the principle of no fine-tuning—and hence to the
possibility of achieving satisfactory causal explana-
tions of correlations—by replacing the framework of
classical causal models with an intrinsically nonclas-
sical generalization thereof.
As is shown in Ref. [43], because the correlations
in a Bell experiment do not provide a means of send-
ing superluminal signals between the wings, the only
causal structure that is a candidate for explaining
these correlations without fine-tuning is one wherein
there is a purely common-cause relation between the
wings, that is, one which admits no causal influences
between the wings. Therefore, the new approach to
achieving a causal explanation of Bell inequality vi-
olations is one that posits a common cause mech-
7
A less common view on how to maintain realism in the face
of Bell inequality violations is to hold fast to locality but give
up on a different assumption that goes into the derivation of
Bell inequalities, namely, that the hidden variables are sta-
tistically independent of the setting variables. This is known
as the “superdeterministic” response to Bell’s theorem [
39
].
Advocates of this approach would presumably find it natural
to quantify the resource of Bell inequality violations in terms
of the deviation from such statistical independence that is
required to explain a given violation. In particular, the results
of Refs. [
40
] and [
41
] seeking to quantify the nonindependence
needed to explain a given Bell inequality violation might be
framed within a resource-theoretic framework. However, given
that the setting variables can no longer be considered as freely
specifiable inputs within such an approach, it would be inap-
propriate to conceptualize a Bell experiment as a box-type
process as we have done here.
5
anism but replaces the usual formalism for causal
models with one which allows for more general pos-
sibilities on how to represent its components [44].
8
We refer to this attitude as the causal modelling
paradigm.
The causal modelling paradigm implies not only
a novel attitude towards Bell’s theorem, but also a
change in how one conceives of the resource that
powers the information-theoretic applications of
Bell-inequality violations. The resource is not taken
to be some abstract notion of distance from the set
of Bell-inequality-satisfying correlations within the
space of all nonsignalling correlations, as advocates
of the strictly operational paradigm seem to favour,
nor to consist of the strength of superluminal causal
influences, as advocates of the superluminal causa-
tion paradigm would presumably have it. Rather, we
take the resource to be the nonclassicality required
by any generalized causal model which can explain
the Bell inequality violations without fine-tuning.
We shall show that in the resource theory that
emerges by adopting this attitude, the nonclassical-
ity of common-cause processes in Bell experiments
cannot be captured solely by the degree of violation
of facet-defining Bell inequalities. That is, there are
distinctions among such common-cause processes—
different ways for these to be nonclassical—which
do not correspond to distinctions in the degree of
violation of any facet-defining Bell inequality.
2.2
The resource theory suggested by the
causal modelling paradigm
2.2.1 Generalized causal models
We will work with the notion of a generalized (i.e.,
not necessarily classical) causal model that has been
developed in Refs. [45, 46] using the framework of
generalized probabilistic theories (GPTs) [4749]),
and refer to it as a GPT causal model.
9
Since
we are interested in the distinction between classical
8
For instance, for the notion of a quantum causal model pro-
posed in Ref. [
44
], reversible deterministic causal dependencies
are represented by unitaries rather than bijective functions,
and lack of knowledge is represented by density operators
rather than by classical probability distributions.
9
In the language of operational probabilistic theories [
50
,
51
],
we are considering
free
and
causal
GPTs. A GPT is said to
be free if, for any mathematically well-formed closed circuit,
it specifies a joint probability distribution over the outcomes
of the instruments. A GPT is said to be causal if there exists
a unique deterministic effect (which is often interpreted as
excluding backwards-in-time signalling in any circuit).
and nonclassical, without specifically distinguishing
quantum and post-quantum types of nonclassical-
ity, we will not be making use of any of the recent
work [44, 52, 53] on devising an intrinsically quan-
tum notion of a causal model.
10
Definition 1.
A
GPT causal model
consists of
a causal structure, represented by a directed acyclic
graph (DAG), and a set of GPT parameters. The
parameters specify, for each node in the DAG, a GPT
operation from the composite system associated to
the parents of that node to the system associated to
the node.
One can approach the study of nonclassicality in
arbitrary causal structures from within the scope of
these GPT causal models, and pursue the develop-
ment of a resource theory of such nonclassical fea-
tures.
We focus on experimental scenarios that are mul-
tipartite. The different wings of the experiment
are commonly conceptualized as the laboratories
of different parties, particularly when discussing
information-theoretic tasks that may be undertaken
by the parties. If one restricts attention to scenar-
ios wherein locally, at each wing, systems can be
put into arbitrary causal relations with one another
(consistent with the absence of backwards-in-time
causal influences), then the only freedom in stipulat-
ing the causal structure is in stipulating the causal
relations that hold among the wings of the exper-
iment. Causal relations among the wings come in
two forms: (i) a relation indicating the potential for
causal influence from one wing to another, corre-
sponding to having access to a GPT channel from
one to the other, and (ii) a relation indicating the po-
tential for a common cause to act on a set of wings,
corresponding to having a source which distributes a
multipartite GPT state among them. The GPT op-
erations and GPT states representing, respectively,
these cause-effect and common-cause relations, to-
gether with the GPT operations representing causal
influences between systems at a given wing, consti-
tute the parameters of the GPT causal model.
The possible operational statistics that one can
observe in this scenario hence arise from all the pos-
sible ways one may assign values to the parameters
10
However, we will consider the question of when certain cor-
relations that arise in a GPT causal model can be quan-
tumly realized. Moreover, the follow-up work described in
Ref. [
54
] explicitly explores the distinction between resources
that are quantumly realizable and those that necessitate a
post-quantum GPT.
6
of the GPT causal model both to those pertain-
ing to the causal structure among the wings, and to
those pertaining to the local actions in each wing.
In this article, we focus on a particular type
of causal structure among the wings, namely, one
wherein there is a common cause that acts on all
of the wings, but no causal influences between any
of them, which we term a Bell scenario. However,
in Appendix A.3, we do include some discussion re-
garding other possible causal structures among the
wings. Details about how entangled states and oper-
ations are represented in a GPT causal model can
be found in Refs. [45, 47, 48], and explicit descrip-
tions of these for the Bell scenario are provided in
Sec. 3.1 and in Appendix A.3.
2.2.2
The distinction between free and nonfree re-
sources in the causal modelling paradigm
We conceptualize any experimental configuration as
a process from its inputs to its outputs. In the frame-
work of GPT causal models, one has the capacity to
consider processes that have GPT systems as inputs
and outputs at the various wings. However, we will
restrict our attention to processes that have only
classical inputs and outputs. Such processes can be
conceptualized as black-box processes, to which one
inputs classical variables and from which classical
variables are output. They are therefore precisely
the sorts of processes considered in the strictly op-
erational paradigm. We further restrict our atten-
tion to processes with a classical input and classical
output at each wing, where the input temporally
precedes the output.
11
In the strictly operational
paradigm, the term “box” is generally used as jar-
gon for such processes (for instance, as it is used in
the term “PR box” [57]). We therefore refer to such
processes as box-type processes or simply boxes.
Definition 2.
A
box
is a process with a classical
input and a classical output at each wing, represented
formally by a stochastic map from the tuple of inputs
to the tuple of outputs.
We use the term common-cause box to refer
to box-type processes which can be realized using a
causal structure consisting of a common cause act-
ing on all of the wings. These will be the resources
11
Thus, we do not consider processes which involve a sequence
over time of classical input variables and classical output
variables; that is, in the language of Refs [
55
,
56
], we do not
consider general n-combs.
that we focus on in this article. In GPT causal mod-
els, all common-cause boxes can be decomposed into
the preparation of a GPT state on a multipartite
system, followed by the distribution of the compo-
nent subsystems among the wings, followed by each
subsystem being subjected to a GPT measurement,
chosen from a fixed set according to the classical
input variable at that wing (the local setting vari-
able), and the result of which is the classical output
variable at that wing (the local outcome variable). In
short, such processes can be decomposed in the same
manner in which a multipartite Bell experiment is
decomposed in quantum theory.
Definition 3.
A
common-cause box
(or, equiv-
alentely, a
GPT-realizable common-cause box
)
is a box that can arise from a GPT causal model of
a multipartite Bell scenario, so that the inputs and
outputs correspond, respectively, to the setting and
outcome variables associated to a set of local GPT
measurements implemented on a multipartite GPT
state.
The distinction between common-cause boxes
that are classically realizable and those that are
not (illustrated for the bipartite case in Fig. 1) is sim-
ply the distinction between whether there is a classi-
cal causal model underlying the process, or whether
it is only realizable by a causal model which invokes
a nonclassical GPT.
Classical causal models are causal models wherein
all systems mediating causal influences are repre-
sented by classical variables, so that every common-
cause source is represented by a joint probability dis-
tribution (referred to as shared randomness) and ev-
ery channel is represented by a conditional probabil-
ity distribution. Equivalently, classical causal mod-
els can be understood to arise as a subset of GPT
causal models wherein the systems are presumed
to be nonclassical (for instance, they might be pre-
sumed to be quantum), but every common-cause
source is represented by a GPT-unentangled state
and every channel is taken to be GPT-entanglement-
breaking. Particularizing to the case of common-
cause boxes, we have:
Definition 4.
A
classically realizable common-
cause box
is a common-cause box that admits of a
classical causal model, such that the common-cause
source consists of shared randomness. Equivalently,
it is a common-cause box that admits of a GPT causal
model wherein the common-cause source consists of
a GPT-unentangled state.
7
(a)
(b)
(a)
(a)
(b)
(b)
Figure 1: In the bipartite scenario, the distinction between
(a) a generic GPT-realizable common-cause box and (b) a
classically realizable common-cause box. Here and through-
out this article, single-line edges denote classical systems,
and single-line boxes denote processes that have only classi-
cal inputs and outputs (depicted in light blue). Double-line
edges denote nonclassical systems and double-line boxes
denote processes that have one or more nonclassical in-
puts or outputs (depicted in pink). Any common-cause
box whose input-output functionality is consistent with an
internal structure of the type indicated in (b) (regardless of
its actual internal structure) is termed classically realizable
and is considered free, while a common-cause box whose
input-output functionality is not consistent with the struc-
ture of (b) but instead is only consistent with an internal
structure of the type indicated in (a) is considered nonfree.
It follows that the free common-cause boxes are
precisely the nonsignalling boxes that satisfy all
the Bell inequalities, while the costly common-cause
boxes are the nonsignalling boxes that violate some
Bell inequality
12
.
2.2.3
Quantifying resourcefulness in the causal mod-
elling paradigm
In order to quantify the nonclassicality of common-
cause boxes (that is, the extent to which they fail
to be classically realizable), we will use an approach
to resource theories described in Ref. [26], namely,
the framework of partitioned process theories. An en-
veloping theory of processes must be specified,
together with a subtheory of processes that can be
implemented at no cost, called the free subtheory
of processes. This partitions the set of all processes
in the enveloping theory into free and costly (i.e.,
nonfree) processes. One can then ask of any pair of
12
Indeed, since the GPT colloquially known as Boxworld realizes
all and only the nonsignalling boxes in Bell scenarios [
48
], it
follows that all nonsignalling boxes admit of a GPT causal
model.
processes in the enveloping theory of a given type
whether the first can be converted to the second by
composing it with a process (of the appropriate type)
from the free subtheory. If interconversion between
processes of type T require composition with a pro-
cess of type T
0
, then the set of free operations
on processes of type T are the elements of the free
subtheory of processes that are of type T
0
. Pairwise
convertibility relations under the set of free opera-
tions define a pre-order on the set of the resources
of interest, and a partial order over the equivalence
classes of such resources. One can then quantify the
relative worth of different resources by their rela-
tive positions in this partial order. Functions over
resources that preserve ordering relations, termed
monotones, provide a particularly simple means of
quantifying the worth of resources.
The resource theory considered in this article will
be described in full detail in Sec. 3. Nonetheless, we
provide a sketch of its definition here in order to be
able to highlight the ways in which it contrasts with
other approaches.
We take the enveloping theory of processes to
include all GPT-realizable common-cause boxes as
well as the GPT-realizable processes that take ev-
ery such box to another such box while only making
use of a common cause (depicted in Fig. 2(a)).
13
A
process that takes a box to a box we will refer to
as a clamp because it is a process that has the form
of a comb with two teeth (relative to the notion of
“comb” introduced in [55, 56]). More precisely, a pro-
cess taking boxes to boxes is a clamp with classical
inputs and outputs. Those that only make use of a
common cause, we refer to as common-cause clamps
with classical inputs and outputs. Such clamps are
the most general type of process required in our en-
veloping theory because common-cause boxes are a
special case of these (for instance, when all the sys-
tems at the inputs and outputs on the bottom teeth
of the clamp are trivial).
We take the free subtheory of processes in our re-
source theory to consist of the subset of common-
cause clamps with classical inputs and outputs that
can be realized in a classical causal model, termed
classically realizable. The distinction between a
generic GPT-realizable common-cause clamp and a
13
This is a general approach to determining a pre-order over
resources of a given type—define the enveloping theory to
include processes corresponding to the resource type of interest
as well as the processes that are required to interconvert
between such resources.
8
classically realizable one is depicted in Fig. 2. By
virtue of boxes being a special type of clamp, this
definition is consistent with Definition 4.
14
(a) (b)
Figure 2: In the bipartite scenario, the distinction between
(a) a generic GPT-realizable common-cause clamp with
classical inputs and outputs and (b) a classically realiz-
able common-cause clamp with classical inputs and outputs.
Any common-cause clamp whose input-output functionality
is consistent with an internal structure of the type indi-
cated in (b) (regardless of its actual internal structure) is
termed classically realizable and is considered free, while a
common-cause clamp whose input-output functionality is
not consistent with the structure of (b) but instead is only
consistent with an internal structure of the type indicated
in (a) is considered nonfree.
To determine the ordering relations that hold
among these common-cause boxes, one must deter-
mine the convertibility relations among them. Given
the definition of our resource theory, whether one
GPT-realizable common-cause box can be converted
to another is determined by whether this can be
achieved by processing it with a classically realiz-
able common-cause clamp, as depicted in Fig. 3. This
subsumes correlated local processings of the inputs
and outputs of the box, as we describe in Section 3.2.
14
For both the GPT-realizable and classically realizable varieties
of these processes, one can define notions of sequential and
parallel composition such that the set of processes, together
with these composition relations, satisfy the formal definition
of a process theory [
26
], thereby justifying the claim that
the resource theory we have defined is formally a partitioned
process theory. The proof of this fact, however, is not rele-
vant to any of the results in this article and is postponed to
forthcoming work [58].
Figure 3: In the bipartite scenario, the most general form of
a free operation (in blue) taking a GPT-realizable common-
cause box (in pink) to another.
2.2.4 A note about nomenclature
In this article, we avoid describing the resource be-
hind Bell inequality violations as “nonlocality”. This
is because we believe that it is only for those who
take the lesson of Bell’s theorem to be the existence
of superluminal causal influences that it is appro-
priate to describe violations of Bell inequalities by
this term. Researchers in the operationalist camp
have not, generally speaking, avoided using the term
“nonlocality”, but seem instead to use it as a syn-
onym for “violation of a Bell inequality” rather than
to imply a commitment to superluminal causal influ-
ences. However, we believe that such a usage invites
confusion and so we opt instead to avoid the term
altogether. Nevertheless, our project is very much in
line with earlier projects that describe themselves as
developing a “resource theory of nonlocality”, such
as Refs. [1518].
2.3
Contrast to the strictly operational
paradigm
As noted in the introduction and as will be demon-
strated in Section 3.2, in the special case of Bell
scenarios—the focus of this article—the natural
set of free operations within our causal modelling
paradigm is equivalent to one of the proposals for the
set of free operations made in earlier works within
the strictly operational paradigm, namely, local oper-
ations and shared randomness (LOSR), as the latter
is defined in Refs. [16, 17]. Additionally, the nat-
ural enveloping theory adopted in the strictly op-
erational approach, namely, the set of no-signalling
9
boxes, also coincides with that of our enveloping the-
ory for the case of Bell scenarios, namely, the set
of GPT-realizable common-cause boxes (where the
equivalence of these two sets can be inferred from the
results of Ref. [48]). Therefore, in spite of the differ-
ence in the attitude we take towards Bell’s theorem,
the resource theory that we define for Bell scenarios
is the same as the one studied in Refs. [16, 17].
Nonetheless, the difference in our attitude towards
Bell’s theorem is not inconsequential. We presently
outline its significance for the project of this article
as well as for potential future generalizations of this
project.
Most importantly, the causal modelling approach
diverges sharply from any strictly operational ap-
proach once one considers causal structures beyond
Bell scenarios. As discussed in Appendix A.3, in
a resource theory of nonclassicality for more gen-
eral causal structures, both the free subtheory and
the enveloping theory proposed by the causal mod-
elling approach are radically different from those
suggested by the strictly operational approach. In
particular, the free subtheory need not be LOSR in
a general causal structure and the enveloping the-
ory need not be the set of all nonsignalling opera-
tions. Our approach allows us to define a resource
theory that is specific to a scenario in which only
strict subsets of the wings are connected by common
causes [46, 59] (such as the triangle-with-settings sce-
nario described in Appendix A.3) and this provides
a concrete example of a case where the free subthe-
ory is not LOSR and the enveloping theory is not
all nonsignalling operations. In these cases, the free
operations are“local operations and causally admiss-
able shared randomness”,wherein only those subsets
of wings that are connected by a common cause have
shared randomness. This is distinct from the LOSR
operations, which assume that randomness is shared
between all the wings. It seems unlikely that the re-
source theory we propose in these cases can be mo-
tivated (or even fully characterized) in the strictly
operational paradigm.
Even for Bell scenarios, however, the causal mod-
elling approach offers advantages over its competi-
tors. In particular, it singles out a unique set of free
operations, while the strictly operational approach
does not. From our perspective, the resource under-
lying Bell inequality violations is the nonclassical-
ity of the causal model required to explain them
with a common cause, so clearly the free operations
should involve only classical common causes act-
ing between the wings. In the strictly operational
paradigm, by contrast, any operation which pre-
serves no-signalling and takes local boxes to local
boxes might constitute a legitimate candidate for
a “free” operation. This ambiguity is reflected in
the existence of distinct proposals for the set of free
operations in strictly operational resource theories.
Aside from LOSR, there is also a proposal called
wirings and prior-to-input classical communication
(WPICC) [18] which allows for classical causal influ-
ences between the wings prior to when the parties
receive their inputs (See Appendix A.1). If one be-
lieves that there is a singular concept which under-
lies the violation of Bell inequalities, then at most
one of these proposals (LOSR or WPICC) can be
taken as the relevant set of free operations.
15
Al-
though WPICC operations meet all desired opera-
tional criteria, they are immediately ruled out as
candidates for the free operations within the causal
modelling paradigm, on the grounds that they in-
volve nontrivial cause-effect influences between the
wings.
Another advantage of our approach for the Bell
scenario is that it highlights the fact that LOSR is
by construction a convex set, a fact which is criti-
cal for the algorithmic method that we derive for
determining the ordering relation between any two
resources. In highlighting this fact, our approach led
us to notice an oversight in some previous attempts
to formalize LOSR, as discussed in Appendix A.2.
Finally, we note that prior work of Geller and
Piani [17] departs from the strictly operational
paradigm through their use of the unified operator
formalism [60, 61], which is analogous to the quan-
tum formalism, but where nonpositive Hermitian op-
erators are allowed to represent states. They do not
characterize boxes primarily by their input-output
functionality, but rather as a composition of a bi-
partite source with local measurements. Indeed in
their Fig. 4, they explicitly depict the internal struc-
ture of the box. It is in this sense that their approach
does not quite fit the mould of a strictly operational
approach but is rather somewhat more in the flavour
of the causal modelling approach we have described
here.
Nonetheless, the unified operator formalism dif-
fers significantly from the GPT formalism of
15
Competing sets of free operations may be interesting for study-
ing phenomena other than the resource that powers violations
of Bell inequalities, but this is not the issue at stake in this
article.
10
Refs. [45, 46] with respect to the independence of the
nonclassical common cause from the measurements
employed in realizing nonclassical boxes. In the uni-
fied operator formalism, the Hermitian operator de-
scribing the shared state cannot be chosen freely for
a given set of quantum measurements, because some
choices would yield negative numbers rather than
valid probabilities. By contrast, in the GPT formal-
ism that we adopt here, the set of GPT states is
contained within the dual of the set of GPT product
measurements, and hence any measurement scheme
can be paired with any shared state while yielding
valid probabilities. The causal modelling paradigm
must reject any dependence of the shared state on
the choice of measurements, while such dependence
is unavoidable within the unified operator formalism.
As defined in Ref. [42], a causal model is a directed
acyclic graph, or equivalently, a circuit of causal pro-
cesses, wherein the distinct processes in the circuit
are required to be autonomous (i.e., independently
variable). We therefore classify Ref. [17] as neither
within the causal modelling paradigm nor within the
strictly operational paradigm, while still exhibiting
some features of each of these approaches.
2.4
Contrast to the superluminal causation
paradigm
To our knowledge, advocates of the superluminal
causation paradigm have not attempted to develop
a resource theory for Bell inequality violations (al-
though Refs. [35, 36] are related in spirit). If it were
attempted (within the framework of Ref. [26]), then
the commitments of the approach suggest that it
would also be done differently from the way we have
done so here. Those who endorse the superluminal
causation paradigm do not shy away from the notion
of causation, and hence a resource theory developed
within their paradigm could be presented using the
same framework that we use here that of causal
models. However, such an approach would likely be
framed entirely in terms of classical causal models,
rather than introducing the notion of GPT causal
models.
Advocates of the superluminal causation
paradigm would naturally define the free boxes
to be those that involve only subluminal causes.
Hence, in scenarios wherein the inputs and the
outputs at one wing are space-like separated from
those at the other wings, so that subluminal causal
influences cannot act between the wings, a box is
free if and only if it can be realized by a classical
common cause. Thus, the natural choice of the free
subtheory in