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 [3–5]. 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 [6–14].
Consequently, much previous effort has been made
to quantify resourcefulness of correlations within
Bell scenarios [15–22].
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. [23–25].
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) [47–49]),
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. [15–18].
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 the superluminal causation paradigm
coincides with the free subtheory in the causal mod-
elling paradigm. On the other hand, the natural
choice of the enveloping theory in the superluminal
causation paradigm consists of the set of boxes that
are classically realizable given superluminal causal
influences between the wings. This differs from the
enveloping theory in the causal modelling paradigm
because it includes boxes that are signalling. In the
superluminal causation paradigm, therefore, it is
natural to try and quantify the resource in terms
of the strength of the superluminal causal influence
between the wings that is required to explain it in
a classical causal model.
16
Because the enveloping theory within this
paradigm includes not only non-signalling boxes
that violate Bell inequalities but signalling boxes
as well, the resource theory is rich enough to de-
scribe communication between the wings. Therefore,
defining the resource theory in this way would not
distinguish common-cause resources that are classi-
cally realizable from those that are not (as we pro-
pose to do here), but would instead draw a line be-
tween common-cause resources that are classically
realizable and everything else — including classical
signalling resources.
17
If one were to go this route,
then all of classical Shannon theory would be sub-
sumed in the resource theory. A potential response
to this expansion in the scope of the project might
be to try to eliminate such signalling resources by
hand, by demanding that the enveloping theory was
constrained to those boxes that are non-signalling
among the wings. Such a response, however, seems
to compromise the ideals of the superluminal cau-
sation paradigm, because no-signalling is an opera-
tional notion rather than a realist one.
18
16
It should be noted that no finite speed of superluminal causal
influences can satisfactorily account for the predictions of
quantum theory, per Ref. [
62
], so such influences would need
to be assumed to be of infinite speed.
17
That is, if one seeks to partition resources of a given type
into classical and nonclassical varieties, then defining the
enveloping theory correctly is just as important as defining
the free subtheory correctly.
18
John Bell famously argued against the idea that no-signalling
could embody an assumption of locality in a fundamental
physical theory on the grounds that it was too anthropocen-
tric [63]:
...the “no signaling” notion rests on concepts which
are desperately vague, or vaguely applicable. The
11
3 Details of the resource theory
3.1 Free and nonfree common-cause boxes
We begin by formalizing the relevant definitions
from Sec. 2.2.2 and 2.2.3, and providing more details
about the definition of the resource theory. For ease
of presentation, we focus throughout on the bipar-
tite Bell scenario, but the multipartite Bell scenario
can be formalized analogously.
Fig. 1(a) depicts the structure of a generic GPT-
realizable common-cause box. The classical variables
that range over the (fixed) choices of local measure-
ments are termed the setting variables, denoted
S (left wing) and T (right wing), while the classi-
cal variables that range over the possible results of
these measurements are termed the outcome vari-
ables, denoted X (left wing) and Y (right wing). In
this article, we will refer to the cardinality of the set
over which a variable X varies as “the cardinality of
X” and we will also sometimes refer to the cardinal-
ity of the setting (outcome) variable as simply “the
cardinality of the setting (outcome)”.
Let us label the system distributed to the left wing
by A and the one to the right wing by B. In the
GPT framework, states and effects on A (B) are
represented by vectors in a real vector space of di-
mension d
A
(d
B
), that is, in R
d
A
(R
d
B
). States and
effects on the composite AB are represented by vec-
tors in the tensor product of these vector spaces
19
,
R
d
A
⊗R
d
B
. If the GPT representation of the X
=
x
outcome of the S
=
s measurement on system A
is r
A
x|s
∈ R
d
A
and that of the Y
=
y outcome of
the T
=
t measurement on system B is r
B
y|t
∈ R
d
B
,
and if s
AB
∈ R
d
A
⊗ R
d
B
denotes the GPT state
assertion that “we cannot signal faster than light”
immediately provokes the question: Who do we
think we are? We who can make “measurements”,
we who can manipulate “external fields”, we who
can “signal” at all, even if not faster than light? Do
we include chemists, or only physicists, plants, or
only animals, pocket calculators, or only mainframe
computers?
19
Strictly speaking, we do allow for a GPT where states and
effects may correspond to vectors outside of the tensor product
of the local vector spaces. That is, we do not assume local
tomography. However, since the causal structure of a Bell
scenario is such that the measurements are assumed to be local,
we can focus on the effects within
R
d
A
⊗ R
d
B
without loss of
generality, and therefore also on the states within
R
d
A
⊗ R
d
B
.
In other words, any so-called holistic degrees of freedom in
the GPT play no role in Bell scenarios, and therefore we can
ignore them without loss of generality.
of the composite AB, then the conditional proba-
bility distribution associated to this GPT-realizable
common-cause box is
P
XY |ST
(xy|st) = (r
A
x|s
⊗ r
B
y|t
) · s
AB
, (1)
where · denotes the Euclidean inner product.
By virtue of their internal causal structure,
all GPT-realizable common-cause boxes satisfy
the no-signalling conditions P
Y |ST
= P
Y |T
and
P
X|ST
= P
X|S
. It is straightforward to verify that
this follows from Eq. (1) using the fact that
P
x
r
A
x|s
=
u
A
, where u
A
is the unique determin-
istic effect on A, which is independent of value s of
the setting variable, and using the analogous fact for
B.
The common-cause boxes that are considered to
be free in our resource theory are those that can
be realized when the GPT governing the internal
workings of the box is classical probability theory,
as depicted in Fig. 1(b).
In such cases, the scope of possibilities for the
overall functionality of the common-cause box can
be characterized as follows. The systems A and B
are described by classical variables,
Λ
A
and
Λ
B
(here assumed to be discrete). Classically, the com-
posite system AB is prepared in a joint distribu-
tion over these, P
Λ
A
Λ
B
. The GPT state in this
case is
[
s
AB
]
λ
A
λ
B
=
P
Λ
A
Λ
B
(
λ
A
, λ
B
)
, where
[
v
]
i
de-
notes the ith component of a vector v living in
vector space R
|Λ
A
|
⊗ R
|Λ
B
|
. Without loss of gen-
erality, we can take systems A and B to be per-
fectly correlated (by incorporating any noise into
the measurements), corresponding to the case where
P
Λ
A
Λ
B
(
λ
A
, λ
B
) =
P
λ
δ
λ
A
,λ
δ
λ
B
,λ
P
Λ
(
λ
)
for some dis-
tribution P
Λ
(
λ
)
, and where δ denotes the Kronecker-
delta function. This distribution over
Λ
A
and
Λ
B
can be conceptualized as follows: sample a variable
Λ
from some distribution, then let
Λ
A
and
Λ
B
be
copies of it.
Classically, the X=x outcome of the S=s mea-
surement on system A is modelled by a conditional
probability distribution P
X|SΛ
A
. The GPT effect as-
sociated to this measurement on A is r
A
x|s
with com-
ponents
[
r
A
x|s
]
λ
A
=
P
X|SΛ
A
(
x|sλ
A
)
. Similarly, the
GPT effect associated to the measurement on B is
r
B
y|t
and has components
[
r
B
y|t
]
λ
B
=
P
Y |T Λ
B
(
y|tλ
B
)
.
Substituting these expressions into Eq. (1), we con-
clude that a classically realizable common-cause box
12
satisfies
P
XY |ST
(xy|st)
=
X
λ
A
λ
B
P
X|SΛ
A
(x|sλ
A
)P
Y |T Λ
B
(y|tλ
B
)P
Λ
A
Λ
B
(λ
A
λ
B
)
=
X
λ
P
X|SΛ
(x|sλ)P
Y |T Λ
(y|tλ)P
Λ
(λ). (2)
This is recognized to be the expression for a condi-
tional probability distribution P
XY |ST
that satisfies
the Bell inequalities.
3.2
The free operations on common-cause
boxes
The most general free operation taking a bipartite
common-cause box with settings S, T and outcomes
X, Y to a bipartite common-cause box with settings
S
0
, T
0
and outcomes X
0
, Y
0
is the clamp depicted
in blue in Fig. 3. It is the most general processing
which makes use of a classical common cause that
can act on the local pre-processings and the local
post-processings at each of the wings. It subsumes as
special cases processings wherein classical common
causes act on any of the subsets of these four local
processings.
Note that the most general free operation allows
arbitrary feed-forward of classical information on
each wing, since this does not require any causal
influences between the wings.
20
But any such op-
eration can always also be put into the canonical
form depicted in blue in Fig. 4. It suffices to note
that the system that mediates the action of the com-
mon cause on the post-processings on a given wing
can always be passed down the classical side-channel.
Henceforth, we use this canonical form when describ-
ing the most general free operation.
Formally, such an operation transforms the condi-
tional probability distribution P
XY |ST
to P
X
0
Y
0
|S
0
T
0
as
P
X
0
Y
0
|S
0
T
0
=
X
XY ST
P
X
0
Y
0
ST |XY S
0
T
0
P
XY |ST
(3)
20
Because the only physical restriction we are imagining is that
no cause-effect influences are present between wings, feed-
forward of nonclassical information (that is, of arbitrary GPT
systems) at each wing is also a free LOSR operation. Without
loss of generality, however, we consider only feed-forward of
classical systems in this work, because this is already suf-
ficient to generate any conditional probability distribution
P
X
0
Y
0
ST |XY S
0
T
0
consistent with the causal structure, i.e.,
satisfying Eqs. (7-8).
where the conditional probability distribution
P
X
0
Y
0
ST |XY S
0
T
0
satisfies certain constraints, which
we specify below.
Figure 4: The canonical form of a generic bipartite LOSR op-
eration
P
X
0
Y
0
ST |XY S
0
T
0
(in blue) taking a common-cause
box
P
XY |ST
(in pink) to a common-cause box
P
X
0
Y
0
|S
0
T
0
.
Circuit fragments that map processes to processes
(such as the ones depicted in blue in Figs. 3 and 4)
have been studied extensively in recent years in a
variety of frameworks, most notably the quantum
combs framework of Refs. [55, 56], and the process
matrix framework of Refs. [64, 65]. If the source and
target resources are denoted by R and R
0
, respec-
tively, and the free operation is denoted by τ , we
represent Eq. (3) as
R
0
= τ ◦R, (4)
where ◦ is a particular instance of the link product
of Ref. [55].
On the left wing, the most general local pre-
processing takes as input the setting variable of
the target resource (S
0
) and the variable originating
from the common cause, and it generates as output
the setting variable of the source resource (S) as well
as an arbitrary variable which propagates down the
side-channel. The most general post-processing on
the left wing takes as input the outcome variable of
the source resource (X) and the side-channel vari-
able, and it generates as output the outcome vari-
able of the target resource (X
0
). Included as spe-
cial cases among these pre- and post-processings are
maps from S
0
to S and from X to X
0
that constitute
relabellings, coarse-grainings, or fine-grainings of the
variable, where the possibilities are constrained by
the cardinalities of these variables. Also included as
13
special cases are instances where the map from S
0
to S or the map from X
0
to X is chosen probabilisti-
cally, and instances where these two maps are corre-
lated (by making use of the side-channel). The anal-
ogous pre- and post-processings at the right wing
are also possible. Finally, the choices of maps on the
left can also be correlated with the choices of maps
on the right, by leveraging the common cause.
The free operations are characterized by those
P
X
0
Y
0
ST |XY S
0
T
0
which can be achieved via the type
of circuit fragment depicted in Fig. 4, namely, those
such that
P
X
0
Y
0
ST |XY S
0
T
0
(x
0
y
0
st|xys
0
t
0
) = (5)
X
λ
A
λ
B
P
X
0
S|XS
0
Λ
A
(x
0
s|xs
0
λ
A
)P
Y
0
T |Y T
0
Λ
B
(y
0
t|yt
0
λ
B
)
×P
Λ
A
Λ
B
(λ
A
λ
B
)
for some joint distribution P
Λ
A
Λ
B
and for
some P
X
0
S|XS
0
Λ
A
and P
Y
0
T |Y T
0
Λ
B
satisfying no-
retrocausation conditions
P
S|XS
0
Λ
A
= P
S|S
0
Λ
A
P
T |Y T
0
Λ
B
= P
T |T
0
Λ
B
.
(6)
One can directly check that any P
X
0
Y
0
ST |XY S
0
T
0
admitting of a decomposition as in Eq. (5) satisfies
the operational no-signalling constraints
P
X
0
S|XY S
0
T
0
= P
X
0
S|XS
0
P
Y
0
T |XY S
0
T
0
= P
Y
0
T |Y T
0
(7)
and the operational no-retrocausation conditions
P
S|XS
0
= P
S|S
0
P
T |Y T
0
= P
T |T
0
.
(8)
The parts of the circuit fragment in Fig. 4 that
are associated to P
X
0
S|XS
0
Λ
A
and P
Y
0
T |Y T
0
Λ
B
we
refer to as local operations. The part associated
to P
Λ
A
Λ
B
corresponds to a joint distribution on
the variables distributed to the two wings and can
therefore be conceived of as shared randomness.
Consequently, the free operations we are endorsing
here can indeed be described as local operations and
shared randomness (LOSR), as noted earlier.
Definition 5.
An operation is in the set
LOSR
(and termed an
LOSR operation
) if and only if it
is associated to a conditional probability distribution
P
X
0
Y
0
ST |XY S
0
T
0
that admits of the sort of decompo-
sition specified by Eqs. (5) and (6).
Previous resource-theoretic approaches to Bell-
inequality violations have also endorsed the intu-
itive notion that local operations supplemented with
shared randomness should constitute the free opera-
tions. Different works, however, have made different
proposals for how this notion ought to be formal-
ized. The correct formalization, in our opinion, is
the one provided in Geller and Piani [17] and inde-
pendently in deVincente [16], which coincides with
the one given above
21
. Therefore, in this article we
are endorsing the proposal of Refs. [16, 17] to take
LOSR as the free operations. On the other hand,
Refs. [18, 20, 21] have formalized the notion of lo-
cal operations supplemented with shared random-
ness differently, defining a strict subset of the set
LOSR defined above (a subset that can be shown
to be nonconvex). Nonetheless, we believe that this
discrepancy was an oversight and that it is unlikely
anyone would defend taking this subset rather the
full set to define the resource theory. We discuss the
issue in depth in Appendix A.2.
As a final comment, note that, without loss of
generality, we can take the joint distribution to be
P
Λ
A
Λ
B
(
λ
A
λ
B
) =
P
λ
δ
λ
A
,λ
δ
λ
B
,λ
P
Λ
(
λ
)
for some dis-
tribution P
Λ
, and hence express Eq. (5) as
P
X
0
Y
0
ST |XY S
0
T
0
(x
0
y
0
st|xys
0
t
0
) = (9)
X
λ
P
X
0
S|XS
0
Λ
(x
0
s|xs
0
λ)P
Y
0
T |Y T
0
Λ
(y
0
t|yt
0
λ)P
Λ
(λ).
As a consequence, the conditional probability distri-
bution P
X
0
Y
0
ST |XY S
0
T
0
can be conceptualized as the
more familiar object P
˜
X
˜
Y |
˜
S
˜
T
for setting variables
˜
S,
˜
T and outcome variables
˜
X,
˜
Y that are defined
as follows. We take the composite of the outputs of
the circuit fragment on the left wing, X
0
and S, as a
composite outcome variable
˜
X, so that
˜
X
:
= (
X
0
, S
)
.
Similarly, we take the composite of the inputs on the
left wing, X and S
0
, as a composite setting variable
˜
S, so that
˜
S
:
= (
X, S
0
)
. Making the analogous def-
initions for
˜
Y and
˜
T in terms of Y, T, Y
0
, T
0
on the
right wing, Eq. (9) can be rewritten as
P
˜
X
˜
Y |
˜
S
˜
T
(˜x˜y|˜s
˜
t) = (10)
X
λ
P
˜
X|
˜
SΛ
(˜x|˜sλ)P
˜
Y |
˜
T Λ
(˜y|
˜
tλ)P
Λ
(λ).
Recalling Eq. (2), it is clear that P
˜
X
˜
Y |
˜
S
˜
T
satisfies
all of the Bell inequalities. This illustrates the con-
21
The definition of
LOSR
given in Geller and Piani [
17
] is very
similar to the one provided here (see Fig. 4 therein), while
the one provided in de Vicente[
16
] is much more cumbersome.
14
sistency of our proposal for the free operations, for
we have just shown that the free operations on a re-
source P
XY |ST
are those that are achieved by taking
a link product [55] with a process P
X
0
Y
0
ST |XY S
0
T
0
:
=
P
˜
X
˜
Y |
˜
S
˜
T
which satisfies all of the Bell inequalities.
3.2.1
Cardinality-based types for boxes and for opera-
tions
Definition 6.
We define the
type of a common-
cause box
as the collection of cardinalities of the
setting and outcome variables, and we denote the
type of a resource
R
as [
R
]. We introduce the fol-
lowing notational convention to specify types: the
cardinalities of the setting variables for all
n
wings
and the cardinalities of the outcome variables for all
n
wings are specified as the bottom and top rows,
respectively, of a 2
×n
matrix. For example, for the
2-wing common-cause box depicted in Fig. 1, the
type is
(
|X| |Y |
|S | |T |
)
, where
|O|
denotes the cardinality of
a variable O.
If we further particularize to the CHSH scenario,
where the cardinalities of both setting and outcome
variables is 2, then the type is
(
2 2
2 2
)
.
Definition 7.
Consider a source resource
R
1
of
type [
R
1
] and a target resource
R
2
of type [
R
2
]. We
denote the
type of an operation τ
taking any re-
source of type [
R
1
] to any resource of type [
R
2
] by
[τ]
:
= [R
1
] → [R
2
]
, and we denote the set of all free
operations of type [R
1
] → [R
2
] by LOSR
[R
1
]→[R
2
]
.
Note that operations — including free operations
— can change the type of a resource, and hence spec-
ifying the type of an operation requires specifying
both the type of the initial resource as well as the
type of the final resource. This reflects the fact that
we have not restricted the cardinalities of X
0
, Y
0
, S
0
,
or T
0
in Eq. (3) in any way.
3.2.2
Locally deterministic operations and local sym-
metry operations
It is valuable to consider two special finite-
cardinality subsets of LOSR operations: those that
are deterministic and those that are invertible. Note
that the invertible LOSR operations are included
among the deterministic ones because any indeter-
minism in the operation would be an obstacle to
invertibility.
Definition 8.
An LOSR operation is in the set
LDO
(i.e., it is a
locally deterministic oper-
ation
) if and only if the conditional probabilities
P
X
0
Y
0
ST |XY S
0
T
0
which define the operation take val-
ues in
{
0
,
1
}
for all values of
X
0
,
Y
0
,
S
,
T
,
X
,
Y
,
S
0
and
T
0
. We denote the complete set of LDO operations
of type [R
1
] → [R
2
] by LDO
[R
1
]→[R
2
]
.
Deterministic LOSR operations—i.e., LDO
operations—factorize in the sense that every LDO
operation can be expressed as the product of two
local deterministic operations such that
P
det
X
0
Y
0
ST |XY S
0
T
0
= P
det
X
0
S|XS
0
P
det
Y
0
T |Y T
0
. (11)
This follows from the fact that the deterministic de-
pendences preclude any dependence on the shared
random variables λ
A
and λ
B
in Eq. (5), which then
reduces to Eq. (11). Furthermore, the no retrocau-
sation assumption of Eq. (8) implies that these de-
terministic dependencies are of the following form:
P
det
X
0
S|XS
0
= δ
S,f
A
(S
0
)
δ
X
0
,g
A
(X,S
0
)
,
P
det
Y
0
T |Y T
0
= δ
T,f
B
(T
0
)
δ
Y
0
,g
B
(Y,T
0
)
(12)
for some functions f
A
, g
A
, f
B
and g
B
. Specifically,
on the left wing, S is generated deterministically
as a function of S
0
(the pre-processing) and X
0
is
generated deterministically as a function of X and
S
0
(the post-processing, which is setting-dependent),
and similarly for the right wing. A generic bipartite
locally deterministic operation is depicted in Fig. 5.
The cardinality of the set LDO for a given type
can be easily deduced. Let |S|, |X|, . . . denote
the cardinalities of the variables S, X, . . . The to-
tal number of possibilities for the function g
A
is
|X
0
|
|X|·|S
0
|
, and the total number of possibilities for
the function f
A
is |S|
|S
0
|
, so that the total number
of possibilities for a deterministic operation on the
left wing is
|S| · |X
0
|
|X|
|S
0
|
. An analogous decom-
position holds for the deterministic operations on
the right wing, and the total number of possibilities
for these is
|T | · |Y
0
|
|Y |
|T
0
|
. Consequently, the car-
dinality of the set LDO in this bipartite case is
|LDO| =
|S| · |X
0
|
|X|
|S
0
|
|T | · |Y
0
|
|Y |
|T
0
|
. (13)
The other important subset of LOSR are those
type-preserving operations which are invertible (and
hence also deterministic). We refer to this subset of
LOSR operations as the local symmetry operations
and denote it LSO.
15
Figure 5: A generic bipartite locally deterministic operation
P
X
0
Y
0
ST |XY S
0
T
0
∈
LDO consists of a product of deter-
ministic operations at each wing. The black dots in the
figure represent classical copy operations, and the output
variables for each gate are deterministic functions of the
input variables for that gate.
Definition 9.
The set
LSO
(i.e., the
local sym-
metry operations
) is the subset of type-preserving
operations in LDO that are invertible.
Every local symmetry operation, P
sym
X
0
Y
0
ST |XY S
0
T
0
,
has the form of a locally deterministic operation,
P
det
X
0
Y
0
ST |XY S
0
T
0
, specified in Eqs. (11)-(12). That is,
P
sym
X
0
Y
0
ST |XY S
0
T
0
= P
sym
X
0
S|XS
0
P
sym
Y
0
T |Y T
0
. (14)
where
P
sym
X
0
S|XS
0
= δ
S,f
A
(S
0
)
δ
X
0
,g
A
(X,S
0
)
,
P
sym
Y
0
T |Y T
0
= δ
T,f
B
(T
0
)
δ
Y
0
,g
B
(Y,T
0
)
,
(15)
but where f
A
, g
A
are such that P
sym
X
0
S|XS
0
defines an
invertible map from
(
X, S
0
)
to
(
X
0
, S
)
, and where f
B
and g
B
are such that P
sym
Y
0
T |Y T
0
defines an invertible
map from
(
Y, T
0
)
to
(
Y
0
, T
)
. Unlike general LDO op-
erations, LSO operations are always type-preserving,
and hence the type
(
|X
0
| |Y
0
|
|S
0
| |T
0
|
)
always matches the type
(
|X| |Y |
|S | |T |
)
.
Note that an exchange of the parties is a symme-
try operation (i.e., invertible), but it cannot be im-
plemented by local operations, and so it is not part
of LSO.
As a final remark, notice that the set of LSO oper-
ations forms a group. This follows from the fact that
the properties of being deterministic and invertible
persist under composition, and that the inverse of
every LSO operation is in LSO. This group is gen-
erated by the permutations of the value of a set-
ting variable, and the permutations of the value of
an outcome variable, where the choice of the latter
permutation might depend also on the value of the
setting variable on the same wing.
In the bipartite case, the LSO group is a finite
group of order
22
|LSO| = (|S|!) · (|X|!)
|S|
· (|T |!) · (|Y |!)
|T |
, (16)
corresponding to the
(
|S|
!)
relabelings for the set-
tings of the left wing, multiplied by the
(
|X|
!)
re-
labelings of outcomes for each of the |S| differ-
ent settings, and similarly for the right wing. The
group can be generated by the relabelings of only
adjacent settings or outcomes, and hence the LSO
group admits a natural representation in terms of
(|S|−1) + |S|(|X|−1) + (|T |−1) + |T |(|Y |−1) gener-
ators (see Ref. [66, App. B]).
For a concrete example, consider the operations
transforming type
(
2 2
2 2
)
into type
(
2 2
2 2
)
. Throughout
this work, we index the values a variable X can
take as x ∈ {
0
,..., |X| −
1
}. Accordingly, in the
(
2 2
2 2
)
scenario, X, Y, S, T take values in {
0
,
1
}. Using
this notation, the group of LSO can be generated
explicitly by the four operations which interconvert
P
XY |ST
(
x, y|s, t
)
with either P
XY |ST
(
x, y|s⊕
1
, t
)
,
P
XY |ST
(
x, y|s, t⊕
1)
, P
XY |ST
(
x⊕s, y|s, t
)
, or
P
XY |ST
(
x, y⊕t|s, t
)
, where ⊕ denotes summation
modulo two.
23
One can readily verify [67] that the
order of this group is 64.
Suppose that a resource R is represented as a
real-valued vector
~
R of conditional probabilities
P
XY |ST
(
xy|st
)
, or any linear transformation thereof
(such as the representation in terms of correlators
used in Section 6). LSO operations act as invertible
linear maps on such a representation. Assuming f is
a linear function over
~
R, then its action can be repre-
sented as f
(
~
R
) =
~
f ·
~
R for some
~
f. Hence, it is equally
as meaningful to speak about
~
f being transformed
under LSO group elements as it is to speak about
~
R
being so transformed. The action of an LSO opera-
tion on
~
f can be thought of as applying the inverse
transformation to
~
R, i.e.,
(π
~
f) ·
~
R = f · (π
-1
~
R). (17)
22
The order of a group is the cardinality of the set of group
elements, i.e., the order of the
LSO
group quantifies the total
number of invertible LDO operations.
23
A second generating set of operations for this group is given
by τ
1
,..., τ
6
defined in Proposition 16.
16
Note that many type-changing LOSR operations
are equally well-defined as transformations on lin-
ear functions. The critical requirement is that the
operation be left-invertible, i.e., it should act as an
injective function on the set of conditional probabili-
ties. See Refs. [66, 68, 69] for discussions on the topic
of converting linear functions (and Bell inequalities
in particular).
3.3 Convexity of the set of free operations
We now show that the set of free operations is con-
vex, and that the extremal elements are determin-
istic, and enumerable for fixed type of the source
resource and of the target resource. This implies
that the set of free operations mapping from a given
source resource type to a given target resource type
is a polytope.
We begin by proving convexity.
Proposition 10.
The set LOSR is convex,
i.e., if
τ
0
∈ LOSR
and
τ
1
∈ LOSR
, then
wτ
0
+ (1 − w)τ
1
∈ LOSR for 0 ≤ w ≤ 1.
This follows from the fact that the resources re-
quired to achieve such a mixing are achievable using
LOSR. Suppose β is a binary variable that decides
whether τ
0
or τ
1
will be implemented. It suffices to
imagine that β is sampled from a distribution P
β
where P
β
(0) =
w, that it is copied and distributed to
both wings (with a copy sent down the side-channel
at each wing), and that the local processings that
are implemented on each wing are made to depend
on β (chosen so that if β
=
b, then τ
b
is implemented
overall). Because β can be incorporated into the def-
inition of the shared randomness, the procedure just
described is itself achievable using LOSR.
The convexity of the set of LOSR operations is
crucial for the technique we develop in the next
section to answer questions about resource conver-
sion. Recognizing the full potential of this convex-
ity is one of the key contributions of our work. In
Appendix A.2, we discuss convexity further, in par-
ticular noting that previous formulations of LOSR
did not seem to recognize the physical realizability
of convex mixing within LOSR, but rather imposed
convexity mathematically.
Next, we highlight features of the extremal free
operations.
Proposition 11.
The set of convexly extremal oper-
ations in LOSR are precisely the subset of operations
comprising LDO, namely the deterministic LOSR
operations.
This proposition is a minor generalization of
Fine’s argument [70], since the latter states that
locally deterministic models can generate any con-
ditional distribution that arises in a locally inde-
terministic model. As in Fine’s argument, here too
any indeterminism in the local operations can be
absorbed into the shared randomness, and hence al-
lowing indeterministic local operations provides no
more generality than considering only deterministic
local operations.
Proof.
It suffices to run Fine’s argument [
70
] for
the composite variables
˜
S,
˜
T,
˜
X
and
˜
Y
. To see this
explicitly, note that the constituent factors in the
expression for an LOSR operation in Eq.
(10)
can
be rewritten as
P
˜
X|
˜
SΛ
(˜x|˜sλ) =
X
λ
A
∈Λ
A
P
det,λ
A
˜
X|
˜
S
(˜x|˜s)P
Λ
A
|Λ
(λ
A
|λ),
P
˜
Y |
˜
T Λ
(˜y|
˜
tλ) =
X
λ
B
∈Λ
B
P
det,λ
B
˜
Y |
˜
T
(˜y|
˜
t)P
Λ
B
|Λ
(λ
B
|λ),
where for each value of
λ
A
, the conditional
P
det,λ
A
˜
X|
˜
S
describes a deterministic operation on the left wing
specifying the value of
˜
X
= (
X
0
, S
) for every value
of
˜
S
= (
X, S
0
), and similarly for
P
det,λ
B
˜
Y |
˜
T
on the right
wing. Plugging these back into Eq.
(10)
, we have
that
P
˜
X
˜
Y |
˜
S
˜
T
(˜x˜y|˜s
˜
t) = (18)
X
λ
A
,λ
B
P
det,λ
A
˜
X|
˜
S
(˜x|˜s)P
det,λ
B
˜
Y |
˜
T
(˜y|
˜
t)P
Λ
A
Λ
B
(λ
A
λ
B
),
where we have defined
P
Λ
A
Λ
B
(
λ
A
λ
B
)
:
=
P
λ∈Λ
P
Λ
A
|Λ
(
λ
A
|λ
)
P
Λ
B
|Λ
(
λ
B
|λ
)
P
Λ
(
λ
)
.
Eq.
(18)
shows that a generic indeterministic LOSR oper-
ation can always be decomposed into a convex
combination of products of deterministic operations
on each wing. Not only is it the case that the
convexly extremal LOSR operations are included
within the LDO operations, but there is actually
precise equality between these two sets: all LDO
operations are convexly extremal because every
LDO operation is a deterministic map.
What we have shown above is that any element
of LOSR
[R
1
]→[R
2
]
admits of a convex decomposition into el-
ements of LDO
[R
1
]→[R
2
]
. This implies the following useful
geometric fact:
17
Proposition 12 (Polytope of free operations).
The set of all free operations of a given type is a
polytope whose vertices are the locally deterministic
operations of that type,
LOSR
[R
1
]→[R
2
]
= ConvexHull
LDO
[R
1
]→[R
2
]
. (19)
The number of vertices of this polytope corre-
sponds to the cardinality of the set of LDO oper-
ations, as given in Eq. (13).
4 Resource theory preliminaries
A central question in any resource theory is whether
one resource can be converted to another via the free
operations. Many notions of conversion are studied:
single-copy deterministic conversion, single-copy in-
deterministic conversion (where the probability of
success need only be nonzero), multi-copy conver-
sion (where one is given more than one copy of the
resource), asymptotic conversion (where one is given
arbitrarily many copies), and catalytic conversion
(where one has access to another resource that must
be returned intact after the conversion). We here
focus on single-copy deterministic conversion.
As noted earlier, we denote the application of an
operation τ to a resource R by τ ◦ R. If R
1
can
be converted to R
2
by free operations, one writes
R
1
7−→R
2
, otherwise one writes R
1
Y7−→R
2
. Explic-
itly,
R
1
7−→R
2
denotes that ∃ τ ∈ LOSR
[R
1
] 7−→[R
2
]
such that R
2
= τ ◦R
1
,
and R
1
Y7−→R
2
denotes that @ τ ∈ LOSR
[R
1
] 7−→[R
2
]
such that R
2
= τ ◦R
1
.
If one can determine, for any pair of resources
R
1
and R
2
, whether R
1
can be converted to R
2
us-
ing a free operation, then one can determine the
pre-order over all resources that is induced by the
conversion relation. A pre-order, by definition, is a
transitive and reflexive binary relation between re-
sources. The conversion relation is reflexive because
the identity operation is free and maps a resource to
itself, while it is transitive because if R
1
7−→R
2
and
R
2
7−→R
3
then R
1
7−→R
3
.
There are four possible ordering relations that
might hold between a pair of resources.
R
1
is strictly above R
2
if:
R
1
7−→R
2
and R
2
Y7−→R
1
,
R
1
is strictly below R
2
if:
R
1
Y7−→R
2
and R
2
7−→R
1
,
R
1
is incomparable to R
2
if:
R
1
Y7−→R
2
and R
2
Y7−→R
1
,
R
1
is equivalent to R
2
if:
R
1
7−→R
2
and R
2
7−→R
1
.
If R
1
is either strictly above or strictly below R
2
, we
say that R
1
and R
2
are strictly ordered.
We pause to comment on the notion of equivalence
of resources. By definition, if R
1
is equivalent to R
2
then the conversion from one to the other is free in
both directions,
∃ τ
1
∈ LOSR
[R
1
] 7−→[R
2
]
such that R
2
= τ
1
◦ R
1
,
and ∃ τ
2
∈ LOSR
[R
2
] 7−→[R
1
]
such that R
1
= τ
2
◦ R
2
.
It need not be the case, however, that either of the
free operations τ
1
or τ
2
is invertible, nor that one is
the inverse of the other. For instance, if R
1
and R
2
are both free resources, then τ
1
can be the operation
which discards R
2
and prepares R
1
, while τ
2
can be
the operation which discards R
1
and prepares R
2
.
The conversion relation between resources implies
a corresponding conversion relation between equiv-
alence classes of resources (relative to the equiv-
alence relation defined above), wherein for any two
equivalence classes, they are either strictly ordered
or incomparable. The conversion relation between
equivalence classes is therefore antisymmetric and
describes a partial order relation rather than a
pre-order relation. One can therefore conceptualize
the project of characterizing the pre-order as a char-
acterization of the equivalence classes and of the par-
tial order that holds among these. In this work, we
do not provide a characterization of the equivalence
classes, and so our focus will be on directly charac-
terizing features of the pre-order of resources.
4.1 Global features of a pre-order
To have a complete understanding of deterministic
single-copy conversion in a resource theory, one must
have an understanding of the pre-order that this
conversion relation defines. In this section, we de-
scribe some of the basic features that characterize
pre-orders.
18
Perhaps the most basic question about a pre-order
of resources is whether or not it is totally pre-
ordered, meaning that every pair of elements in
the pre-order is strictly ordered or equivalent (i.e.,
the pre-order has no incomparable elements). Equiv-
alently, we say that a pre-order is totally pre-ordered
if and only if the partial order over equivalence
classes that it defines is totally ordered (i.e., has no
incomparable elements).
If there do exist incomparable resources, one can
ask if the binary relation of incomparability is tran-
sitive, in which case the pre-order is termed weak.
A chain is a subset of the pre-order in which ev-
ery pair of elements is strictly ordered. The height
of a pre-order is the cardinality of the largest chain
contained therein. An antichain is a subset of the
pre-order in which every pair of elements is incom-
parable. The width of a pre-order is the cardinality
of the largest antichain contained therein.
Other important properties of the pre-order refer
to the interval between a pair of resources, where
R is in the interval of R
1
and R
2
if and only if both
R
1
7−→R and R 7−→R
2
. If the number of equivalence
classes which lie in the interval between a pair of
resources is finite for every pair of inequivalent re-
sources, then the pre-order is said to be locally fi-
nite, otherwise it is said to be locally infinite.
4.2 Features of resource monotones
A resource monotone is a real-valued function
24
over
resources whose value cannot increase under any free
operation in the resource theory. Formally,
Definition 13.
A function
M
from resources to the
reals is called a
resource monotone
if and only if
R
1
7−→R
2
implies M(R
1
) ≥ M (R
2
), (20a)
or equivalently,
M(R
1
) < M (R
2
) implies R
1
Y7−→R
2
. (20b)
In other words, a resource monotone is an order-
preserving map from the pre-order of resources to
the total order of real numbers. Whenever some
monotone M and a pair of resources R
1
and R
2
satis-
fies M
(
R
1
)
< M
(
R
2
)
, we will say that the monotone
M witnesses the fact that R
1
Y7−→R
2
.
If the pre-order is not totally pre-ordered (i.e., if
there exist incomparable resources), then no single
24
Technically, it is an extended-real-valued function, where the
set of extended real numbers is obtained by adding
−∞
and
+∞ to the set of real numbers.
monotone can completely characterize the pre-order.
A complete characterization may be achieved, how-
ever, by a family of monotones. Specifically, a fam-
ily of monotones {M
i
}
i
is said to be complete if it
completely characterizes the pre-order, that is, if
∀R
1
, R
2
: R
1
7−→R
2
if and only if ∀i : M
i
(R
1
) ≥ M
i
(R
2
).
(21)
A complete set of monotones is therefore an alterna-
tive way of describing the pre-order.
Strictly speaking, monotones should be functions
from resources of any type in the resource theory
to the reals. However, many natural functions are
only defined for particular types of resources. For in-
stance, the function P
XY |ST
(00
|
00)
P
XY |ST
(11
|
01) +
P
XY |ST
(20
|
02)
is only defined for common-cause
boxes where the cardinalities of X and T are three.
To accommodate this, we define the notion of a
monotone relative to a set S: M is a monotone
relative to a set S of resources if and only if for all
{R
1
, R
2
} ∈ S, R
1
7−→R
2
implies M
(
R
1
)
≥ M
(
R
2
)
.
A family of monotones {M
i
}
i
is said to be complete
relative to a set S if it holds that
∀R
1
, R
2
∈ S : R
1
7−→R
2
if and only if ∀i : M
i
(R
1
) ≥ M
i
(R
2
).
(22)
If S is any set of resources all of which are of a
particular type, a monotone relative to S is said to
be type-specific.
4.3
Monotone constructions for any resource
theory
Here we review a variety of approaches to construct-
ing resource monotones. We will make use of these
versatile constructions to define an especially use-
ful pair of monotones for the resource theory of
common-cause boxes in Section 6.
4.3.1 Cost and yield monotones
It is possible to upgrade a type-specific monotone
to a type-independent monotone using either a cost
construction or a yield construction. In fact, a
cost or yield construction takes any function (mono-
tone or not) together with a set of resources and
induces a type-independent monotone from it, as fol-
lows.
Given any function f which maps some set S of
resources to the real numbers, one can define associ-
ated monotones which are applicable to all resources,
19
as follows:
M[f -yield, S](R)
:
= (23)
max
R
?
∈S
{f(R
?
) s.t. R 7−→R
?
},
M[f -cost, S](R)
:
= (24)
min
R
?
∈S
{f(R
?
) s.t. R
?
7−→R}.
If there does not exist any R
?
∈ S such that
R 7−→R
?
, then the yield is defined to be −∞. Sim-
ilarly, if there does not exist any R
?
∈ S such that
R
?
7−→R, then the cost is defined as ∞ [71].
In words, M
[
f-yield, S
]
is a monotone which asks
for the most valuable resource in the set S (as mea-
sured by the function f) that one can create from
the given resource R.
25
Meanwhile, M
[
f-cost, S
](
R
)
is a monotone which asks for the least valuable re-
source in the set S (as measured by the function
f) that one can use to create the given resource R.
Note that in both cases, many different functions
may yield the same monotone, so there is a conven-
tional element to one’s choice of function. Note also
that S may be restricted to resources of a particu-
lar type (in which case f need only be defined on
resources of that type), and yet the type of the re-
source R for which the monotones may be evaluated
is unrestricted.
4.3.2 Weight and robustness monotones
Various functions have been used as measures of
the distance of a resource from the set of clas-
sically realizable common-cause boxes in previous
work [16, 17, 22, 73–76]. In what follows, we high-
light some of these which are monotones in our re-
source theory.
25
The maximum of a function
f
over the set of boxes to which
R
can be converted can also be thought of as the performance
of
R
over the so-called ‘nonlocal game’ defined by the ‘payoff
function’
f
. Since the set of boxes to which
R
can be con-
verted (of any given type) is a polytope, it follows that all
forbidden conversions (those from
R
to a resource outside the
polytope) can be witnessed by a suitable set of payoff functions,
namely, whatever linear functions pick out the facets of
R
’s
polytope (for any given target type). In other words, any
resource outside the polytope will attain a higher value on at
least one of these functions. It follows, then, that the set of
yield monotones induced by all possible linear functions con-
stitutes a complete set of monotones. While this observation
may not be useful in practice, it does pose an interesting con-
trast with the findings of Ref. [
72
]: For common-cause boxes,
we find that ‘nonlocal games’ constitute a complete set of
monotones; whereas [
72
] shows that for the resource theory of
quantum states under LOSR it is semiquantum games instead
of nonlocal games that form a complete set of monotones.
The nonlocal fraction, which we denote here by
M
NF
, is the minimum weight of the nonfree fraction
in any convex decomposition of the resource,
M
NF
(R)
:
= (25)
min
0≤λ≤1
R
∗
∈S
[R]
L∈L
[R]
{λ s.t. R = λ R
∗
+ (1−λ)L}.
The nonlocal fraction was proven to be a resource
monotone relative to (a superset of) the LOSR free
operations in Ref. [16, Sec. 5.2], though it is there
termed the ‘EPR2’ measure.
Next, there is the case of robustness measures
26
which quantify the minimum weight of a resource
from some particular class that must be added con-
vexly with the original resource for the mixture to be
free. The two robustness measures that we consider
differ by the class of resources that are mixed with
the original resource. The first, which we denote by
M
RBST,L
(
R
)
, considers mixing the original resource
R with any element in the set L
[R]
of free resources
of the same type:
M
RBST,L
(R)
:
= (26)
min
0≤λ≤1
L∈L
[R]
λ s.t. λ L + (1−λ)R ∈ L
[R]
.
This robustness measure was shown to be a resource
monotone relative to LOSR in Ref. [17, Sec. 3].
The second robustness measure, which we denote
simply by M
RBST
(
R
)
considers mixing the original
resource R with any element in the set S
[R]
of all
resources of the same type:
M
RBST
(R)
:
= (27)
min
0≤λ≤1
R
∗
∈S
[R]
λ s.t. λ R
∗
+ (1−λ)R ∈ L
[R]
.
The unified resource theory formalism of Ref. [71]
implies that all three of these distance measures are
resource monotones in any resource theory wherein
all of the operations in the free set are convex-
linear
27
operations, including our resource theory
here. Additionally, in Corollary 18, we show that
each of these three distance measures can be ex-
plicitly related to a monotone for which we pro-
vide a closed-form expression relative to
(
2 2
2 2
)
-type
26
Note that in Ref. [76] these were termed ‘visibilities’.
27
An operation
τ
is convex-linear if the image
τ ◦
(
R
3
) is a given
mixture of
τ ◦
(
R
1
) and
τ ◦
(
R
2
) whenever the preimage
R
3
is the same mixture of
R
1
and
R
2
. All linear operations are
convex-linear.
20
resources. By extension, we therefore also provide
closed-form expressions for these three distance mea-
sures relative to
(
2 2
2 2
)
-type resources.
5 A linear program for determining
the ordering of any pair of resources
Next, we provide a linear program which allows one
to determine the ordering relation that holds be-
tween any two resources in our enveloping theory.
To do so, it is convenient to set up some useful no-
tation.
Definition 14.
Let the bold symbol
S
refer to any
set of resources. We use subscripts to specify the type
of the resources in the set, such as
S
(
|X| |Y |
|S | |T |
)
or
S
[R]
.
We use superscripts to specify further properties of
a set. For example, the set of all GPT-realizable
common-cause boxes is denoted by
S
G
, the set of all
nonfree resources is denoted by
S
nonfree
, and the set
of all free resources is denoted by
S
free
. Whenever
we wish to emphasize that a specific set is discrete,
we denote it
V
, and whenever we wish to emphasize
that a specific set is a polytope, we denote it P .
Let P
LOSR
[R
2
]
(
R
1
)
denote the continuous set of re-
sources of type
[
R
2
]
into which R
1
can be converted
under LOSR, that is, the image of R
1
under LOSR
[R
1
]→[R
2
]
.
Similarly, let V
LDO
[R
2
]
(
R
1
)
denote the discrete set of re-
sources of type
[
R
2
]
into which R
1
can be converted
under LDO, that is, the image of R
1
under LDO
[R
1
]→[R
2
]
.
From Propositions 10 and 12, and the finite cardi-
nality of V
LDO
[R
2
]
(
R
1
)
, it follows that P
LOSR
[R
2
]
(
R
1
)
is
a convex set with a finite number of vertices, and
hence is a polytope:
Proposition 15
(The polytope of resources obtain-
able from a given resource by LOSR).
The set of all resources of type [
R
2
] obtainable from
R
1
by LOSR forms a polytope,
P
LOSR
[R
2
]
(R
1
) = ConvexHull
V
LDO
[R
2
]
(R
1
)
. (28)
We can express the content of Proposition 15
equivalently as
R
1
7−→R
2
if and only if
R
2
∈ ConvexHull
V
LDO
[R
2
]
(R
1
)
.
(29)
Therefore, to determine whether R
1
is higher than
R
2
in the pre-order of resources, it suffices to imple-
ment the following computational test:
1. Enumerate all of the locally deterministic oper-
ations which take resources of type
[
R
1
]
to type
[R
2
]. (They are finite in number.)
2. Compute the images of R
1
under all of these
locally deterministic operations.
3. Determine whether or not R
2
can be expressed
as a convex combination of these images. (This
is a linear program.)
To determine which of the four possible ordering
relations holds for a given pair of resources, R
1
and
R
2
, it suffices to determine whether R
1
7−→R
2
or not
and whether R
2
7−→R
1
or not. This requires just two
instances of the linear program.
28
According to Proposition 15, the image of a re-
source under the set of all LOSR free operations is
equivalent to the convex closure of the image of the
resource under only the extremal operations. Replac-
ing the set of all operations with only the extremal
ones is a dramatic shortcut.
In principle, the linear program just described al-
lows one to characterize the pre-order completely.
For instance, this linear program defines a complete
set of monotones for a given set of resources S,
namely, {M
R
0
:
R
0
∈ S} where the monotone M
R
0
is defined as follows: for all R ∈ S, M
R
0
(
R
) = 1
if R → R
0
by LOSR and M
R
0
(
R
) = 0
otherwise.
M
R
0
(
R
)
reports the answer returned by the linear
program for the question of whether R → R
0
by
LOSR, and if one has the answer for all R
0
∈ S, then
one has located R within the pre-order. However,
such a brute-force characterization of the pre-order
requires one to apply the linear program to every
pair of resources, which is not possible in practice.
Rather, the linear program is primarily useful for
answering questions about conversions among pairs
(or finite sets) of resources.
To characterize the full pre-order more generally,
one would ideally have a finite set of resource mono-
tones that characterize the pre-order completely.
Furthermore, in order to determine certain global
properties of the pre-order, such as those described
earlier, knowledge of a few carefully chosen resource
monotones will typically suffice. This is the strategy
we will adopt hereafter in the article. Specifically,
over the next few sections, we define a pair of re-
source monotones and use these to prove that the
pre-order of single-copy deterministic conversion is
not totally pre-ordered (i.e., there exist incompara-
ble resources), that it is not weak (the incompara-
28
In the language of Ref. [
77
], these linear programs constitute
a complete witness for conversion.
21
bility relation is not transitive), that it has both in-
finite width and infinite height, and that it is locally
infinite.
6 Two useful monotones
We will define two monotones, one a cost construc-
tion and the other a yield construction, where the
sets of resources relative to which these costs and
yields are evaluated (to be described below) contain
only resources of type
(
2 2
2 2
)
. It is useful to first review
some facts about the set of all common-cause boxes
of type
(
2 2
2 2
)
, that is, about S
G
(
2 2
2 2
)
.
6.1
Preliminary facts regarding CHSH inequal-
ities and PR boxes
We adopt the convention of Ref. [78] of parametriz-
ing common-cause boxes of type-
(
2 2
2 2
)
in terms of out-
come biases and two-point correlators. The outcome
biases are
hA
s
i :=
X
x∈{0,1}
(−1)
x
P
X|S
(x|s)
= P
X|S
(0|s) − P
X|S
(1|s)
and hB
t
i :=
X
y∈{0,1}
(−1)
y
P
Y |T
(y|t)
= P
Y |T
(0|t) − P
Y |T
(1|t),
and the two-point correlators are
hA
t
B
s
i :=
X
x,y∈{0,1}
(−1)
(x⊕y)
P
XY |ST
(xy|st).
Recalling that the set of common-cause boxes co-
incides with the set of no-signalling boxes in the Bell
scenario, S
G
(
2 2
2 2
)
constitutes what is conventionally re-
ferred to as the “no-signalling” set for this type.
29
This set is well-known to be a polytope defined by
16 positivity inequalities [74, 79].
The set of classical (free) resources of type
(
2 2
2 2
)
is a subset therein, conventionally termed the “local
set”, and is defined by the same 16 positivity inequal-
ities together with eight additional facet-defining
Bell inequalities, namely, the canonical CHSH in-
equality and its seven variants [80]. A resource is
29
However, as noted in Appendix A.3, for causal structures
different from the Bell scenario, the set
S
of processes that
can be realized by a GPT causal model on the causal structure
is typically distinct from the no-signalling set.
therefore nonclassical (nonfree) if and only if it vio-
lates a facet-defining Bell inequality.
The eight variants of the canonical CHSH function
are
CHSH
0
(R)
:
= +hA
0
B
0
i+hA
1
B
0
i+hA
0
B
1
i−hA
1
B
1
i,
CHSH
1
(R)
:
= +hA
0
B
0
i+hA
1
B
0
i−hA
0
B
1
i+hA
1
B
1
i,
CHSH
2
(R)
:
= +hA
0
B
0
i−hA
1
B
0
i+hA
0
B
1
i+hA
1
B
1
i,
CHSH
3
(R)
:
= −hA
0
B
0
i+hA
1
B
0
i+hA
0
B
1
i+hA
1
B
1
i,
CHSH
4
(R)
:
= −hA
0
B
0
i−hA
1
B
0
i−hA
0
B
1
i+hA
1
B
1
i,
CHSH
5
(R)
:
= −hA
0
B
0
i−hA
1
B
0
i+hA
0
B
1
i−hA
1
B
1
i,
CHSH
6
(R)
:
= −hA
0
B
0
i+hA
1
B
0
i−hA
0
B
1
i−hA
1
B
1
i,
CHSH
7
(R)
:
= +hA
0
B
0
i−hA
1
B
0
i−hA
0
B
1
i−hA
1
B
1
i.
(31)
The canonical CHSH function is CHSH
0
, which we
will sometimes denote simply as CHSH.
In terms of these, the eight facet-defining Bell in-
equalities are
CHSH
k
(R) ≤ 2 for k ∈ {0, . . . , 7}. (32)
Note that the regions defined by strict violation of
each of the eight inequalities are nonoverlapping [74].
It follows that one and only one of the eight CHSH
inequalities can be violated by a given resource, i.e.,
for nonfree R there is precisely one value of k such
that CHSH
k
(R) > 2.
There are eight extremal nonfree vertices of the
full polytope S
G
(
2 2
2 2
)
. One of these is the canonical
PR box [57, 81], denoted R
PR
and defined explicitly
in Table 2; the other seven are variants of this PR
box. For each k, we denote the associated variant
of the PR-box by R
PR,k
(so that the canonical PR
box is associated to k
= 0
, R
PR
=
R
PR,0
). R
PR,k
is the unique resource that maximally violates the
kth CHSH inequality, i.e., that achieves its algebraic
maximum, CHSH
k
(R
PR,k
) = 4.
Unsurprisingly, the variants of the facet-defining
Bell inequalities are interconvertible under LSO op-
erations, as are the variants of the extremal vertices.
To illustrate this, it is convenient to factorize the
(
2 2
2 2
)
LSO group into a subgroup which stabilizes CHSH
0
and a subgroup which does not, as follows.
Proposition 16.
Consider the following invertible
operations, i.e., elements of the LSO group for
(
2 2
2 2
)
-type resources:
τ
1
: P
XY |ST
(x, y|s, t) ↔ P
XY |ST
(x, y⊕1|s, t)
τ
2
: P
XY |ST
(x, y|s, t) ↔ P
XY |ST
(x, y|s⊕1, t)
τ
3
: P
XY |ST
(x, y|s, t) ↔ P
XY |ST
(x, y|s, t⊕1)
τ
4
: P
XY |ST
(x, y|s, t) ↔ P
XY |ST
(x⊕1, y⊕1|s, t)
τ
5
: P
XY |ST
(x, y|s, t) ↔ P
XY |ST
(x⊕s, y|s, t⊕1)
τ
6
: P
XY |ST
(x, y|s, t) ↔ P
XY |ST
(x, y⊕t|s⊕1, t)
Then,
22
(16a)
The order-64 group
G
123456
generated by
{τ
1
, τ
2
, τ
3
, τ
4
, τ
5
, τ
6
}
is the entire LSO group
for
(
2 2
2 2
)
resources.
(16b)
The order-8 subgroup
G
123
generated by
{τ
1
, τ
2
, τ
3
}
has no elements in common with
the subgroup
G
456
generated by
{τ
4
, τ
5
, τ
6
}
other than the identity operation.
(16c)
The order-8 subgroup
G
456
generated by
{τ
4
, τ
5
, τ
6
}
stabilizes the canonical PR box and
the CHSH
0
inequality.
(16d)
For any
k ∈ {
0
...
7
}
, the orbit of
CHSH
k
under
G
123
is
{CHSH
0
, ..., CHSH
7
}
, and the orbit of
R
PR,k
under G
123
is {R
PR,0
, ..., R
PR,7
}.
Proof.
The first two claims in Proposition 16 are
readily verified by standard group theory algo-
rithms [
67
]. The latter two claims become self-
evident by explicitly examining the actions of the
operations on expectation values (and hence, their
action on resources or functions on resources), per
Table 1. In light of Table 1, the third claim is
easily verified. The fourth claim simply captures the
fact that the eight CHSH functions are related by
LSO
, and similarly the eight PR boxes are also inter-
convertible under
LSO
. We can explicitly show how
the interconversions are accomplished by
G
123
by de-
scribing the actions of
{τ
1
, τ
2
, τ
3
}
as permutations on
the ordered set of
CHSH
functions, or equivalently,
on the ordered set of PR boxes.
• τ
1
flips the sign of every correlator, so the action
of
τ
1
on the ordered set of
CHSH
functions is
the permutation (0, 4)(1, 5)(2, 6)(3, 7).
• τ
2
exchanges the roles of
A
0
and
A
1
, so the ac-
tion of
τ
2
on the ordered set of
CHSH
functions
is the permutation (0, 1)(2, 3)(4, 5)(6, 7).
• τ
3
exchanges the roles of
B
0
and
B
1
, so the ac-
tion of
τ
3
on the ordered set of
CHSH
functions
is the permutation (0, 2)(1, 3)(4, 6)(5, 7).
Therefore the orbit of
CHSH
k
under
G
123
is easily
checked to be {CHSH
0
, ..., CHSH
7
}, as claimed.
The ordered set of PR boxes transforms under
LSO operations in exactly the same manner as the
ordered set of CHSH functions, since the values of
the marginals and the correlators for resource
R
PR,k
coincide with the coefficients of the associated terms
in the linear function
CHSH
k
(compare, e.g., the
expression for CHSH
0
in Eq.
(31)
with the values of
the marginals and correlators for
R
PR
in Table
(2)
).
Hence, the argument just given also establishes that
the orbit of
R
PR,k
under
G
123
is
{R
PR,0
, ..., R
PR,7
}
.
6.2 Defining the two useful monotones
Monotone 1: The yield of a resource with re-
spect to the set of resources of type
(
2 2
2 2
)
, as
measured by the CHSH function.
To define our first monotone, consider the canon-
ical CHSH function
CHSH(R)
:
= hA
0
B
0
i + hA
0
B
1
i + hA
1
B
0
i − hA
1
B
1
i.
The CHSH function is type-specific
30
and further-
more is not a monotone [16]. However, we can apply
the prescription of Eq. (23) to this function, taking
the set S to be S
G
(
2 2
2 2
)
, i.e., the set of all common-
cause boxes of type
(
2 2
2 2
)
. Doing so, we define the
following (type-independent) yield-based monotone,
which we will denote by M
CHSH
:
M
CHSH
(R)
:
= M[CHSH-yield, S
G
(
2 2
2 2
)
](R)
= max
R
?
∈S
G
(
2 2
2 2
)
{CHSH(R
?
) s.t. R 7−→R
?
}.
(33)
Note that one can always find some R
?
∈ S
G
(
2 2
2 2
)
such that R 7−→R
?
regardless of the type or details
of R, simply because free resources of type
(
2 2
2 2
)
may
always be freely generated after discarding R. Hence,
the value of this monotone is never less than 2, which
is the maximum of the CHSH function when applied
to the subset of free resources.
If one applies this procedure to any of the eight
variants of the CHSH functions in Eq. (31), the
monotones one thereby obtains all turn out to be
equivalent to M
CHSH
. This follows from the fact that
all variants of the CHSH function are interconvert-
ible under LSO and therefore the maximum of any
one in an optimiziation over all LOSR operations is
the same as any other, as noted in Proposition 16d.
Monotone 2: The cost of a resource with re-
spect to a set of noisy PR box resources, as
measured by the CHSH function.
Our second monotone also involves optimizing the
CHSH function, but it is a cost-based monotone, and
the set of resources over which one optimizes is re-
stricted to a particular one-parameter family of re-
sources of type
(
2 2
2 2
)
(rather than the full set S
G
(
2 2
2 2
)
).
To define this family, we need to highlight a par-
ticular resource in the free set, which we denote
30
The CHSH function is well-defined only for resources of type
(
2 2
2 2
)
.
23
hA
0
i hA
1
i hB
0
i hB
1
i hA
0
B
0
i hA
1
B
0
i hA
0
B
1
i hA
1
B
1
i
τ
1
hA
0
i hA
1
i −hB
0
i −hB
1
i −hA
0
B
0
i −hA
1
B
0
i −hA
0
B
1
i −hA
1
B
1
i
τ
2
hA
1
i hA
0
i hB
0
i hB
1
i hA
1
B
0
i hA
0
B
0
i hA
1
B
1
i hA
0
B
1
i
τ
3
hA
0
i hA
1
i hB
1
i hB
0
i hA
0
B
1
i hA
1
B
1
i hA
0
B
0
i hA
1
B
0
i
τ
4
−hA
0
i −hA
1
i −hB
0
i −hB
1
i hA
0
B
0
i hA
1
B
0
i hA
0
B
1
i hA
1
B
1
i
τ
5
hA
0
i −hA
1
i hB
1
i hB
0
i hA
0
B
1
i −hA
1
B
1
i hA
0
B
0
i −hA
1
B
0
i
τ
6
hA
1
i hA
0
i hB
0
i −hB
1
i hA
1
B
0
i hA
0
B
0
i −hA
1
B
1
i −hA
0
B
1
i
Table 1: Action of each of the six specified symmetry operations in terms of marginal expectation values and correlators.
L
b
NPR
.
31
L
b
NPR
can be defined as the uniform mix-
ture of the PR box with the maximally mixed re-
source L
∅
(defined in Table 2), namely L
b
NPR
=
1
2
R
PR
+
1
2
L
∅
, as enumerated in Table 2. The su-
perscript
b
in the notation L
b
NPR
denotes the fact
that this resource sits on the boundary of the free
set, namely, that it saturates the canonical CHSH
inequality, CHSH(L
b
NPR
) = 2.
The one-parameter family of resources defining
our cost construction are the convex mixtures of
R
PR
and L
b
NPR
. We denote the set of these by C
NPR
.
Formally,
C
NPR
:
= {C(α) : α ∈ [0, 1]}, (34)
where
C(α)
:
= α R
PR
+ (1−α)L
b
NPR
. (35)
We use “C” because the set of resources forms a
chain (defined in Section 4.1) and “NPR” because
each resource in the chain is a noisy version of the
PR box.
Geometrically, the chain C
NPR
describes a line
segment of resources with endpoints R
PR
and L
b
NPR
,
and α parametrizes the distance from C
(
α
)
to L
b
NPR
(the bottom of the chain). To see that the elements
of C
NPR
do indeed form a chain in the partial order,
it suffices to note that one can move downwards (de-
creasing α) starting from any C
(
α
)
by mixing C
(
α
)
with L
b
NPR
, but one cannot move upwards (increas-
ing α) from any C
(
α
)
, as doing so would require
increasing the value of the monotone M
CHSH
.
Table 2 provides an explicit characterization of a
generic resource on the chain, as well as its endpoints
and the maximally-mixed free resource.
31
The use of
L
instead of
R
when describing the resource
L
b
NPR
is a nod to the conventional terminology wherein the classically
realizable common-cause boxes are often called the local boxes.
See the discussion in the introduction for why we explicitly
avoid the local-nonlocal terminology here.
Using this one-parameter family of resources, we
define the following cost-based monotone, which we
denote M
NPR
,
M
NPR
(R)
:
= M[CHSH -cost, C
NPR
](R) (36)
= min
R
?
∈C
NPR
{CHSH(R
?
) s.t. R
?
7−→R},
where if for some R there is no R
?
∈ C
NPR
such
that R
?
7−→R, then we define M
NPR
= ∞.
Critically, note that the CHSH function is an in-
jective (one-to-one) mapping from points on the line
segment C
NPR
to the real numbers, with
CHSH
C(α)
= 2α+2. (37)
Thus, the problem of minimizing the CHSH function
over R
?
∈ C
NPR
such that R
?
7−→R is exactly the
same as minimizing the function
2
α+
2
under the
constraint C(α) 7−→R, that is,
M
NPR
= min
α∈[0,1]
{2α+2 s.t. C(α) 7−→R}. (38)
For each variant R
PR,k
of the PR box, where
k ∈ {
0
, . . . ,
7
}, we can define the chain of noisy ver-
sions thereof, that is, C
NPR,k
:
=
{C
k
(
α
) :
α ∈
[0
,
1]
}
where C
k
(
α
)
:
=
α R
PR,k
+ (1
−α
)
L
b
NPR,k
, with
L
b
NPR,k
=
1
2
R
PR,k
+
1
2
L
∅
. One can of course define a
cost-based monotone for each such chain. However,
all eight of these chains define the same monotone,
because the local symmetry operations allow one
to move among these, as a consequence of Propo-
sition 16d and the fact that L
∅
is stable under all
(
2 2
2 2
)
-type local symmetry operations.
32
32
As an aside, note that, unlike the cost with respect to the
chain
C
NPR
, Eq.
(36)
, the cost with respect to the set
S
G
(
2 2
2 2
)
of all resources of type
(
2 2
2 2
)
, as measured by the CHSH func-
tion, is utterly uninformative with regards to distinguishing
the elements of
S
G
(
2 2
2 2
)
. This is because the resource
R
PR,4
can be converted to any other
(
2 2
2 2
)
-type resource, and yet
24
hA
0
i hA
1
i hB
0
i hB
1
i hA
0
B
0
i hA
1
B
0
i hA
0
B
1
i hA
1
B
1
i CHSH
R
PR
= C(1) 0 0 0 0 +1 +1 +1 −1 4
L
b
NPR
= C(0) 0 0 0 0
+1
/2
+1
/2
+1
/2
−1
/2 2
C(α) 0 0 0 0
α+1
2
α+1
2
α+1
2
−α−1
2
2α+2
L
∅
0 0 0 0 0 0 0 0 0
Table 2: An explicit description of the resources referenced in our definitions.
6.3
Closed-form expressions for
M
CHSH
and
M
NPR
for
(
2 2
2 2
)
-type resources
The definitions of M
CHSH
and M
NPR
both involve
an optimization over a continuous set of states. In
this section, we derive closed-form expressions for
these monotones for resources of type
(
2 2
2 2
)
.
Consider first M
CHSH
.
Proposition 17.
For any free resource
R
of type
(
2 2
2 2
)
,
M
CHSH
(
R
) = 2. For any nonfree resource
R
of
type
(
2 2
2 2
)
, there is a unique
k ∈ {
0
, . . . ,
7
}
for which
CHSH
k
(R) > 2 and such that
M
CHSH
(R) = CHSH
k
(R). (39)
Equivalently, each function
CHSH
k
is a monotone
relative to the subset of
(
2 2
2 2
)
-type resources for which
CHSH
k
(R) ≥ 2.
Proof.
We already noted in Section 6 that
M
CHSH
(
R
) = 2 for all resources
R
that are free,
so it suffices to consider the case of nonfree resources.
As noted above, the fact that there is precisely one
value of
k
such that
CHSH
k
(
R
)
>
2 for a nonfree
resource
R
follows from the results in Ref. [
74
]. Thus,
we must show that
M
CHSH
(
R
) =
CHSH
k
(
R
) for this
value of k.
To prove this, we invoke Theorem 2.2 of
Ref. [
74
], which informs us that every resource
R
which violates the
k
th
CHSH
inequality ad-
mits a convex decomposition in terms of the
k
th variant of the PR box and some free re-
source that saturates the
k
th
CHSH
inequality,
denoted
L
b
k
, such that
R = λ R
PR,k
+ (1−λ)L
b
k
for some
λ ∈
[0
,
1]. Further,
λ
is spec-
ified uniquely by the linearity of the
CHSH
functions and the fact that
CHSH
k
(R
PR,k
) = 4
and
CHSH
k
(L
b
k
) = 2
, which together imply that
CHSH
(
R
PR,4
) =
−
4, the algebraic minimum of the canonical
CHSH function. Consequently, the value of this CHSH-cost
with respect to the set of all resources of type
(
2 2
2 2
)
is
−
4. Since
this monotone is constant on all resources in the scenario, it
is completely uninformative.
CHSH
k
(
R
) =
CHSH
k
λ R
PR,k
+ (1−λ)L
b
k
=
4λ + 2(1−λ)
. Again leveraging this unique decompo-
sition together with linearity of the
CHSH
k
function
and the linearity of LOSR transformations, it follows
that for any LOSR operation
τ
, we have
CHSH
k
(
τ ◦
R
) =
λ CHSH
k
(τ ◦R
PR,k
) + (1−λ) CHSH
k
(τ ◦L
b
k
)
.
Clearly
CHSH(τ ◦R
PR,k
) ≤ 4
, since four is the al-
gebraic maximum of the
CHSH
k
function, and
CHSH
k
(τ ◦L
b
k
) ≤ 2
, since every LOSR operation
takes a free resource
L
b
k
to a free resource
L
0
k
,
for which
CHSH
k
(
L
0
k
)
≤
2. For
R
such that
CHSH
k
(
R
)
>
2, then, it follows that free opera-
tions on
R
cannot increase its
CHSH
k
value, and
hence the maximum in Eq.
(33)
is achieved by
R
itself. This proves Eq. (39).
Using the closed-form expression for M
CHSH
, we
can additionally provide closed-form expressions for
the weight and robustness monotones introduced in
Section 4.3.2 for
(
2 2
2 2
)
-type resources:
Corollary 18.
For resources of type
(
2 2
2 2
)
, the nonlo-
cal fraction and the robustnesses to mixing are related
to M
CHSH
as follows:
M
NF
(R) =
M
CHSH
(R) − 2
2
, (40a)
M
RBST,L
(R) =
M
CHSH
(R) − 2
M
CHSH
(R) + 2
, (40b)
M
RBST
(R) =
M
CHSH
(R) − 2
M
CHSH
(R) + 4
. (40c)
Proof.
The relationship of these distance measures to
the extent by which the
CHSH
inequality is violated
was derived in Appendix E of Ref. [
76
]. We simply
recast those results in terms of
M
CHSH
(
R
) instead
of CHSH(R) by means of Proposition 17.
The values of the four monotones M
CHSH
(
R
)
,
M
NF
(
R
)
, M
RBST,L
(
R
)
, and M
RBST
(
R
)
are therefore
all expressible as strictly-increasing functions of one
another when applied to resources of type
(
2 2
2 2
)
. That
is, if any one of these monotones increases (respec-
tively decreases) between a given pair of resources
25
of type
(
2 2
2 2
)
, then all of monotones will similarly in-
crease (respectively decrease) between that pair of
resources. As we will focus on the
(
2 2
2 2
)
type below,
and the three distance-function monotones are no
more informative than M
CHSH
in this case, we will
not discuss them further.
We now turn to providing a closed-form expres-
sion for M
NPR
for resources of type
(
2 2
2 2
)
. We first
recall some more details of the geometry of S
G
(
2 2
2 2
)
.
Recall that we use the superscript b to denote
that a resource lies on the particular boundary of
the free set that is defined by the CHSH inequal-
ity (and thus that it saturates this inequality). We
further use the superscript bb to denote that a re-
source both saturates the CHSH inequality and ad-
ditionally lies on the boundary of the full polytope of
resources, S
G
(
2 2
2 2
)
. The set L
b
k
of CHSH
k
-inequality-
saturating resources is 7-dimensional, and the set
L
bb
k
of CHSH
k
-inequality-saturating resources on
the boundary of the full polytope S
G
(
2 2
2 2
)
is 6-
dimensional.
33
It follows that L
bb
k
⊆ L
b
k
.
Proposition 19.
For any free resource
R
of type
(
2 2
2 2
)
,
M
NPR
(
R
) = 2. For any nonfree resource
R
of
type
(
2 2
2 2
)
, there is a unique
k ∈ {
0
, . . . ,
7
}
for which
CHSH
k
(
R
)
>
2. Within this region, if
R ∈ C
NPR,k
,
then we have simply
M
NPR
(
R
) =
CHSH
k
(
R
). If, on
the other hand, R 6∈ C
NPR,k
, we have
M
NPR
(R) = 2α+2,
where
α
is the value appearing in the decomposition
R
=
γ L
bb
R
+ (1
−γ
)
C
k
(
α
), where
C
k
(
α
)
∈ C
NPR,k
,
L
bb
R
∈ L
bb
k
and
γ ∈
[0
,
1]. This value of
α
is un-
ambiguous (and computable from simple geometry)
because there exists a unique resource
L
bb
R
∈ L
bb
k
and
a unique choice of
γ ∈
[0
,
1] and of
α ∈
[0
,
1] such
that R = γ L
bb
R
+ (1−γ)C
k
(α).
33
L
b
is a facet of the 8-dimensional
(
2 2
2 2
)
local polytope, and
facets of polytopes are always one dimension lower than the
dimension of the polytope itself. A resource is within
L
bb
k
if it is both a member of the facet defined by the CHSH
k
inequality and also a member of some other facet defined by a
positivity inequality. The regions defined by the intersection
of adjacent facets are generally termed ‘ridges’, and a ridge
always has dimensionality
d −
2, where
d
is the dimension
on the polytope.
L
bb
k
is a collection of all the eight ridges
adjacent to the
L
b
k
facet. Equivalently,
R ∈ L
bb
k
if and only
if
R
can be convexly decomposed as a mixture over seven-or-
fewer (out of eight) deterministic boxes which saturate the
CHSH inequality. Each possible size-seven subset of CHSH-
inequality-saturating deterministic boxes defines one of the
eight 6-dimensional ridges comprising L
bb
k
.
L
bb
R
R
PR
L
b
NPR
R
C(α)
γ
(1 − γ)
(1 − α)
α
Figure 6: A depiction of a family of resources parametrized
by
α
and
γ
, and the unique decomposition of a particular
point
R
(
α,γ
) in terms of a point
C
(
α
) on the chain
C
NPR
and a (unique) CHSH-saturating resource
L
bb
R
that lies in
the boundary of the set of GPT-realizable common-cause
boxes. Note that the parameters
α
and (1
− α
) indicate
the fraction of the full line segment attributed to each
sub-segment, and similarly with γ and (1 − γ).
The (unique) relevant decomposition is shown in
Fig. 6 (for the case where k
= 0
). The proof of this
proposition is given in Appendix B.1.
7 Properties of the pre-order of
common-cause boxes
We now leverage the two monotones just introduced
to prove multiple interesting features of the pre-
order of common cause boxes.
7.1
Inferring global properties of the pre-order
Important properties of the pre-order over all re-
sources can already be learned by considering just
these two monotones (M
CHSH
and M
NPR
) and just
resources of type
(
2 2
2 2
)
, indeed, just a specific kind
of two-parameter family of resources within this set.
The kind of two-parameter family that we consider,
denoted S
L
bb
?
(
2 2
2 2
)
⊂ S
G
(
2 2
2 2
)
, is
S
L
bb
?
(
2 2
2 2
)
:
= {R(α,γ) : α ∈ [0, 1], γ ∈ [0, 1]}, (41)
where
R(α,γ)
:
= γ L
bb
?
+ (1−γ)C(α), (42)
26
with C
(
α
)
∈ C
NPR
. There are many such families,
one for each choice of a resource L
bb
?
∈ L
bb
. Each
such family S
L
bb
?
(
2 2
2 2
)
is the convex hull of the chain
C
NPR
and the associated point L
bb
?
, i.e.,
S
L
bb
?
(
2 2
2 2
)
= ConvexHull
{L
bb
?
, R
PR
, L
b
NPR
}
. (43)
Evaluating M
NPR
for resources in this family
is straightforward, thanks to Proposition 19. The
proposition directly implies that for any R
(
α,γ
)
∈
S
L
bb
?
(
2 2
2 2
)
,
M
NPR
R(α,γ)
= 2α+2. (44)
We now consider the value of M
CHSH
for resources
in this family. Noting that CHSH
R(α,γ)
≥ 2
for all R(α,γ) ∈ S
L
bb
?
(
2 2
2 2
)
, Proposition 17 states
that M
CHSH
R(α,γ)
= CHSH
R(α,γ)
. Substitut-
ing the definition of C
(
α
)
from Eq. (35) into Eq. (42),
we obtain
R(α,γ) = γ L
bb
?
+ (1−γ)α R
PR
+ (1−γ)(1−α)L
b
NPR
.
Recalling that the CHSH function is linear and
that it satisfies CHSH(L
b
) = 2 for all L
b
∈ L
b
and
CHSH(R
PR
) = 4, it follows that
M
CHSH
R(α,γ)
= CHSH
R(α,γ)
= 2γ + 4(1−γ)α + 2(1−γ)(1−α)
= 2α(1−γ) + 2. (45)
In Fig. 7(a), we plot some of the level
curves
34
for M
NPR
and M
CHSH
over any such
two-parameter family of resources. The level curve
defined by M
NPR
(R) = 2α+2 is a diagonal line
in Fig. 7(a), extending from the (implicit) point
C(α) to the point L
bb
?
. The level curve defined
by M
CHSH
(R) = 2α(1−γ) + 2 is a horizontal line in
Fig. 7(a), extending between the two implicit points
C(α) and α R
PR
+ (1−α)L
bb
?
.
From these level curves, we can immediately de-
duce a number of features of the pre-order of re-
sources. In particular, we consider those features of
the pre-order that were defined in Section 4.1.
First, we see that the pre-order is locally infi-
nite, simply by virtue of the fact that there exist
chains which are represented by continuous sets of
34
A level curve of a function
f
is a set of points that yield the
same value of f; e.g., {x | f(x)=c}.
distinct resources, such as the chain C
NPR
. The in-
terval between any two resources in such a continu-
L
bb
?
R
PR
L
b
NPR
R
1
R
2
R
3
(a)
2
4
4
R
P R
M
NPR
M
CHSH
R
1
R
2
R
3
(b)
Figure 7: (a) A plot of the 2-parameter family of resources
S
L
bb
?
(
2 2
2 2
)
(defined in Eq.
(41)
), with values for
M
CHSH
depicted
by a set of level curves (light blue, horizontal lines) and values
for
M
NPR
depicted by another set of level curves (orange,
diagonal lines). (b) A plot of the same 2-parameter family of
resources, but in a Cartesian coordinate system with
M
CHSH
and
M
NPR
as the coordinates. Because all resources on the
bottom border in plot (a) are free, these all map to a single
point in (b), namely (
M
CHSH
, M
NPR
) = (2
,
2). The fact
that there are no resources with
M
CHSH
= 2 and
M
NPR
>
2
is represented by the use of a dashed line at the base of the
plot in (b). Similarly, the hatched region in (b) describes
joint values of the two monotones that are not achieved by
any resource in the family, as
M
CHSH
(
R
)
≤ M
NPR
(
R
) for
all
R
. Pictured in both plots are three illustrative resources.
The points
R
1
and
R
2
are incomparable, as are
R
3
and
R
2
,
while
R
1
and
R
3
are strictly ordered. This implies that the
incomparability relation in the pre-order is not transitive.
27
ous chain contains a continuous infinity of inequiva-
lent resources.
Second, one can also see that the pre-order of re-
sources is not totally pre-ordered. For instance, the
two resources R
1
and R
2
in Fig. 7(a) are incompa-
rable, as witnessed by the fact that R
1
has a larger
value of M
CHSH
than R
2
does, but R
2
has a larger
value of M
NPR
than R
1
does. More generally, the
level curves for the two monotones allow one to im-
mediately construct (by inspection) a continuous in-
finity of such incomparable pairs.
Furthermore, the binary relation of incomparabil-
ity is not transitive, so the partial order is not weak.
This can be seen by the example of the three re-
sources in Fig. 7(a): R
1
and R
2
are incomparable
(as just argued) and R
3
and R
2
are incomparable
(by the same logic), yet R
1
and R
3
are comparable,
as evidenced by the fact that one can obtain R
3
from R
1
, by mixing R
1
with any free resource that
intersects the line defined by the points R
1
and R
3
.
In addition, one can also see that the height of the
pre-order is infinite. It suffices to note that the chain
C
NPR
is totally ordered and contains a continuum of
elements. The width of the pre-order is also infinite.
Consider, for example, the line segment defined by
the points R
1
and R
2
in Fig. 7(a). This subset of
resources constitutes an antichain, as every resource
in it is incomparable to every other: each resource
has a higher M
NPR
value and lower M
CHSH
value
than any of its neighbors towards the left, and has
a lower M
NPR
value and higher M
CHSH
value than
any of its neighbors towards the right. Because this
subset also forms a continuum, it follows that the
width of the pre-order is infinite.
Also by inspection, for a given nonfree resource,
there are a continuum of chains and antichains
which contain it. In order to see this, let us first intro-
duce some terminology. Within the plane of the two-
parameter family of resources, depicted in Fig. 8(a),
we refer to a direction from a given point R as an
“antichain direction” relative to that point, if this di-
rection lies strictly clockwise from the direction de-
fined by the M
CHSH
level curve that passes through
R and strictly counterclockwise from the direction
defined by the M
NPR
level curve that passes through
R. Otherwise, it is called a “chain direction”. Thus
an antichain direction relative to R is defined by any
vector originating in R and terminating at a point
strictly within either yellow region in Fig. 8(a), while
a chain direction relative to R is defined by any vec-
tor originating in R and terminating in either blue
region.
A one-dimensional curve of resources in this sub-
set defines a chain (antichain) if and only if at every
point on the curve, the tangent to the curve at that
L
bb
?
R
PR
L
b
NPR
R
(a)
2
4
4
R
P R
M
NPR
M
CHSH
R
(b)
Figure 8: (a) and (b) provide the same pair of depictions
of the 2-parameter family of resources
S
L
bb
?
(
2 2
2 2
)
as were intro-
duced in Fig. 7. We consider a particular resource
R
. In
(a), we depict the level curves of
M
CHSH
(horizontal) and
M
NPR
(angled) which include
R
. By monotonicity of the
two monotones,
R
cannot be freely converted into any re-
source in the upper light-blue region or in the pair of yellow
regions. As we prove in Section 7.2.1, the two monotones
are complete for this subset, which is equivalent to the
fact that an arbitrary resource
R
can be freely converted
to any resource in the lower dark-blue region; namely, the
entire region wherein
M
CHSH
and
M
NPR
do not have a
value greater than the one they have on
R
. Resources in
the upper light-blue region can be converted to
R
, while
resources in the pair of yellow regions are incomparable to
R.
28
point is aimed
35
in a chain direction (antichain di-
rection) relative to that point.
A final lesson we learn from these two mono-
tones is that the set of all monotones induced (via
Eq. (33)) by the facet-defining Bell inequalities for a
given type do not yield a complete set of monotones
for the resources of that type. We have shown that
the set of resources is not totally pre-ordered, and as
stated in Section 4.3.1, the eight facet-defining Bell
inequalities for the
(
2 2
2 2
)
-scenario induce only a single
monotone: M
CHSH
. Since no single monotone can be
complete for a pre-order of resources that includes
incomparable resources, it follows immediately that
the monotones induced by the facet-defining Bell in-
equalities for the
(
2 2
2 2
)
type are not sufficient for fully
characterizing the pre-order of resources of that type.
Since such resources trivially can be lifted to any
nontrivial Bell scenario (where the lifted resource
will violate no facet-defining Bell inequalities other
than CHSH), it follows that:
Proposition 20.
The pre-ordering of resources rel-
ative to LOSR operations cannot be resolved solely
using the degree of violations of facet-defining Bell
inequalities.
Proof.
By definition, any complete set of mono-
tones allows one to compute the values of any other
monotone from them
36
. However, although the
value of M
CHSH
(R) can be computed (for any type-
(
2 2
2 2
)
resource
R
) from the eight values of the facet-
defining
CHSH
functionals in Eq.
(31)
, the value of
M
NPR
(
R
) cannot. This implies that any complete
set of monotones must include at least one mono-
tone (like
M
NPR
(
R
)) which depends on information
beyond the values of the eight
CHSH
functionals.
Proposition 20 shows that the nonclassicality of
common-cause processes is not completely charac-
35
More precisely: a line defines two opposing directions, and
both of these directions will point in a chain direction, or both
will point in an antichain direction.
36
If one has a set of monotones
{M
i
}
i
which is complete, then
for a given resource
R
, the set of values
{M
i
(
R
)
}
i
is sufficient
for (in principle) computing the value
M
(
R
) of any monotone
M
on resource
R
. First, one can deduce the equivalence class
of
R
from
{M
i
(
R
)
}
i
; this is possible by the completeness of
the set
{M
i
}
i
. Then, one can select any resource
R
0
from the
equivalence class of
R
and can evaluate
M
(
R
0
) for the given
monotone
M
. Because a monotone must assign the same
value to all resources within an equivalence class, it holds
that
M
(
R
0
) =
M
(
R
). (Note that our argument here does not
imply that one can in practice compute the value
M
(
R
); this
computation might involve solving a hard problem.)
terized by the monotones that are naturally associ-
ated to facet-defining Bell functionals, despite the
fact that such Bell functionals are sufficient to wit-
ness whether or not a resource is nonclassical.
7.2 Incompleteness of the two monotones
In this section, we prove that the two-element set of
monotones {M
CHSH
, M
NPR
} is not a complete set.
We do so by showing that it is not complete even
for resources of type
(
2 2
2 2
)
.
A simple proof is as follows. Consider resources of
the form R
=
1
2
L
bb
?
+
1
2
C(½) for different choices of
the CHSH-saturating resource L
bb
?
that lies in the
boundary of S
G
(
2 2
2 2
)
. We will show that there are pairs
of resources of this form which are strictly ordered,
and other pairs of resources of this form which are
incomparable. These facts cannot be captured by
the two monotones, which see all resources of this
form as equivalent, with M
NPR
= 3
and M
CHSH
=
2.5.
Consider for example the resources L
bb
1
, L
bb
2
, and
L
bb
3
defined in Table 3. Using the pairwise compar-
ison algorithm described in Section 5, one can ver-
ify that the resource
1
2
L
bb
1
+
1
2
C(½) is strictly higher
in the order than
1
2
L
bb
2
+
1
2
C(½), while the two re-
sources
1
2
L
bb
2
+
1
2
C(½) and
1
2
L
bb
3
+
1
2
C(½) are incom-
parable. Note that L
bb
1
is a convexly extremal re-
source, while L
bb
2
and L
bb
3
are not.
As an aside, it is worth noting that because
the nonlocal fraction and the two standard robust-
ness measures witness exactly the same ordering
relations as M
CHSH
does (as demonstrated in Sec-
tion 4.3.2), one gains nothing by supplementing
M
CHSH
and M
NPR
with them. Rather, new mono-
tones are needed.
The incompleteness of the two-element set
{M
CHSH
, M
NPR
} is also established directly from
the argument presented in Section 7.3.
7.2.1
Completeness of the two monotones for certain
families of resources
Although M
CHSH
and M
NPR
do not form a complete
set of monotones for the set of all resources of type
(
2 2
2 2
)
, it turns out that they do form a complete set
of monotones for certain subsets thereof.
Proposition 21.
The pair of monotones
{M
CHSH
, M
NPR
}
are a complete set relative
to the subset of resources
S
L
bb
?
(
2 2
2 2
)
(defined in Eq.
(41)
)
for any L
bb
?
∈ L
bb
.
29
hA
0
i hA
1
i hB
0
i hB
1
i hA
0
B
0
i hA
1
B
0
i hA
0
B
1
i hA
1
B
1
i M
CHSH
M
NPR
L
bb
1
1 1 1 1 1 1 1 1 2 2
L
bb
2
0 0 0 0 1 1 0 0 2 2
L
bb
3
0 0 0 0 1 0 1 0 2 2
C(½) 0 0 0 0
3
/4
3
/4
3
/4
−3
/4 3 3
1
2
L
bb
1
+
1
2
C(½)
1
/2
1
/2
1
/2
1
/2
7
/8
7
/8
7
/8
1
/8
5
/2 3
1
2
L
bb
2
+
1
2
C(½) 0 0 0 0
7
/8
7
/8
3
/8
−3
/8
5
/2 3
1
2
L
bb
3
+
1
2
C(½) 0 0 0 0
7
/8
3
/8
7
/8
−3
/8
5
/2 3
Table 3: An explicit description of the resources which demonstrate the incompleteness of the pair of monotones
{M
CHSH
, M
NPR
}
. The fact that
hA
0
B
0
i =
1 for the free boxes immediately proves that these do indeed lie on the boundary
of the full set of GPT-realizable common-cause boxes of this type,
S
G
(
2 2
2 2
)
(since it implies that
p
(0
,
1
|
0
,
0) = 0 =
p
(1
,
0
|
0
,
0),
and hence these boxes saturate positivity inequalities).
Proposition 21 is proven in Appendix B.2. The
logic of the proof is quite simple: we prove that there
always exists a free operation τ
erase−γ
which con-
verts an arbitrary resource R
(
α
1
, γ
1
)
in the family
to some resource R
(
α
2
,
0)
lying on the chain C
NPR
without changing the value of M
CHSH
. By convexity,
it follows that R
(
α
1
, γ
1
)
can be converted to any re-
source in the convex hull of R
(
α
1
, γ
1
)
, R
(
α
2
,
0)
, L
bb
?
,
and L
b
NPR
; namely, the dark-blue region in Fig. 8.
This region corresponds to the set of all resources
with a lower value of both M
CHSH
and M
NPR
. It
follows that if a conversion is not forbidden by con-
sideration of this pair of monotones, then it is achiev-
able. By the definition of completeness for a set of
monotones (see Eq. (21)), this implies that the two
monotones are indeed a complete set for this family
of resources.
7.3
At least eight independent measures of
nonclassicality
In this section, we tackle the question of how many
independent continuous monotones are required to
fully specify the partial order of resources. This is
the content of Theorem 26. Along the way to prov-
ing this result, we also prove a powerful result about
the equivalence classes under LOSR for nonfree re-
sources of type
(
2 2
2 2
)
, stated in Proposition 23.
We begin by drawing a distinction among re-
sources.
Definition 22.
A resource is said to be
orbital
if
its equivalence class under type-preserving LOSR is
equal to its equivalence class under LSO.
It follows that if all the resources in a set S are
orbital, then the quotient space [82] of S under the
group LSO provides a representation of the partial
order of LOSR-equivalence classes of resources in S
(despite the fact that the LOSR operations do not
themselves form a group).
37
This property of resources is pertinent to the dis-
cussion here because of the following result:
Proposition 23.
All nonfree resources of type
(
2 2
2 2
)
are orbital.
The proof is provided in Appendix B.3.
Note that for free resources, LOSR-equivalence
is distinct from LSO-equivalence because the LSO-
equivalence class of any resource (including a free
resource) is of finite cardinality, while the LOSR-
equivalence of a free resource is the entire set of
free resources, which is of infinite cardinality. Thus,
free resources are not orbital. Moreover, the coinci-
dence between being nonfree and being orbital does
not generalize beyond the
(
2 2
2 2
)
scenario. For instance,
note that a pair of
(
2 2
2 2
)
resources, R
1
and R
2
, which
are implemented in parallel can be conceptualized
as a
(
4 4
4 4
)
resource, R
1⊗2
, by composing the two bi-
nary setting variables on the left wing into a single
4-valued setting variable on the left wing, and simi-
larly for the other setting variable and the outcome
variables. If R
1
is free and R
2
is nonfree, then R
1⊗2
is nonfree, and yet because R
1
’s equivalence class
is not generated by LSO, neither is the equivalence
37
For practical purposes, Ref. [
66
, App. B] provides a technical
discussion regarding how to efficiently select a representative
Bell inequality under a finite symmetry group; the procedure
discussed there is equally applicable for the task of efficiently
selecting canonical form resources. Note, however, that the
LSO symmetry group differs from the Bell-polytope automor-
phism group considered in Ref. [
66
], in that LSO does not
include the symmetry of exchange-of-parties.
30
class of R
1⊗2
. Thus, R
1⊗2
is a nonfree resource that
is not orbital.
To express the next proposition, we require the
following definition.
Definition 24.
The
intrinsic dimension
of a set
of resources
S
, denoted
IntrinsicDim
(
S
), is the small-
est cardinality of continuous functions from the set
to the real numbers required to uniquely identify a
resource within S.
Proposition 25.
For any compact set
S
of resources
that are all orbital, the intrinsic dimension of the set
S
is a lower bound on the cardinality of a complete set
of continuous monotones for
S
(and for any superset
of S).
The proof is provided in Appendix B.4.
Recognizing that the set of nonfree resource of
type
(
2 2
2 2
)
has intrinsic dimension equal to eight,
38
then Propositions 23 and 25 together imply the fol-
lowing theorem:
Theorem 26.
For resources of type
(
2 2
2 2
)
, the cardi-
nality of a complete set of continuous monotones is
no less than 8.
8 Properties of the pre-order of quan-
tumly realizable common-cause boxes
The bulk of this article has considered the resource
theory which is defined by taking the enveloping the-
ory of resources to be the GPT-realizable common-
cause boxes, and the free subtheory of resources
to be the classically realizable common-cause boxes.
In this section, we consider a slightly different re-
source theory, wherein the enveloping theory of re-
sources is taken to be the common-cause boxes that
are realizable in a quantum causal model, which
we term quantumly realizable, while the free
subtheory is chosen to be, as before, the common-
cause boxes that are classically realizable. Effec-
tively, the new resource theory concerns the nonclas-
sicality of common-cause boxes within the scope of
38
That
IntrinsicDim(S
nonfree
(
2 2
2 2
)
) = 8
is evidenced by the char-
acterization of such resources in terms of outcome bi-
ases and two-point correlators. If
T
indicates any type,
then
IntrinsicDim(S
nonfree
T
) = IntrinsicDim(S
G
T
)
whenever
S
G
T
6= S
free
T
(think of subtracting one polytope from a circum-
scribing polytope of the same dimension). See Refs.[
66
,
68
,
80
,
83
] for discussions on the intrinsic dimension of no-signalling
polytopes.
nonclassicality that can be achieved quantumly. In
other words, it concerns the intrinsic quantumness
of common-cause boxes.
Formally, the conditional probability distribution
associated to a quantumly realizable common-cause
box is of the same form as Eq. (1), that is,
P
XY |ST
(xy|st) = (r
A
x|s
⊗ r
B
y|t
) · s
AB
, (46)
but where the vector s
AB
is a real vector represen-
tation of a quantum state on the bipartite system
composed of quantum systems A and B, and the
sets of vectors {r
A
x|s
}
x
and {r
B
y|t
}
y
are real vector
representations of POVMs on A and on B respec-
tively. (See, e.g., Ref. [45].)
Although the conclusions we drew in Section 7.1
concerned the pre-order of GPT-realizable common-
cause boxes, analogous results hold true for the pre-
order of quantumly realizable common-cause boxes.
This is because the kind of two-parameter family
of GPT-realizable common-cause boxes that was
used to establish global features of the pre-order of
such boxes in Section 7.1 contains a two-parameter
family of quantumly realizable common-cause boxes
that can be used for the same purpose. A carica-
ture of one such quantumly realizable family is pro-
vided in Fig. 9. Specifically, if one reviews the argu-
ments that were used in Section 7.1 to establish the
various global properties of the pre-order of GPT-
realizable common-cause boxes, it becomes appar-
ent that these apply equally well to the quantumly
realizable common cause boxes.
It is also straightforward to show that the lower
bound on the cardinality of a complete set of mono-
tones, obtained in Section 7.3, also applies to the
resource theory of quantumly realizable common-
cause boxes. It suffices to consider the case of the
quantumly realizable resources of type
(
2 2
2 2
)
, here-
after S
Q
(
2 2
2 2
)
, and to note that the set of nonfree re-
sources therein, that is, the set S
nonfree
(
2 2
2 2
)
T
S
Q
(
2 2
2 2
)
, still
has intrinsic dimension equal to eight.
In the rest of this section, we consider properties
of the pre-order of quantumly realizable common-
cause boxes that are particular to the quantum case.
Unlike for the set S
G
(
2 2
2 2
)
, where the partial or-
der of equivalence classes has a unique element
at the top of the order (the equivalence class
of R
PR
), in S
Q
(
2 2
2 2
)
there is no unique element at
the top of the order. An easy way to see this is
by considering the example of the Tsirelson box
31
hA
0
i hA
1
i hB
0
i hB
1
i hA
0
B
0
i hA
1
B
0
i hA
0
B
1
i hA
1
B
1
i M
CHSH
M
NPR
R
Tsirelson
0 0 0 0
√
2
/2
√
2
/2
√
2
/2
−
√
2
/2 2
√
2 2
√
2
≈0.707 ≈0.707 ≈0.707 ≈−0.707 ≈2.828 ≈2.828
R
Hardy
5−2
√
5
√
5−2 5−2
√
5
√
5−2 6
√
5−13 3
√
5−6 3
√
5−6 2
√
5−5 10(
√
5−2) 4
≈0.528 ≈0.236 ≈0.528 ≈0.236 ≈0.416 ≈0.708 ≈0.708 ≈−0.528 ≈2.361
R
Tilt
(θ) cos(θ) 0
cos(θ)
ξ(θ)
cos(θ)
ξ(θ)
1
ξ(θ)
sin
2
(θ)
ξ(θ)
1
ξ(θ)
−sin
2
(θ)
ξ(θ)
2 ξ(θ)
(
see
caption
)
R
Tilt
(0) 1 0 1 1 1 0 1 0 2 2
Table 4: An explicit description of the Tsirelson resource, the Hardy resource, and a family of extremal quantum resources
(parametrized by
θ
) which are exposed by tilted Bell inequalities [
84
,
85
]. We employ the shorthand
ξ
(
θ
)
:
=
p
sin
2
(θ)+1
to allow all definitions to fit within the table. We also analytically derived
M
NPR
R
Tilt
(
θ
)
=
ξ(θ)(ξ(θ)−1)
2(1−cos(θ))−ξ(θ)(ξ(θ)−1)
, for
0
< θ ≤ π/
2. One can readily verify that
M
NPR
R
Tilt
(
θ
)
increases with the amount of tilt (i.e.,
hA
0
i
=
cos
(
θ
)), whereas
M
CHSH
R
Tilt
(
θ
)
= 2
p
2 − cos
2
(θ)
decreases with added tilt. The opposite behavior of the two monotones implies that
every resource in the tilted family
θ ∈
(0
, π/
2] is incomparable to every other.
R
Tilt
(0) is a free resource, not violating any
Bell inequality; at the other end of the family, R
Tilt
(
π
2
) = R
Tsirelson
.
(R
Tsirelson
) and the Hardy box (R
Hardy
), each of
which is defined explicitly in Table 4. Noting that
M
CHSH
(
R
Tsirelson
) =
M
NPR
(
R
Tsirelson
) = 2
√
2 ≈
2
.
828
, and that M
CHSH
(
R
Hardy
) = 10(
√
5−
2)
≈
2
.
361
and M
NPR
(
R
Hardy
) = 4
, it follows im-
mediately that the two boxes are incomparable
since M
CHSH
(
R
Tsirelson
)
> M
CHSH
(
R
Hardy
)
while
M
NPR
(R
Tsirelson
) < M
NPR
(R
Hardy
).
We show these two resources in Fig. 9(a), to-
gether with an approximate sketch
39
of the extremal
quantumly realizable resources which interpolate be-
tween them (the light-blue curve). The values of
M
CHSH
and M
NPR
on all of these resources is plotted
in Fig. 9(b). From the figure, one can immediately
infer that R
Tsirelson
and R
Hardy
are incomparable.
Recall that no quantumly realizable resource can
achieve the algebraic maximum of M
CHSH
, while
some GPT-realizable (such as R
PR
) can achieve the
maximum. In contrast to M
CHSH
, M
NPR
is such
that some quantumly realizable resources (such as
R
Hardy
) violate it maximally. Furthermore, whereas
R
PR
maximizes both M
CHSH
and M
NPR
, no sin-
gle quantumly realizable resource maximizes both
those monotones. Therefore, a unique feature of the
enveloping theory of quantumly realizable common-
cause boxes is that inequivalent resources can si-
multaneously be maximally nonclassical (according
39
An analytic characterization of the set of all extremal quan-
tumly realizable resources within
S
Q
(
2 2
2 2
)
is not known. In
Fig. 9(a), the endpoints and the slope of the curve at the
endpoints are exact, and the rest of the curve is merely an
interpolation.
to distinct monotones), even among
(
2 2
2 2
)
-type re-
sources.
The interpolated curve in Figs. 9(a) and 9(b)
furthermore suggests that perhaps all extremal
quantum-realizable resources depicted therein are
relatively incomparable. The following lemma gives
a powerful result regarding maximally nonclassical
resources:
Lemma 27.
If a nonfree resource
R
is convexly
extremal in the set
S
Q
(
2 2
2 2
)
of quantumly realizable re-
sources of type
(
2 2
2 2
)
, then
R
is at the top of the pre-
order among quantumly realizable resources of type
(
2 2
2 2
)
.
Proof.
Let
R ∈ S
Q
(
2 2
2 2
)
be nonfree and extremal in
S
Q
(
2 2
2 2
)
. Then, to prove the proposition, we need only
prove that any quantumly realizable
R
0
∈ S
Q
(
2 2
2 2
)
that
can be freely converted to
R
cannot be higher in
the order than
R
(rather, it must be equivalent).
Assume the existence of some quantumly realizable
R
0
such that
R
0
7−→R
. Since
R
is extremal in the
image of
R
0
under LOSR,
40
it must be that
R
0
is
converted to
R
through extremal operations: that
is, through LDO. But as follows from Lemma 35 in
40
This is justified as follows: from
R
0
7−→ R
it follows that
R ∈ P
LOSR
[R]
(
R
0
), and from the fact that quantumly realizable
boxes remain quantumly realizable under LOSR, it follows that
P
LOSR
[R]
(
R
0
)
⊂ S
Q
(
2 2
2 2
)
. Finally,
R
is by assumption extremal
in S
Q
(
2 2
2 2
)
; hence, it is extremal in P
LOSR
[R]
(R
0
) as well.
32
Appendix B.3, or as can be explicitly checked,
41
the
L
bb
?
R
PR
L
b
NPR
R
Tsirelson
R
Hardy
(a)
2
4
4
R
PR
M
NPR
M
CHSH
R
Tsirelson
R
Hardy
2
√
2
2
√
2
(b)
Figure 9: (a) and (b) provide the same pair of depictions
of the 2-parameter family of resources
S
L
bb
?
(
2 2
2 2
)
as were in-
troduced in Fig. 7. Here, we provide a caricature of some
ordering relations among quantumly realizable common-
cause boxes within this 2-parameter family. We depict the
Tsirelson and Hardy boxes (with scaled-up values of the
monotones, but accurate ordering of these values), together
with a guess of what the boundary of the set of quantumly
realizable resources within this 2-parameter family might be
(dotted blue curves). In (b), we also depict the values of the
two monotones for the set of convexly extremal, quantumly
realizable resources which are self-tested by the tilted Bell
inequalities (smooth black curve).
41
One can explicitly check that all extremal
(
2 2
2 2
)
-type resources
are mapped to the free set by any deterministic operation
which is not a symmetry, which implies by convexity that all
(
2 2
2 2
)
-type resources are also mapped to the free set by these
operations.
image of any
(
2 2
2 2
)
-scenario resource is free under any
deterministic operation which is not a symmetry!
Put another way, there is no preimage of any nonfree
(
2 2
2 2
)
-scenario resource among
(
2 2
2 2
)
-scenario resources
under deterministic nonsymmetry operations. This
means that the only
τ ∈ LDO
(
2 2
2 2
)
→
(
2 2
2 2
)
such that con-
ceivably
τ ◦ R
0
=
R
are symmetry operations. As
such, if
R
is a nonfree extremal quantumly realizable
resource of type
(
2 2
2 2
)
, the only quantumly realizable
resources (of the same type) which can be converted
to
R
are symmetries of
R
. Since resources related by
a symmetry operation are in the same equivalence
class, there are no
(
2 2
2 2
)
-type quantumly realizable
resources strictly above R in the partial order.
Lemma 27 allows us to conclude the following:
Proposition 28.
There exists a continuous set of
resources that are at the top of the pre-order of quan-
tumly realizable
(
2 2
2 2
)
resources, and wherein each re-
source is incomparable to every other resource in the
set.
Proof.
Lemma 27 states that any subset of resources
which are extremal in
S
Q
(
2 2
2 2
)
are at the top of the
pre-order of quantumly realizable
(
2 2
2 2
)
resources. The
fact that one can find a continuous set of such re-
sources follows from the well-known fact that
S
Q
(
2 2
2 2
)
is not a polytope. By furthermore choosing such
a set of extremal resources for which
M
CHSH
takes
a distinct value for every resource in the set, one
additionally guarantees that no two of these top-of-
the-order resources are in the same equivalence class,
and hence each must be incomparable to every other
in the set. Refs. [
76
,
80
,
86
,
87
] provide some explicit
sets of resources satisfying these criteria.
As one concrete example, consider the one-
parameter family of quantumly realizable resources
which are self-tested by the tilted Bell inequalities.
We denote this family by {R
Tilt
(
θ
) :
θ ∈
(0
, π/
2]
}.
The definition of R
Tilt
(
θ
)
is given in Table 4. These
resources are related to a corresponding family of
tilted Bell functionals [84, 85, 88, 89], parametrized
33
by β ∈ [0, 2], namely,
TiltedCHSH
β
(R)
:
= βhA
0
i + hA
0
B
0
i
+ hA
1
B
0
i + hA
0
B
1
i − hA
1
B
1
i,
where max
R
?
∈S
free
(
2 2
2 2
)
TiltedCHSH
β
(R
?
) = 2 + β,
and where max
R
?
∈S
Q
(
2 2
2 2
)
TiltedCHSH
β
(R
?
) =
p
8 + 2β
2
.
Note that the only value of β for which the maxi-
mum value of this function over the quantumly real-
izable set S
Q
(
2 2
2 2
)
coincides with the maximum value
over the free set S
free
(
2 2
2 2
)
is β
= 2
. Whenever β <
2
, the
resource R
Tilt
(
θ
)
for θ defined implicitly by the equa-
tion β
=
2
√
1+2 tan
2
(θ)
is the unique maximizer over
S
Q
(
2 2
2 2
)
of the corresponding tilted Bell functional.
Formally,
β =
2
p
1 + 2 tan
2
(θ)
< 2 implies
TiltedCHSH
β
(R
?
) < TiltedCHSH
β
R
Tilt
(θ)
for any R
?
∈ S
Q
(
2 2
2 2
)
R
Tilt
(
θ
)
. It follows that every
resource R
Tilt
(
θ
)
is convexly extremal in the set
of quantumly realizable resources, and its extremal-
ity is exposed by the corresponding tilted Bell func-
tional.
In fact, every resource in this family is incompa-
rable to every other in the family, as can be shown
directly by considering the values of M
CHSH
and
M
NPR
. In Fig. 9(b), we show a plot of the values
of the two monotones evaluated on this family. The
points form a continuous antichain, shown in black.
Note that the family of resources {R
Tilt
(
θ
) :
θ ∈
(0
, π/
2]
} does not lie in any plane in the linear space
of resources, and as such we do not attempt to plot
the family directly (rather we only plot its valua-
tions with respect to the two monotones).
9 Conclusions and outlook
We have conceptualized Bell experiments as
common-cause ‘box-type’ processes: bipartite or
multipartite processes with classical variables as in-
puts and outputs, the internal causal structure of
which is a common-cause acting on all of the wings
of the experiment. We have argued in favour of this
conceptualization by appeal to the fact that Bell’s
theorem can be regarded as implying the need for
nonclassicality in the causal model that underlies
the process. We have begun to quantify the nonclas-
sicality of such common-cause box-type processes by
developing a resource theory thereof. We have ar-
gued in favour of a particular choice of the free oper-
ations for this resource theory, namely, those which
can be achieved by embedding the resource into a cir-
cuit consisting of box-type processes realizable with
a classical common cause, and we have shown that
this set is equivalent to the set of local operations
and shared randomness.
We have focused here on characterizing the pre-
order defined by single-copy deterministic conver-
sion of resources under the free operations. We have
provided a linear program that decides how any two
resources are ordered. By leveraging a pair of func-
tions that we have proven to be monotones, we have
also established a number of properties of this pre-
order, such as the fact that it contains incompara-
ble resources, that it has infinite width and height,
that it is locally infinite, and that the incompara-
bility relation is not transitive. Moreover, despite
the fact that the values of the facet-defining Bell
functionals are necessary and sufficient for witness-
ing the nonclassicality of a common-cause box, we
have shown that they are not sufficient for quanti-
fying the nonclassicality of a common-cause box. In
other words, there are aspects of the nonclassical-
ity of such boxes relevant to resource conversions
that are not captured by the degree of violation of
the facet-defining Bell inequalities. For the particu-
lar case of resources with two binary inputs and two
binary outputs, we moreover showed that at least
eight continuous monotones are required to fully
specify the pre-order among resources. We have also
derived some interesting facts about the pre-order of
resources when one restricts attention to common-
cause boxes that can be realized in quantum theory.
In particular, we have shown that for quantumly re-
alizable resources of type
(
2 2
2 2
)
, all convexly extremal
resources are at the top of the pre-order of such re-
sources, and that there are an infinite number of
incomparable resources at the top of this pre-order.
There is much scope for advancing and generaliz-
ing our work, some examples of which we now de-
scribe.
One of the most fundamental problems that is
yet to be solved is that of characterizing the equiv-
alence classes of resources in the pre-order induced
by single-copy deterministic conversion. That is, one
34
would like a compressed representation of each re-
source that includes all and only information that
is relevant to determining its equivalence class in
this pre-order. Finding such a representation would
be the analogue within our resource theory of prov-
ing that the equivalence classes of pure bipartite en-
tangled states under LOCC [90] are given by the
Schmidt coefficients of the state. All resource mono-
tones could then be efficiently expressed in terms of
this compressed representation, while all other pa-
rameters of a resource could be safely ignored.
Even among resources of type
(
2 2
2 2
)
(much less for
resources of arbitrary type), we do not have a com-
plete set of monotones for this pre-order.
42
Another
interesting open problem is to connect the existing
monotones to figures of merit for interesting opera-
tional tasks. E.g., does the value of the monotone
M
CHSH
determine the extent to which a given re-
source can be used for key distribution or random-
ness generation [6–14]? Since the monotone M
NPR
is maximized for high-bias boxes from the R
Tilt
(
θ
)
family (and by the Hardy box) as opposed to by the
Tsirelson box, M
NPR
is likely a figure of merit for
operational tasks where the advantage is provided
by such correlations [88, 91].
Note that in deriving our results about properties
of this pre-order, we have not needed to consider any
types of resource beyond
(
2 2
2 2
)
, that is, it has sufficed
to consider Bell experiments of the CHSH type. It
may be that more nuanced features of this pre-order
only become apparent for more general types of re-
sources.
An obvious generalization of our work is to con-
sider the pre-order induced by different sorts of
conversion relations, such as indeterministic single-
copy conversion
43
, multi-copy conversion, asymp-
42
Although considerations of the examples given in Section 7.2
might provide the intuition necessary to find such a complete
set for resources of type
(
2 2
2 2
)
.
43
Indeterministic single-copy conversion is single-copy conver-
sion that makes use of a post-selection. Therefore, to contem-
plate this notion of conversion for our resource theory is to
contemplate expanding the set of free operations from LOSR
to LOSR with post-selection. However, LOSR with postse-
lection can map a correlation
P
XY |ST
that satisfies the Bell
inequalities to one that violates them, and even to one that
violates the no-signalling condition. (This is in contrast to
the situation with LOCC, where allowing postselection does
not change the set of states that one can prepare for free.)
Consequently, what sort of correlation is consistent with a
classical common cause—and hence what should be deemed
free in a resource theory of nonclassicality of common cause
boxes—becomes contingent on what sort of postselection was
implemented. For example, in a Bell experiment wherein de-
totic conversion, and conversion in the presence of
a catalyst (see Refs. [26, 92, 93] for a discussion of
these different notions, and Refs. [94–98] for relevant
examples of such generalized conversions).
Other generalizations require changes to the en-
veloping theory of resources one is considering. We
have noted that our definition of the free opera-
tions can easily be extended to define a resource
theory of nonclassicality for box-type processes in
more general causal structures, distinct from that
of a Bell experiment. For example, as discussed
in Appendix. A.3, it can be extended to a sce-
nario we term the triangle-with-settings scenario [59,
Fig. 8], of which the much-studied ‘triangle sce-
nario’ [99–102] is a special case. Another example
would be to extend our definition to the ‘bilocality
scenario’ [59, 103–107]. The analysis of such cases
is complicated by the fact that our proposal im-
plies that the set of free operations is not convex
for them. Another such generalization would be to
causal structures wherein there are cause-effect re-
lations between different parts of the experiment,
for instance, experiments involving sequences of non-
destructive measurements on parts of a shared re-
source, such as the causal structure known as the
‘instrumental scenario’ [45, 108–112].
A generalization of our resource theory in a differ-
ent direction is to consider processes whose inputs
and outputs are not classical (i.e., processes that
are not ‘box-type’), but rather describe quantum or
post-quantum systems. For the case of the common-
cause structure which we focused on here, a quan-
tum resource theory of this sort would subsume en-
tanglement theory, but where quantum correlation
is defined relative to the set of local operations and
shared randomness (LOSR) rather than local oper-
ations and classical communication (LOCC).
10 Acknowledgments
The authors acknowledge useful discussions with
Jonathan Barrett, Tobias Fritz, Tom´aˇs Gonda and
tectors are not perfectly efficient, postselecting on detection
can induce Bell inequality violations even in the absence of
a nonclassical common cause. However, for a given value of
the detection efficiency, this might only be able to explain a
particular degree of violation, while any higher violation would
still attest to the presence of a nonclassical common cause. In
such a context, the boundary between the correlations that
are consistent with a classical common cause and those that
are not would no longer coincide with the facets of the Bell
polytope. Consequently, even defining the free set of resources
becomes quite complicated when postselection is allowed.
35
Denis Rosset. D.S. is supported by a Vanier Canada
Graduate Scholarship. R.K. is supported by the
Program of Concerted Research Actions (ARC) of
the Universit´e libre de Bruxelles. This research
was supported by Perimeter Institute for Theo-
retical Physics. Research at Perimeter Institute is
supported in part by the Government of Canada
through the Department of Innovation, Science and
Economic Development Canada and by the Province
of Ontario through the Ministry of Colleges and
Universities. This publication was made possible
through the support of a grant from the John Tem-
pleton Foundation. The opinions expressed in this
publication are those of the authors and do not
necessarily reflect the views of the John Temple-
ton Foundation. ABS acknowledges support by the
Foundation for Polish Science (IRAP project, IC-
TQT, contract no. 2018/MAB/5, co-financed by EU
within Smart Growth Operational Programme).
36
Appendices
A Comparing our framework with prior work
Correlations that violate Bell inequalities have become an important object of study, not only for their rele-
vance in foundational aspects of quantum theory, but also for their role as a resource in quantum information-
processing tasks [6–14]. Hence, particular effort has been devoted to the formulation of a resource theory
describing them [15, 16, 18, 19]. Two sets of free operations have previously been proposed to define such
a resource theory, namely LOSR [16–19] which we have developed in the main text, but also wirings and
prior-to-input classical communication (WPICC) [15].
In this section, we assess WPICC from the lens of our resource theory, and we identify an inconsistency
among previous proposals for the definition of LOSR. The primary differences between our approach and
previous approaches become most evident when one considers the question of how to develop such a resource
theory for more general causal structures, as we discuss further on in Appendix A.3.
A.1 WPICC versus LOSR as the set of free operations
The set of WPICC operations allows for classical causal influences among the wings prior to when the parties
receive their inputs. An example of a free operation in the WPICC approach is depicted in Fig. (10). If one
seeks to understand the resource as nonclassicality of common-cause processes, as we do here, then it is
clear that the free operations should not include any cause-effect influences between the wings, and therefore
should not include any classical communication between the wings. In other words, in our approach, WPICC
is not a viable choice for the set of free operations, as wirings that connect different wings of the experiment
cannot be part of any free operation.
One might think that the choice to take WPICC or LOSR to be the set of free operations is not a
particularly consequential one, since WPICC and LOSR define the same partial order for boxes [18] (see
the discussion of this point in Sec. A.2.1). This equivalence breaks down, however, when one considers more
general resources, e.g., bipartite quantum states. Since bipartite quantum states have no inputs, allowing
classical communication prior to inputs means allowing arbitrary classical communication. Hence, WPICC
coincides with LOCC in this case, and LOCC defines a partial order on bipartite quantum states that is
distinct from the partial order defined by LOSR [72].
A.2 An oversight in the literature concerning how to formalize LOSR
As we noted in the Introduction and in Section 2, the intuitive notion that the set of free operations should
constitute local operations supplemented by shared randomness is widely agreed upon in previous work [15–
22]. Nonetheless, some prior work seems to have formalized this intuitive notion incorrectly. Specifically, the
set of free operations defined in Ref. [18] (and repeated in Refs. [20, 21]) does not coincide with the set
of free operations defined in Refs. [16, 17] and which we endorse here as the correct choice. Rather, it is a
nonconvex subset thereof, as we will show here. (Note that Ref. [18] referred to their set of free operations
as “LOSR” but we will here reserve that term for the set of operations described in Definition 5.)
We suspect that the discrepancy in the definitions introduced in these papers was merely an oversight,
and in particular, that none of the authors of these articles would advocate for this nonconvex subset over
the full set. Nonetheless, we think that it is important to highlight this oversight, so that it may be avoided
in future work.
It is easiest to see the difference between the definition of the free operations given in Ref. [18] and the
one endorsed here (which coincides with the definitions of Refs. [16, 17]) by considering the diagrammatic
representation of a generic operation in each case. The most general free operation proposed by Ref. [18]
is depicted in their Fig. 1(a), which we reproduce here as Fig. 11, which should be compared with Figs. 3
and 4 of our article. The difference is that in Fig. 11, the side-channels on each wing that carry information
forward from the pre-processing to the post-processing are limited to carry information only about the setting
37
Figure 10: An example of a free operation in the WPICC approach, using the diagrammatic conventions of this article.
(Compare with Fig. 1(b) of Ref. [
18
].) Here, we see an example in which there is communication from the left wing to the
right wing, which (in contrast to our approach) is allowed for free in the WPICC approach, for all times prior to when the
wings receive the inputs S
0
and T
0
.
Figure 11: A depiction of Fig. 1(a) of Ref. [
18
] using the diagrammatic conventions of this article. The set of operations
having this form is not as general as those depicted in our Fig. 4 because the post-processing does not have complete access
to the shared randomness available at the pre-processing. One can explicitly show that the set of operations having this
form is not convex. It is only after taking the convex closure of the set of operations depicted here that one recovers LOSR.
variables (S, S
0
, T and T
0
), while in Figs. 3 and 4, they can also carry information about the common cause
that acts on the local pre-processings.
This difference is also reflected in the equations. The most general free operation proposed by Ref. [18] is
defined via their Eq. (7). In terms of the notation of this article, their Eq. (7) asserts that
P
X
0
Y
0
ST |XY S
0
T
0
= P
X
0
Y
0
|XY ST S
0
T
0
P
ST |S
0
T
0
, (47)
38
where P
ST |S
0
T
0
(denoted I
(L)
in Ref. [18]) and P
X
0
Y
0
|XY ST S
0
T
0
(denoted O
(L)
in Ref. [18]) represent, respec-
tively, the pre- and post-processings (depicted in Fig. 11). Consistently with their Fig. 1(a), the expression
for the post-processing stipulates that the side-channel between the pre- and post-processings only carries
information about the setting variables S, S
0
, T and T
0
. The analogue of this equation for the proposal
endorsed here is
P
X
0
Y
0
ST |XY S
0
T
0
=
X
Z
A
,Z
B
P
X
0
Y
0
|XY Z
A
Z
B
P
ST Z
A
Z
B
|S
0
T
0
, (48)
where Z
A
and Z
B
represent the variables propagated along the side-channels in Fig. 4. This is more general,
given that Z
A
and Z
B
can encode information about the common cause in the pre-processing.
To see the difference more explicitly, consider how these expressions appear if one includes the common
causes. Because the pre- and the post-processings in the proposal of Ref. [18] must depend on independent
sources of shared randomness (by virtue of the restriction on the side-channels), we distinguish the common
causes notationally using primed and unprimed variables. The post-processing is given by
P
X
0
Y
0
|XY ST S
0
T
0
=
X
Λ
0
P
X
0
|XSS
0
Λ
0
P
Y
0
|Y T T
0
Λ
0
P
Λ
0
, (49)
and the pre-processing is given by
P
ST |S
0
T
0
=
X
Λ
P
S|S
0
Λ
P
T |T
0
Λ
P
Λ
. (50)
Putting these together, we have
P
X
0
Y
0
ST |XY S
0
T
0
=
X
ΛΛ
0
P
X
0
|XSS
0
Λ
0
P
Y
0
|Y T T
0
Λ
0
P
S|S
0
Λ
P
T |T
0
Λ
P
Λ
0
P
Λ
(51)
By contrast, the proposal endorsed here distributes a single source of shared randomness between the pre-
and post-processings. If we consider the circuit depicted in Fig. 4 and note that the side-channels can now
feed forward not just S, S
0
, T and T
0
, but the common cause as well, we see that we can express the most
general free operation as follows (which is equivalent to Eq. (9))
P
X
0
Y
0
ST |XY S
0
T
0
=
X
Λ
P
X
0
|XSS
0
Λ
P
Y
0
|Y T T
0
Λ
P
S|S
0
Λ
P
T |T
0
Λ
P
Λ
. (52)
The operational discrepancy between the two proposals is a consequence of the fact that Eq. (51) is strictly
less general than Eq. (52).
One can intuitively expect a failure of convex closure for the set of operations depicted in Fig. 11 and
described in Eq. (51), since the pre-processing and the post-processing have access to independent sources of
shared randomness, and these two sources cannot generally be subsumed into a single source. To explicitly
demonstrate the failure of convexity, we consider the following operations:
τ
1
:
= P
X
0
Y
0
ST |XY S
0
T
0
= δ
Y
0
,0
δ
S,0
δ
T,0
δ
X
0
,0
,
τ
2
:
= P
X
0
Y
0
ST |XY S
0
T
0
= δ
Y
0
,1
δ
S,1
δ
T,0
δ
X
0
,0
,
τ
3
:
=
1
2
(τ
1
+ τ
2
).
While τ
1
and τ
2
are each free operations which can be realized using the circuit in Fig. 11, the transformation
τ
3
defined by their mixture cannot be realized using this circuit. To see this, note that in Fig. 11, any
correlations between S and Y
0
can only be mediated by T , since the only variable in the causal past of both
S and Y
0
is the variable acting as the common cause of the pre-processing, which we will denote by
Λ
1
, and
the only means by which the value of
Λ
1
could be communicated via the side channel is through T . But in
this example, T does not vary and so cannot mediate any correlations; the point distribution on T screens off
any correlation between S and Y
0
. Hence, τ
3
, which exhibits perfect correlation between S and Y
0
, cannot
39
be realized in a circuit of the form of Fig. 11. It follows that the set of operations depicted in Fig. 11 is not
convexly closed.
Despite the definition of the free operations given in Ref. [18], in Appendix A of that article, the authors
avail themselves of convex mixtures of operations of the sort described by their Fig. 1(a) and Eq. (7). However,
a mixing operation is only allowed if the shared randomness required to implement it is present, and given
that the shared randomness available for the pre-selection is independent from that which is available for the
post-selection, an arbitrary mixing operation is not allowed under the free operations proposed by Ref. [18].
The use of convex mixtures in Appendix A of Ref. [18] is therefore inconsistent with the definition of the
free operations provided therein.
The mistake of defining the free operations as this nonconvex subset of LOSR is repeated in Ref. [21]:
Fig. 1 and Eq. (13) therein are reproductions of Fig. 1(a) and Eq. (7) of Ref. [18], and, like the latter,
limits the side-channels to carry information only about the setting variables. It is also repeated in Ref. [20],
where the formalization of a “noncontextual wiring” per Eq. (9) there utilizes a post-processing with random-
ness independent from that of pre-processing, again restricting the side-channels to exclusively information
pertaining to the setting variables.
The above discussion has highlighted the fact that if one wishes the set of free operations to include
arbitrary convex mixtures of some smaller set, it is important that it be stipulated precisely how the shared
randomness is distributed in order to ensure the possibility of such mixing. In this regard, although de
Vicente [16] provided a definition of the free operations that is equivalent to LOSR, the physical justification
for this choice was wanting. Specifically, the definition in Ref. [16] proceeds by enumerating a long list
of nominally ‘elementary’ operations and then stating (in Section 4.1 of that article) that any mixture of
these operations is also allowed. No discussion is provided of why the type of shared randomness necessary
for achieving arbitrary mixtures should be considered freely available. The work of Geller and Piani [17], by
contrast, does stipulate the physical structure of the circuit that defines the free operations, thereby providing
a physical justification for taking LOSR as the set of free operations.
A.2.1 Previous results in light of this oversight
Given that some previous work [18, 20, 21] formally defined the set of free operations by Eq. (47), which
yields a nonconvex subset of LOSR, one might wonder to what extent the results reached by those works
still hold for LOSR proper, as defined in Eq. (48). In the following, we will briefly comment on some results
described in Refs. [18] and [21].
Lemma 6 of Ref. [18] purports to demonstrate that if a function is a monotone relative to a set of operations
that the authors term “LOSR”, then it is also a monotone relative to WPICC. If one interprets the set of
operations termed “LOSR” by the authors of Ref. [18] in the manner of the definition stipulated by their
Fig. 1(a) or their Eq. (7) (which is equivalent to Eq. (47) above), namely, as a nonconvex subset of LOSR
proper, as defined in Eq. (48), then the question would arise as to whether an analogous lemma holds for
LOSR proper rather than simply the nonconvex subset thereof. In fact, however, the proof of Lemma 6 in
Ref. [18] assumes that the set of free operations can map a resource R to a convex combination of R with
any local box. This is not possible if the set of free operations is the one defined by their Fig. 1(a) or Eq. (7)
(or equivalently, by Eq. (47) above). Hence, the proof of Lemma 6 holds only if the set of free operations
termed “LOSR” in the statement of the lemma is taken to be LOSR proper, as defined in Eq. (48), and not
the nonconvex subset of LOSR defined by Eq. (47). This fact provides yet another piece of evidence that the
nonconvexity of the formal definition of the free operations in Ref. [18] was merely an oversight. The bottom
line is that the proofs provided in Ref. [18] do establish that monotonicity relative to LOSR proper implies
monotonicity relative to WPICC.
Kaur et al. [21] state in their Proposition 6 that their proposed “intrinsic non-locality” measure is mono-
tonically nonincreasing under the set of free operations they term “LOSR”. But given that their definition
of this term is precisely the same as the definition provided in Ref. [18], the set of operations in question is
the nonconvex subset of LOSR defined by Eq. (47). This prompts the question of whether this proposition
holds if one considers LOSR proper, as defined in Eq (48), rather than this nonconvex subset thereof.
40
The answer is that it does. Establishing this is nontrivial, however, as an arbitrary monotone relative to
the nonconvex subset of LOSR defined by Eq. (47) need not be a monotone relative to LOSR. Note, however,
that if
(i) a function f is a monotone relative to LDO, and
(ii) f happens to be a convex function,
then f is also a monotone relative to LOSR, as a consequence of Proposition 15. Since LDO is contained
within the nonconvex subset of LOSR defined by Eq. (47), convex monotones relative to those limited
operations are also valid monotones relative to LOSR proper. Finally, we can use this implication to recover
Ref. [21]’s Proposition 6 by leveraging Proposition 7 there regarding the convexity of “intrinsic non-locality”
over box-type resources.
A.3 Generalizing from Bell scenarios to more general causal structures
In the introduction, we contrasted our approach to defining a resource theory, which we termed the causal
modelling paradigm, with a pre-existing approach, which we termed the strictly operational paradigm. Con-
sidering causal scenarios beyond Bell scenarios helps to clarify the differences between these two approaches.
Consider, for instance, a tripartite box-type process, with setting variables for the three wings denoted S,
T , and U , and outcome variables for the three wings denoted X, Y , and Z respectively. One can distinguish
two distinct causal structures that could underlie this sort of process: (i) the tripartite Bell scenario, where
there is a common cause acting on all the three wings, depicted in Fig. 12, and (ii) the triangle-with-settings
scenario [59, Fig. 8], where there is a common cause for each pair of wings, depicted in Fig. 13.
(a) (b)
Figure 12: The distinction between (a) a generic box in the tripartite Bell scenario and (b) a classical box in this scenario.
(a) (b)
Figure 13: The distinction between (a) a generic box in the triangle-with-settings scenario and (b) a classical box in this
scenario.
Consider the case of a generic box in the tripartite Bell scenario, depicted in Fig. 12(a), and label the
systems distributed to the three wings by A, B and C respectively. Let us denote by r
A
x|s
the GPT represen-
tation of the X
=
x outcome of the S
=
s measurement on system A, and similarly define r
b
y|t
and r
C
z|u
. If
s
ABC
denotes the GPT state of the composite ABC, then the conditional probability distribution associated
41
to this box is
P
XY Z|ST U
(xyz|stu) = (r
A
x|s
⊗ r
b
y|t
⊗ r
C
z|u
) · s
ABC
. (53)
When the GPT is classical probability theory, we obtain the classically-realizable box shown in Fig. 12(b),
and the conditional probability distribution associated to it is
P
XY Z|ST U
(xyz|stu)
=
X
λ
A
λ
B
,λ
C
P
X|SΛ
A
(x|sλ
A
)P
Y |T Λ
B
(y|tλ
B
)P
Z|U Λ
C
(z|uλ
C
)P
Λ
A
Λ
B
Λ
C
(λ
A
λ
B
, λ
C
)
=
X
λ
P
X|SΛ
(x|sλ)P
Y |T Λ
(y|tλ)P
Z|U Λ
(z|uλ)P
Λ
(λ). (54)
Now consider a generic box in the triangle-with-settings scenario, depicted in Fig. 13(a). Instead of an
arbitrary joint GPT state s
ABC
on the triple of systems associated to the three wings, each system is
composed of two parts—A is composed of A
1
and A
2
, and similarly for B and C—and the joint GPT state
has the form s
A
1
B
1
⊗ s
A
2
C
1
⊗ s
B
2
C
2
. The conditional probability distribution associated to this box is
P
XY Z|ST U
(xyz|stu) = (r
A
x|s
⊗ r
b
y|t
⊗ r
C
z|u
) · (s
A
1
B
1
⊗ s
A
2
C
1
⊗ s
B
2
C
2
). (55)
When the GPT is classical probability theory, we obtain the classically-realizable box shown in Fig. 13(b).
Taking Λ
A
= (Λ
A
1
, Λ
A
2
), and similarly for Λ
B
and Λ
C
, we have
P
XY Z|ST U
(xyz|stu) =
X
λ
A
λ
B
,λ
C
P
X|SΛ
A
(x|sλ
A
)P
Y |T Λ
B
(y|tλ
B
)P
Z|U Λ
C
(z|uλ
C
)
× P
Λ
A
1
Λ
B
1
(λ
A
1
, λ
B
1
)P
Λ
B
2
Λ
C
1
(λ
B
2
, λ
C
1
)P
Λ
A
2
Λ
C
2
(λ
A
2
, λ
C
2
)
=
X
λ,λ
0
,λ
00
P
X|SΛΛ
00
(x|sλλ
00
)P
Y |T ΛΛ
0
(y|tλλ
0
)P
Z|U Λ
0
Λ
00
(z|uλ
0
λ
00
)
× P
Λ
(λ)P
Λ
0
(λ
0
)P
Λ
00
(λ
00
). (56)
Figure 14: The free operations for a tripartite Bell scenario,
P
X
0
Y
0
Z
0
ST U|XY ZS
0
T
0
U
0
, taking a tripartite common-cause box
P
XY Z|ST U
to a new such box P
X
0
Y
0
Z
0
|S
0
T
0
U
0
.
As we see, the form of the GPT-realizable boxes in the tripartite Bell scenario differs from the form of the
GPT-realizable boxes in the triangle-with-settings scenario. Similarly for the form of the classically realizable
boxes. These differences have consequences when one compares the strictly operational paradigm with our
causal modelling paradigm, as we argue next.
42
Figure 15: The free operations for the so-called triangle-with-settings scenario,
P
X
0
Y
0
Z
0
ST U|XY ZS
0
T
0
U
0
, taking a triangle-
with-settings box P
XY Z|ST U
to a new such box P
X
0
Y
0
Z
0
|S
0
T
0
U
0
.
We begin by considering what each paradigm implies for the definitions of the free and enveloping sets of
resources for each scenario.
For the tripartite Bell scenario, the definitions of both the enveloping process theory and the free subtheory
of processes that are natural from the perspective of the causal modelling paradigm can also be expressed
in a way that is natural within the strictly operational paradigm. Specifically, the boxes in the enveloping
theory, which we take to be those that are realizable in a GPT causal model of this scenario (formalized in
Eq. (53)), can also be characterized as those that are nonsignalling between the wings. Similarly, the boxes
in the free subtheory, which we take to be those that are realizable in a classical causal model of this scenario
(formalized in Eq. (54)), can also be characterized as those that are mixtures of deterministic boxes which
are nonsignalling between the wings.
For the triangle-with-settings scenario, on the other hand, the set of boxes realizable in a GPT causal model
for that scenario (formalized in Eq. (55)), is a strict subset of the boxes that are nonsignalling between the
wings, and the set of boxes that are realizable in a classical causal model for that scenario (formalized in
Eq. (56)) is a strict subset of the set of boxes that are mixtures of deterministic boxes that are nonsignalling
between the wings. In both the enveloping theory and the free subtheory, the set of boxes is characterized via
nontrivial inequalities in addition to merely the equalities that represent the no-signalling constraints. See
Ref. [100] for a discussion of these inequalities in the special case of trivial setting variables. Consequently,
within the causal modelling paradigm, the resource theory associated to the triangle-with-settings scenario
and the resource theory associated to the tripartite Bell scenario differ in both the choice of enveloping
theory and free subhteory. Within the strictly operational paradigm, however, it is unclear whether there is
any natural way to pick out the enveloping theory and free subtheory that the causal modelling paradigm
dictates for the triangle-with-settings scenario because it is unclear whether there is any natural way of
picking these out by referring merely to the input-output functionality of the boxes.
Now, we shift our attention to what each paradigm implies for the definitions of the free operations in each
scenario. We will show that the definitions that are natural within the causal modelling paradigm cannot be
easily motivated within the strictly operational paradigm.
The free operations prescribed by the causal modelling paradigm for the tripartite Bell scenario are depicted
in Fig. 14. They are of the form
P
X
0
Y
0
Z
0
ST U |XY ZS
0
T
0
U
0
(x
0
y
0
z
0
stu|xyzs
0
t
0
u
0
)
=
X
λ
P
X
0
S|XS
0
Λ
(x
0
s|xs
0
λ)P
Y
0
T |Y T
0
Λ
(y
0
t|yt
0
λ)P
Z
0
U|ZU
0
Λ
(z
0
u|zu
0
λ)P
Λ
(λ), (57)
which is clearly a convex set. This, we believe, is the appropriate definition of local operations and shared
randomness for three parties.
43
This scenario does not show much difference with what would be natural in the strictly operational
paradigm, because one can motivate taking this set of operations to be free on the grounds that they take
nonsignalling boxes to nonsignalling boxes (even though, as in the case with the bipartite Bell scenario, the
set of WPICC operations between the three wings can also be motivated in this way).
It is the free operations in the triangle-with-settings scenario that really distinguishes the causal modelling
paradigm from the strictly operational paradigm.
The free operations prescribed by the causal modelling paradigm for the triangle-with-settings scenario
are depicted in Fig. 15. They are of the form
P
X
0
Y
0
Z
0
ST U |XY ZS
0
T
0
U
0
(x
0
y
0
z
0
stu|xyzs
0
t
0
u
0
)
=
X
λ,λ
0
,λ
00
P
X
0
S|XS
0
ΛΛ
0
(x
0
s|xs
0
λλ
0
)P
Y
0
T |Y T
0
Λ
0
Λ
00
(y
0
t|yt
0
λ
0
λ
00
)P
Z
0
U|ZU
0
ΛΛ
00
(z
0
u|zu
0
λλ
00
)
× P
Λ
(λ)P
Λ
0
(λ
0
)P
Λ
00
(λ
00
). (58)
Note that this is not a convex set. Furthermore, since a triple of pairwise common causes can be simulated by
a triplewise common cause, the free operations defined in Eq. (58) are a strict subset of the tripartite LOSR
operations defined in Eq. (57). It follows that, just as we saw for the free boxes in the triangle-with-settings
scenario, one cannot motivate the free operations defined in Eq. (58) by appeal to the no-signalling principle.
And, again just as we noted for the free boxes, it is unclear how such a choice could ever be motivated by a
principle that appealed only to the input-output functionality of the operation.
The triangle-with-settings scenario also illustrates why one should not mathematically impose convex
closure of the set of free operations, as was done in Refs. [18]. Rather, whether or not the set of free
operations is convexly closed depends on the causal structure, which specifies precisely how randomness is
shared among the parties. For Bell scenarios, the set of free operations is convex by construction, whereas for
other causal structures, such as the triangle-with-settings scenario, it is not. Mathematically imposing convex
closure in the triangle-with-settings scenario would be equivalent to asserting that there was a common cause
for all three wings, which would constitute a change in the causal structure being considered. In other words,
imposing convexity in an ad-hoc manner contradicts the foundations of the causal modelling paradigm,
where it is the causal structure that specifies how randomness is shared among the parties, and consequently
specifies whether or not convexity holds.
Note finally that the lack of convexity in general causal structures (such as the triangle-with-settings
scenario) implies that the project of quantifying nonclassicality in these cases will be much more complicated
than it was in the Bell scenario.
B Proofs
B.1 Proof of Proposition 19: closed-form expression for M
NPR
(R)
In this section, we present some arguments that aid in justifying Proposition 19, the proof of which is given
at the end of this appendix. Recall Proposition 19:
Proposition 19.
For any free resource
R
of type
(
2 2
2 2
)
,
M
NPR
(
R
) = 2. For any nonfree resource
R
of type
(
2 2
2 2
)
, there is a unique
k ∈ {
0
, . . . ,
7
}
for which
CHSH
k
(
R
)
>
2. Within this region, if
R ∈ C
NPR,k
, then we
have simply M
NPR
(R) = CHSH
k
(R). If, on the other hand, R 6∈ C
NPR,k
, we have
M
NPR
(R) = 2α+2,
where
α
is the value appearing in the decomposition
R
=
γ L
bb
R
+ (1
−γ
)
C
k
(
α
), where
C
k
(
α
)
∈ C
NPR,k
,
L
bb
R
∈ L
bb
k
and
γ ∈
[0
,
1]. This value of
α
is unambiguous because there exists a unique resource
L
bb
R
∈ L
bb
k
and a unique choice of γ ∈ [0, 1] and of α ∈ [0, 1] such that R = γ L
bb
R
+ (1−γ)C
k
(α).
44
The (unique) relevant decomposition is shown in Fig. 6 (for the case where k = 0).
We first demonstrate the equivalence of three statements which pertain to the value of M
NPR
(
R
)
for the
subset of resources that satisfy CHSH(R) ≥ 2:
Proposition 29.
For any resource
R
of type
(
2 2
2 2
)
such that
CHSH
(
R
)
≥
2, the following definitions are
equivalent to M
NPR
(R):
min
0≤α≤1
{CHSH(C(α)) such that C(α) 7−→R}, (59a)
min
α
n
CHSH(C(α)) such that ∃γ ≥ 0 and ∃L
b
R
∈ L
b
with R = γ L
b
R
+ (1−γ)C(α)
o
, (59b)
(
if R ∈ C
NPR
: CHSH(R), else
if R 6∈ C
NPR
: 2α+2, where α, γ ≥ 0, and L
bb
R
∈ L
bb
are all unique
in the decomposition R = γ L
bb
R
+ (1−γ)C(α).
(59c)
Proof of Eq. (59a).
Eq.
(59a)
is directly equivalent to the definition of the
M
NPR
monotone given in Eq.
(36)
. Hence, we take
that as our starting point, and prove the implications of the subequations in Proposition 29.
Proof that Eq. (59a) ⇔ Eq. (59b).
Section 5 guarantees that
C
(
α
)
7−→R
if and only if we can generate R by convex mixtures of
C
(
α
) with the
images of
C
(
α
) under LDO operations. For any
R /∈ C
NPR
such that
CHSH
(
R
)
≥
2, we simplify the situation
by proving that if
R
can be generated by mixing
C
(
α
) with its images under LDO, then
R
can alternatively
be generated by mixing
C
(
α
) with a local point which saturates the CHSH inequality; namely, a point in
L
b
,
as stated in Eq. (59b). To prove this, it is useful to define the notion of a screening-off inequality.
Definition 30.
The inequality
f
(
R
)
≥ b
is said to
screen-off
the fixed-type set of resources which satisfy it,
i.e.,
n
R : f(R) ≥ b , [R] =
(
|X| |Y |
|S | |T |
)
o
, if the fixed-type set of resources which saturate it is a free set, i.e., if
n
R : f(R) = b , [R] =
(
|X| |Y |
|S | |T |
)
o
consists only of classically realizable common-cause boxes.
For example, the inequality
CHSH
(
R
)
≥
2 screens-off the set
R : CHSH(R) ≥ 2 , [R] =
(
2 2
2 2
)
since
R : CHSH(R) = 2 , [R] =
(
2 2
2 2
)
⊂ S
free
(
2 2
2 2
)
.
Screening-off inequalities are useful when making statements about resource convertibility, as follows.
Consider the case where we ask whether
R
2
7−→R
1
: if
R
1
lies inside some screened-off region, then, given
Proposition 15,
R
1
∈ P
LOSR
[R
1
]
(
R
2
) if and only if
R
1
is in the convex hull of those images of
R
2
under LDO
inside the screened-off region, together with the boundary (where the inequality is saturated). Formally, if
f(R) ≥ b is a screening-off inequality for resources of type [R
1
], then, given Proposition 15,
R
2
7−→R
1
iff ∃L
b
such that f(L
b
) = b and R
1
∈ ConvexHull
L
b
, V
LDO
[R
1
]
(R
2
)
\
{R : f(R) > b}
| {z }
the LDO images of R
2
interior to screened-off region
.
Since
CHSH
(
R
)
≥
2 is a screening-off inequality whose saturation-boundary is given by
L
b
, and since the
only image in V
LDO
(
2 2
2 2
)
(C(α)) which violates the CHSH inequality is C(α) itself, it follows that
C(α) 7−→R if and only if ∃γ ≥ 0 and ∃L
b
R
∈ L
b
such that R = γ L
b
R
+ (1−γ)C(α). (60)
The equivalence Eq.
(59a) ⇔
Eq.
(59b)
follows. As a final comment, notice that this characterization of
convertibility in terms of the existence of a geometric decomposition involves arbitrary points which saturate
the CHSH inequality, and there are typically many such decompositions.
45
Proof that Eq. (59b) ⇔ Eq. (59c).
Recall that Eq.
(59b)
involves a minimization under the constraint that
α
is such that
R = γ L
b
R
+ (1−γ)C(α)
.
We can formally recast it as a constrained optimization problem, as follows:
min
0≤α≤1
CHSH
C(α)
such that L
b
R
:
=
R − (1−γ)C(α)
γ
, under the constraint that
all conditional probabilities in the expression of
~
L
b
R
are nonnegative,
where γ is an implicit function of α according to
CHSH(R) − (1−γ) CHSH
C(α)
γ
= 2, as implied by the fact that CHSH(L
b
R
) = 2.
Essentially, this is a constrained optimization problem with a linear objective subject to one nonlinear
constraint; namely, that the smallest conditional probability in the expression
~
L
b
R
44
must be nonnegative.
For such optimization problems, it is always the case that the objective is maximized when the constraint is
not merely satisfied but saturated. Put another way, the set of achievable
α
arise from points
L
b
R
wherein
all conditional probabilities are nonnegative, but the optimal
α
arises for some unique
L
b
R
=
L
bb
R
where the
smallest conditional probability in
~
L
bb
R
is precisely zero.
Proof of Proposition 19.
Proposition 19 for arbitrary resources follows from Proposition 29 by the symmetry noted in Proposition 16d.
Namely, the argument can be repeated unchanged in each of the eight spaces of resources generated by the
images under LSO of the set of resources satisfying
CHSH
(
R
)
≥
2. Together with the trivial observation that
free resources (which do not violate any of the eight CHSH inequalities) always have value of
M
NPR
equal to
2, Proposition 19 follows.
B.2 Proof of Proposition 21: when the two monotones are complete
We now prove Proposition 21, repeated here:
Proposition 21.
Consider a two-parameter family
R(α,γ) = γ L
bb
?
+ (1−γ)C(α)
, marked by any fixed
L
bb
?
—
that is, a point which saturates the CHSH inequality and is on the boundary of the no-signaling set. The pair
of monotones {M
CHSH
, M
NPR
} is complete relative to such a family of resources.
Proof.
A set of monotones is complete relative to a family of resources if and only if every candidate conversion
among resources in the family which is not ruled out by any of the monotones in the set is in fact possible for
free, as per Eq. (21).
In Fig. 16(a), we depict in blue the set of candidate conversions (from a generic resource
R
(
α
1
, γ
1
) to
another resource in the family) which are not ruled out by
{M
CHSH
, M
NPR
}
; namely, the blue shaded region
contains all resources which have a value for each of the two monotones that is equal to or lower than that
of
R
(
α
1
, γ
1
). To prove the proposition, we argue that
R
(
α
1
, γ
1
) can indeed be converted to any resource
in the blue region. By convexity, it suffices to prove that
R
(
α
1
, γ
1
) can be converted to each of the four
extreme points of the blue region. Since
L
bb
?
and
L
b
NPR
are free resources,
R
(
α
1
, γ
1
) can freely be converted
to either of them, and the resource
R
(
α
1
, γ
1
) can obviously be ‘converted’ to itself, as the identity is free. Our
proof, therefore, focuses on demonstrating that
R
(
α
1
, γ
1
) can indeed be converted to the fourth extreme point
R
(
α
2
,
0), shown as a green star. We now give the explicit free operation which takes a generic initial resource
44
~
L
b
R
denotes the representation of
L
b
R
as a vector whose components are the conditional probabilities
{P
XY |ST
(
xy|st
) :
x, y, s, t ∈
{0, 1}}.
46
L
bb
?
R
PR
L
b
NPR
R(α
1
, γ
1
)
R(α
2
, 0)
= R
α
1
(1−γ
1
), 0
(a)
2
4
4
R
P R
M
NPR
M
CHSH
R(α
1
, γ
1
)
(b)
Figure 16: (a) and (b) provide the same pair of depictions of the two-parameter family of resources
S
L
bb
?
(
2 2
2 2
)
as were introduced
in Fig. 7. We consider a two-parameter family of resources. A generic such resource, specified by
α
1
and
γ
1
, is marked by a
red diamond. Also depicted are some of the level curves of the two monotones
M
CHSH
and
M
NPR
. The solid dark blue
region denotes the set of all resources within this family which have values for both monotones less than or equal to their
values for
R
(
α
1
, γ
1
). To prove Proposition 21, one must show that
R
(
α
1
, γ
1
) can be converted to any resource in the solid
blue region. The critical step in this proof is the demonstration that it is possible to convert any resource to one lying on
the line connecting
R
PR
and
L
b
NPR
without changing the value of
M
CHSH
. Graphically, this corresponds to converting the
generic resource R(α
1
, γ
1
) to the resource R(α
2
, 0) marked by a green star.
R
(
α
1
, γ
1
) and projects it onto the chain leftwards in the two-dimensional coordinate system of Fig. 16(a), i.e.,
to the target resource R(α
2
, 0), where α
2
= α
1
(1 − γ
1
)
45
.
We denote the free operation which enacts this conversion by
τ
erase-γ
; it is the operation which projects
any resource into the subspace of resources that are invariant under the
G
456
subgroup of
LSO
(
2 2
2 2
)
(
G
456
is
defined in Proposition 16c on page 23), i.e., onto the chain
C
NPR
.
46
This operation is indeed free, as it can
be constructed by a uniform mixture of all the elements of
G
456
, each of which is free. Recall that
G
456
is the
subgroup of
LSO
(
2 2
2 2
)
which stabilizes
CHSH
0
, and therefore clearly does not modify the value of the
M
CHSH
monotone.
It remains only to show that the
G
456
-invariant subspace of resources within the set of all
(
2 2
2 2
)
-type resources
for which
CHSH
(
R
)
≥
2 is the chain
C
NPR
, i.e., the line of points between
R
PR
and
L
b
NPR
. This is evident
by confirming that
τ
erase-γ
leaves
R
PR
invariant, but maps each of the 8 deterministic
CHSH
-saturating
boxes to
L
b
NPR
. Those 1
+
8 resources are the extreme points of the set of all
(
2 2
2 2
)
-type resources such that
CHSH
(
R
)
≥
2; since the extreme points map to the line under the action of
τ
erase-γ
, by convex linearity it
follows that the chain is the only space invariant under G
456
within the two-parameter family.
B.3 Proof of Proposition 23: all nonfree resources of type
(
2 2
2 2
)
are orbital
We now prove Proposition 23, repeated here:
Proposition 23. All nonfree resources of type
(
2 2
2 2
)
are orbital.
45
The particular relation between
α
2
and (
α
1
, γ
1
) follows from Eq.
(45)
, by noticing that
R
(
α
2
,
0) and
R
(
α
1
, γ
1
) must have the
same value for M
CHSH
.
46
Equivalently, τ
erase-γ
is the Reynold’s operator of the subgroup G
456
.
47
Before presenting the proof, we introduce some additional concepts and a few lemmas on which our proof
relies. Throughout the following, we are focused on sets of resources of fixed type, and on type-preserving
operations. Hence, we use slightly abbreviated notation; e.g. V
LDO
(
R
)
is used as shorthand for V
LDO
[R]
(
R
)
,
and so on.
Definition 31.
The set of
local deterministic type-preserving nonsymmetry operations
, denoted
LDTNO
, contains all type-preserving operations in LDO which are not in LSO. The image of a resource
R
under LDTNO constitutes a discrete set of resources denoted
V
LDTNO
(
R
). Moreover, we use
HullLDTNO
(
R
)
to indicate the set of all resources in the convex hull of the image of a resource
R
under LDTNO, i.e.,
HullLDTNO(R)
:
= ConvexHull
V
LDTNO
(R)
.
Definition 32.
A resource
R
is said to be
sensitive
if every element of LDTNO removes
R
from its
equivalence class; i.e., if for all
τ∈LDTNO
it holds that
τ ◦R Y7−→R
. Equivalently, a resource
R
is sensitive
if and only if
R
is not in the convex hull of its images under LDTNO, i.e., if
R 6∈ HullLDTNO
(
R
). A set
of resources is called sensitive if every resource in the set is sensitive.
We bring up the property of sensitivity because (i) it is straightforward to test if a given resource is
sensitive or not by means of a linear program, and (ii) eventually we will argue that if a resource is sensitive,
then it is also orbital. Furthermore, we now prove that sensitive resources never appear in isolation. That is,
a single sensitive resource can be used to construct a set of sensitive resources, as follows:
Lemma 33.
For any resource
R
, every resource
R
0
which is below
R
in the pre-order and which cannot be
generated from
R
by mixtures of LDTNO operations is Formally: the set of resources
S
R
sens
:
=
P
LOSR
(
R
)
\
HullLDTNO(R) is always sensitive.
Proof. First, note two related, useful facts:
(1) The composition of any deterministic operation (invertible or not) followed by some deterministic
nonsymmetry operation is precisely some (other) deterministic nonsymmetry operation. Formally, if
τ
LDTNO
∈
LDTNO
and
τ
LDO
∈ LDO
, and defining
τ
0
:
=
τ
LDTNO
◦τ
LDO
, then
τ
0
∈ LDTNO
. A consequence of this is that
the entire set
P
LOSR
(
R
) is mapped to the set
HullLDTNO
(
R
) under
LDTNO
and convex mixtures thereof.
To see this, recall that the image of any convex set of resources under any convex set of operations is identically
the convex hull of the images of the extremal resources under the extremal operations (in the respective sets).
We use this fact to effectively replace
P
LOSR
(
R
) with
V
LDO
(
R
) and to replace convex mixtures of LDTNO
with LDTNO itself, without loss of generality. In summary:
HullLDTNO
(
P
LOSR
(
R
)) =
HullLDTNO
(
R
)
by virtue of the fact that LDTNO ◦ LDO = LDTNO.
(2) The composition of any deterministic nonsymmetry operation followed by some deterministic operation
(invertible or not) is some (other) deterministic nonsymmetry operation. Formally, if
τ
1−LDTNO
∈ LDTNO
and
τ
2−LDO
∈ LDO
, and defining
τ
4
:
=
τ
2−LDO
◦τ
1−LDTNO
, then
τ
4
∈ LDTNO
. A consequence of this is that the
entire set
HullLDTNO
(
R
) is mapped to itself under
LOSR
. To see this, we reuse the shortcut of considering
only extremal resources and extremal operations. Specializing to our objects of interest, we effectively replace
the operations-set
LOSR
by its extremal operations — namely
LDO
— and the resources-set
HullLDTNO
(
R
)
by
V
LDTNO
(
R
) without loss of generality. In summary:
P
LOSR
(
HullLDTNO
(
R
)) =
HullLDTNO
(
R
) by
virtue of the fact that LDO ◦ LDTNO = LDTNO.
Now we are in position to prove Lemma 33. The set of resources
R
0
below
R
in the partial order is
identically
P
LOSR
(
R
). The set of resources which can be generated from
R
by mixtures of deterministic
nonsymmetry operations is identically
HullLDTNO
(
R
). So, a resource
R
0
is below
R
in the partial
order and cannot be generated from
R
by mixtures of deterministic nonsymmetry operations if and only if
R
0
∈ P
LOSR
(R) \ HullLDTNO(R) =: S
R
sens
.
Now, consider any
τ ∈ LDTNO
and any
R
0
∈ S
R
sens
, and define
R
00
:
=
τ ◦ R
0
. Since we have estab-
lished that the entirety of
P
LOSR
(
R
) is mapped to
HullLDTNO
(
R
) under LDTNO, it follows that
R
00
∈ HullLDTNO
(
R
). However, since we have also established that the entirety of
HullLDTNO
(
R
)
is mapped only to itself under LOSR, and since
R
0
/∈ HullLDTNO
(
R
), it further follows that
R
00
Y7−→R
0
.
Evidently, any
R
0
∈ S
R
sens
is removed from its equivalence class by every deterministic nonsymmetry operation,
i.e., S
R
sens
is sensitive. This proves the Lemma.
48
Note that Lemma 33 implies that if R is sensitive, and R
0
is equivalent to R, then R
0
is also sensitive.
Lemma 34.
If a resource is sensitive, then it is also orbital. That is, if two sensitive resources are
interconvertible under type-preserving LOSR, then they are interconvertible under LSO.
Proof.
Let
R
and
R
0
be distinct sensitive resources that are interconvertible under type-preserving LOSR,
i.e.,
R 6
=
R
0
but
R ←→R
0
and [
R
] = [
R
0
]. Any operation which preserves the equivalence class of a sensitive
resource can be expressed as a convex combination of elements of LSO. The assumption of sensitivity thus
dictates that
R
0
is in the convex hull of the images of
R
under LSO, and vice versa. We proceed to show
that this sort of relationship must imply that
R ∈ V
LSO
(
R
0
) and
R
0
∈ V
LSO
(
R
), that is, that
R
and
R
0
are
LSO-equivalent.
47
This can be seen by recognizing that the 2-norm is a convex function invariant under LSO, meaning
R
0
∈ ConvexHull
V
LSO
(R)
implies
k
~
R
0
k
2
≤ k
~
Rk
2
.
48
By symmetry under exchange of
R
and
R
0
, it holds that
k
~
Rk
2
≤ k
~
R
0
k
2
, and hence
k
~
R
0
k
2
=
k
~
Rk
2
. The 2-norm, moreover, strictly decreases under nontrivial stochastic
mixing;
49
hence all interconversions between equivalent sensitive resources must be mediated by deterministic
symmetries. Formally: If
R
0
=
P
n
i=1
w
i
π
i
◦ R
and
R =
P
n
i=1
w
0
i
π
i
◦ R
, where
P
n
i=1
w
i
=
P
n
i=1
w
0
i
= 1
and
{w
1
, ..., w
n
, w
0
1
, ..., w
0
n
} ≥ 0, then k
~
R
0
k
2
= k
~
Rk
2
and w
i
, w
0
i
∈ {0, 1}.
Lemma 35. R
PR
is a sensitive resource, and
P
LOSR
(
R
PR
)
\HullLDTNO
(
R
PR
) is the entire eight dimen-
sional set of all nonfree resources of type
(
2 2
2 2
)
.
Proof by inspection.
One can readily verify that τ ◦ R
PR
∈ S
free
(
2 2
2 2
)
for all type-preserving LDTNO operations τ.
Proof of Proposition 23.
Lemma 35 together with Lemma 33 immediately imply that all nonfree resources of type
(
2 2
2 2
)
are sensitive.
Lemma 34 then directly implies that all these resources are orbital.
A final comment: consider generalizing Proposition 23 in light of the discussion just given. If one desires to
construct an orbital set of resources beyond
(
2 2
2 2
)
-type, one needs only to find some single sensitive resource
R of the desired type. From Lemmas 33 and 34, it then follows that the set of resources P
LOSR
(
R
)
\
HullLDTNO
(
R
)
constitutes an orbital set. It might be the case, for instance, that for any nontrivial choice
of resource type, there is at least one convexly extremal resource that is sensitive, analogous to how the
PR-box is a sensitive resource for type
(
2 2
2 2
)
.
B.4 Proof of Proposition 25: lower bound on the number of monotones in any complete set
Recall that a resource is termed orbital if and only if its LOSR-equivalence class of resources of the same
type is equal to its LSO-equivalence class. We now prove Proposition 25, recalled below:
Proposition 25.
For any compact set
S
of resources that are all orbital, the intrinsic dimension of the set
S
is a lower bound on the cardinality of a complete set of continuous monotones for
S
(and for any superset
of S).
47
The fact that
R
0
is in the convex hull of the images of
R
under permutations of
R
’s probabilities is equivalent to stating that
R
vector majorizes
R
0
. The relationship is reflexive, however. Readers familiar with vector majorization may recall that two
vectors are equivalent under the majorization order if and only if they are related by some reordering, i.e., a (not necessarily
physical) symmetry operation.
48
Recall that
~
R
is shorthand for the representation of the resource in terms of a real-valued vector consisting of all possible
conditional probabilities, i.e.,
~
R =
P
XY |ST
(xy|st) : x, y, s, t ∈ {0, 1}
.
49
Consider the hypersphere consisting of all resources with 2-norm in common with
~
R
. All the images of
R
under LSO lie on
the surface of this hypersphere. Stochastic mixing of symmetry operations (applied to
R
) is equivalent to convexly combining
different points from the surface of the hypersphere. Any convex combination of points from the surface of a hypersphere results
in a final point strictly interior to the sphere. Strictly interior points are closer to the center, in precisely the sense of having a
strictly smaller 2-norm.
49
Proof.
The set of local symmetry operations for a given type has finite cardinality, and hence there are a
finite number of resources in the LSO-equivalence class of any resource. For an orbital resource
R
, this
implies that the LOSR-equivalence class of
R
(over resources of type [
R
]) is precisely
V
LSO
(
R
), which is a
finite set. If a compact set
S
of orbital resources has intrinsic dimension
d
, and the LOSR-equivalence class of
every resource in the set is finite and hence zero-dimensional, then it follows that one can find
d
-dimensional
compact subsets of resources in S in which no two resources are equivalent.
50
Hence, no two resources in such a subset are assigned the same tuple of values by any complete set of
monotones. In other words, a complete set of
n
continuous monotones maps the subset of resources injectively
to
R
n
. But this map can only be injective if
n ≥ d
, which guarantees that the number of continuous monotones
required to identify a resource in the set is at least as large as the intrinsic dimension
d
of the set
S
. Finally,
note that the number of continuous monotones required to identify a resource in any superset of
S
must be
at least as large as for the set S itself, which completes the proof.
References
[1] J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics 1, 195 (1964).
[2] J. S. Bell, “On the Problem of Hidden Variables in Quantum Mechanics,” Rev. Mod. Phys. 38, 447 (1966).
[3]
B. Hensen
et al.
, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,”
Nature 526, 682 EP (2015).
[4]
M. Giustina
et al.
, “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Phys. Rev.
Lett. 115, 250401 (2015).
[5] L. Shalm et al., “Strong Loophole-Free Test of Local Realism,” Phys. Rev. Lett. 115, 250402 (2015).
[6]
J. Barrett, L. Hardy, and A. Kent, “No Signaling and Quantum Key Distribution,” Phys. Rev. Lett.
95
, 010503
(2005).
[7]
A. Ac´ın, N. Gisin, and L. Masanes, “From Bell’s Theorem to Secure Quantum Key Distribution,” Phys. Rev.
Lett. 97, 120405 (2006).
[8]
V. Scarani, N. Gisin, N. Brunner, L. Masanes, S. Pino, and A. Ac´ın, “Secrecy extraction from no-signaling
correlations,” Phys. Rev. A 74, 042339 (2006).
[9]
A. Ac´ın, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum
Cryptography against Collective Attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[10] R. Colbeck and R. Renner, “Free randomness can be amplified,” Nat. Phys. 8, 450 EP (2012).
[11]
S. Pironio, A. Ac´ın, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes,
L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature
464
, 1021 EP
(2010).
[12]
C. Dhara, G. Prettico, and A. Ac´ın, “Maximal quantum randomness in Bell tests,” Phys. Rev. A
88
, 052116
(2013).
[13]
U. Vazirani and T. Vidick, “Fully Device-Independent Quantum Key Distribution,” Phys. Rev. Lett.
113
,
140501 (2014).
[14]
J. Kaniewski and S. Wehner, “Device-independent two-party cryptography secure against sequential attacks,”
New J. Phys. 18, 055004 (2016).
[15]
R. Gallego, L. E. W¨urflinger, A. Ac´ın, and M. Navascu´es, “Operational Framework for Nonlocality,” Phys. Rev.
Lett. 109, 070401 (2012).
[16]
J. I. de Vicente, “On nonlocality as a resource theory and nonlocality measures,” J. Phys. A
47
, 424017 (2014).
[17]
J. Geller and M. Piani, “Quantifying non-classical and beyond-quantum correlations in the unified operator
formalism,” J. Phys. A 47, 424030 (2014).
[18] R. Gallego and L. Aolita, “Nonlocality free wirings and the distinguishability between Bell boxes,” Phys. Rev.
A 95 (2017).
50
Not all compact subsets will necessarily have this property, but some will. For example, consider a nonfree resource
R
asym
which
is not invariant under any LSO operation. Every LSO operation maps such a resource to a distinct resource not in the original
neighborhood for a suitably small neighborhood. Because LSO operations are linear (and hence continuous), they map compact
subspaces to compact subspaces. Hence, every LSO operation takes the entire neighborhood of nonfree resources around
R
asym
to some other nonfree neighborhood; if the original neighborhood is chosen to be small enough, these two neighborhoods will not
intersect. Hence, no two resources in the original neighborhood are interconvertible by LSO.
50
[19]
K. Horodecki, A. Grudka, P. Joshi, W. K lobus, and J.
L
odyga, “Axiomatic approach to contextuality and
nonlocality,” Phys. Rev. A 92, 032104 (2015).
[20]
B. Amaral, A. Cabello, M. T. Cunha, and L. Aolita, “Noncontextual wirings,” Phys. Rev. Lett.
120
, 130403
(2018).
[21]
E. Kaur, M. M. Wilde, and A. Winter, “Fundamental limits on key rates in device-independent quantum key
distribution,” arXiv:1810.05627 (2018).
[22]
S. G. A. Brito, B. Amaral, and R. Chaves, “Quantifying Bell nonlocality with the trace distance,” Phys. Rev.
A 97, 022111 (2018).
[23]
D. Schmid, D. Rosset, and F. Buscemi, “Type-independent resource theory of local operations and shared
randomness,” arXiv:1909.04065 (2019).
[24]
D. Rosset, D. Schmid, and F. Buscemi, “Characterizing nonclassicality of arbitrary distributed devices,”
arXiv:2004.09194 (2020).
[25]
D. Schmid, T. C. Fraser, R. Kunjwal, A. B. Sainz, E. Wolfe, and R. W. Spekkens, “Why standard entanglement
theory is inappropriate for the study of Bell scenarios,” arXiv:1911.12462 (2019).
[26]
B. Coecke, T. Fritz, and R. W. Spekkens, “A mathematical theory of resources,” Info. & Comp.
250
, 59 (2016).
[27]
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable
theories,” Phys. Rev. Lett. 23, 880 (1969).
[28] A. Shimony, “Bell’s Theorem,” in The Stanford Encyclopedia of Philosophy (2017).
[29] B. d’Espagnat, “The Quantum Theory and Reality,” Scientific American 241, 158 (1979).
[30] H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[31] R. F. Werner, “Comment on ‘What Bell did’,” J. Phys. A 47, 424011 (2014).
[32] V. Scarani, “The Device-Independent Outlook on Quantum Physics,” Acta Physica Slovaca 62, 347 (2012).
[33]
T. Maudlin, Quantum Non-Locality and Relativity : Metaphysical Intimations of Modern Physics (Blackwell
Publishers, 2002).
[34] T. Norsen, “Bell Locality and the Nonlocal Character of Nature,” Found. Phys. Lett. 19, 633 (2006).
[35]
R. Chaves, R. Kueng, J. B. Brask, and D. Gross, “Unifying Framework for Relaxations of the Causal Assumptions
in Bell’s Theorem,” Phys. Rev. Lett. 114, 140403 (2015).
[36]
R. Chaves, D. Cavalcanti, and L. Aolita, “Causal hierarchy of multipartite Bell nonlocality,” Quantum
1
, 23
(2017).
[37]
T. Maudlin, “Bell’s Inequality, Information Transmission, and Prism Models,” in Philosophy of Science Associa-
tion, 1 (1992) pp. 404–417.
[38]
B. F. Toner and D. Bacon, “Communication Cost of Simulating Bell Correlations,” Phys. Rev. Lett.
91
, 187904
(2003).
[39] G. Hooft, “The Fate of the Quantum,” arXiv:1308.1007 (2013), report numbers: ITP-UU-13/22, SPIN-13/15.
[40]
M. J. W. Hall, “Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement
Independence,” Phys. Rev. Lett. 105, 250404 (2010).
[41]
J. Barrett and N. Gisin, “How Much Measurement Independence Is Needed to Demonstrate Nonlocality?” Phys.
Rev. Lett. 106, 100406 (2011).
[42] J. Pearl, Causality: Models, Reasoning, and Inference (Cambridge University Press, 2009).
[43] C. J. Wood and R. W. Spekkens, “The lesson of causal discovery algorithms for quantum correlations: causal
explanations of Bell-inequality violations require fine-tuning,” New J. Phys. 17, 033002 (2015).
[44]
J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, “Quantum Common Causes and
Quantum Causal Models,” Phys. Rev. X 7, 031021 (2017).
[45]
J. Henson, R. Lal, and M. F. Pusey, “Theory-independent limits on correlations from generalized Bayesian
networks,” New J. Phys. 16, 113043 (2014).
[46] T. Fritz, “Beyond Bell’s theorem: correlation scenarios,” New J. Phys. 14, 103001 (2012).
[47] L. Hardy, “Quantum Theory From Five Reasonable Axioms,” quant-ph/0101012 (2001).
[48] J. Barrett, “Information processing in generalized probabilistic theories,” Phys. Rev. A 75, 032304 (2007).
[49]
P. Janotta and H. Hinrichsen, “Generalized probability theories: what determines the structure of quantum
theory?” J. Phys. A 47, 323001 (2014).
[50]
G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Probabilistic theories with purification,” Phys. Rev. A
81
,
062348 (2010).
[51]
G. M. Ariano, Quantum Theory from First Principles: An Informational Approach (Cambridge University
Press, 2019).
51
[52] F. Costa and S. Shrapnel, “Quantum causal modelling,” New J. Phys 18, 063032 (2016).
[53] J. Barrett, R. Lorenz, and O. Oreshkov, “Quantum Causal Models,” arXiv:1906.10726 (2019).
[54]
D. Schmid, H. Du, M. Mudassar, G. C. de Wit, D. Rosset, and M. J. Hoban, “Postquantum common-cause
channels: the resource theory of local operations and shared entanglement,” arXiv:2004.06133 (2020).
[55]
G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Quantum Circuit Architecture,” Phys. Rev. Lett.
101
,
060401 (2008).
[56]
G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Theoretical framework for quantum networks,” Phys. Rev. A
80, 022339 (2009).
[57] S. Popescu and D. Rohrlich, “Quantum nonlocality as an axiom,” Found. Phys. 24, 379 (1994).
[58]
J. Selby
et al.
, “Contextuality Quantified: A Resource Theory Encompassing Prepare-and-Measure Processes,”
Forthcoming.
[59]
C. Branciard, D. Rosset, N. Gisin, and S. Pironio, “Bilocal versus nonbilocal correlations in entanglement-
swapping experiments,” Phys. Rev. A 85, 032119 (2012).
[60]
A. Ac´ın, R. Augusiak, D. Cavalcanti, C. Hadley, J. K. Korbicz, M. Lewenstein, L. Masanes, and M. Piani,
“Unified Framework for Correlations in Terms of Local Quantum Observables,” Phys. Rev. Lett.
104
, 140404
(2010).
[61]
S. W. Al-Safi and A. J. Short, “Simulating all Nonsignaling Correlations via Classical or Quantum Theory with
Negative Probabilities,” Phys. Rev. Lett. 111, 170403 (2013).
[62]
J.-D. Bancal, S. Pironio, A. Ac´ın, 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 (2012).
[63]
J. S. Bell, “La nouvelle cuisine,” in Quantum Mechanics, High Energy Physics And Accelerators: Selected Papers
Of John S Bell (With Commentary) (World Scientific, 1995) pp. 910–928.
[64]
O. Oreshkov, F. Costa, and
ˇ
C. Brukner, “Quantum correlations with no causal order,” Nat. Comm.
3
, 1092 EP
(2012).
[65] O. Oreshkov and C. Giarmatzi, “Causal and causally separable processes,” New J. Phys. 18, 093020 (2016).
[66]
D. Rosset, J.-D. Bancal, and N. Gisin, “Classifying 50 years of Bell inequalities,” J. Phys. A
47
, 424022 (2014).
[67] A. Seress, Permutation Group Algorithms (Cambridge University Press, 2003).
[68] S. Pironio, “Lifting Bell inequalities,” J. Math. Phys. 46, 062112 (2005).
[69]
D. Rosset,
¨
Amin Baumeler, J.-D. Bancal, N. Gisin, A. Martin, M.-O. Renou, and E. Wolfe, “Algebraic and
geometric properties of local transformations,” arXiv:2004.09405 (2020).
[70] A. Fine, “Hidden Variables, Joint Probability, and the Bell Inequalities,” Phys. Rev. Lett. 48, 291 (1982).
[71] T. Gonda and R. W. Spekkens, “Monotones in General Resource Theories,” arXiv:1912.07085 (2019).
[72] F. Buscemi, “All Entangled Quantum States Are Nonlocal,” Phys. Rev. Lett. 108, 200401 (2012).
[73]
S. Beigi and A. Gohari, “Monotone Measures for Non-Local Correlations,” IEEE T. Inform. Theory
61
, 5185
(2015).
[74]
P. Bierhorst, “Geometric decompositions of Bell polytopes with practical applications,” J. Phys. A
49
, 215301
(2016).
[75]
D. Cavalcanti and P. Skrzypczyk, “Quantitative relations between measurement incompatibility, quantum
steering, and nonlocality,” Phys. Rev. A 93, 052112 (2016).
[76]
K. T. Goh, J. Kaniewski, E. Wolfe, T. V´ertesi, X. Wu, Y. Cai, Y.-C. Liang, and V. Scarani, “Geometry of the
set of quantum correlations,” Phys. Rev. A 97, 022104 (2018).
[77]
M. W. Girard and G. Gour, “Computable entanglement conversion witness that is better than the negativity,”
New J. Phys. 17, 093013 (2015).
[78]
N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys.
86
, 419
(2014).
[79]
J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, “Nonlocal correlations as an
information-theoretic resource,” Phys. Rev. A 71, 022101 (2005).
[80]
N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys.
86
, 419
(2014).
[81]
J. Barrett and S. Pironio, “Popescu-Rohrlich Correlations as a Unit of Nonlocality,” Phys. Rev. Lett.
95
,
140401 (2005).
[82] V. L. Popov, Algebraic Geometry IV (Springer-Verlag, 1994) Chap. 4: Quotients.
[83]
D. Collins and N. Gisin, “A relevant two qubit Bell inequality inequivalent to the CHSH inequality,” J. Phys. A
37, 1775 (2004).
52
[84]
T. H. Yang and M. Navascu´es, “Robust self-testing of unknown quantum systems into any entangled two-qubit
states,” Phys. Rev. A 87, 050102(R) (2013).
[85]
C. Bamps and S. Pironio, “Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like
inequalities and their application to self-testing,” Phys. Rev. A 91, 052111 (2015).
[86]
L. Masanes, “Necessary and sufficient condition for quantum-generated correlations,” quant-ph/0309137 (2003).
[87]
J. Allcock, N. Brunner, M. Pawlowski, and V. Scarani, “Recovering part of the boundary between quantum and
nonquantum correlations from information causality,” Phys. Rev. A 80, 040103(R) (2009).
[88]
A. Ac´ın, S. Massar, and S. Pironio, “Randomness versus Nonlocality and Entanglement,” Phys. Rev. Lett.
108
,
100402 (2012).
[89]
E. Wolfe and S. F. Yelin, “Quantum bounds for inequalities involving marginal expectation values,” Phys. Rev.
A 86, 012123 (2012).
[90] M. A. Nielsen, “Conditions for a class of entanglement transformations,” Phys. Rev. Lett. 83, 436 (1999).
[91]
C. Bamps, S. Massar, and S. Pironio, “Device-independent randomness generation with sublinear shared quantum
resources,” Quantum 2, 86 (2018).
[92]
G. Gour, M. P. M¨uller, V. Narasimhachar, R. W. Spekkens, and N. Y. Halpern, “The resource theory of
informational nonequilibrium in thermodynamics,” Phys. Rep. 583, 1 (2015).
[93]
T. Fritz, “Resource convertibility and ordered commutative monoids,” Math. Struct. Comp. Sci.
27
, 850–938
(2017).
[94]
N. Brunner and P. Skrzypczyk, “Nonlocality Distillation and Postquantum Theories with Trivial Communication
Complexity,” Phys. Rev. Lett. 102, 160403 (2009).
[95]
B. Lang, T. V´ertesi, and M. Navascu´es, “Closed sets of correlations: answers from the zoo,” J. Phys. A
47
,
424029 (2014).
[96]
Y. R. Sanders and G. Gour, “Necessary conditions for entanglement catalysts,” Phys. Rev. A
79
, 054302
(2009).
[97]
D. Jonathan and M. B. Plenio, “Entanglement-Assisted Local Manipulation of Pure Quantum States,” Phys.
Rev. Lett. 83, 3566 (1999).
[98]
W. van Dam and P. Hayden, “Universal entanglement transformations without communication,” Phys. Rev. A
67, 060302 (2003).
[99] B. Steudel and N. Ay, “Information-Theoretic Inference of Common Ancestors,” Entropy 17, 2304 (2015).
[100]
E. Wolfe, R. W. Spekkens, and T. Fritz, “The Inflation Technique for Causal Inference with Latent Variables,”
J. Causal Inference 7 (2019).
[101]
N. Gisin, “The Elegant Joint Quantum Measurement and some conjectures about N-locality in the Triangle and
other Configurations,” arXiv:1708.05556 (2017).
[102]
T. C. Fraser and E. Wolfe, “Causal compatibility inequalities admitting quantum violations in the triangle
structure,” Phys. Rev. A 98, 022113 (2018).
[103]
C. Branciard, N. Gisin, and S. Pironio, “Characterizing the Nonlocal Correlations Created via Entanglement
Swapping,” Phys. Rev. Lett. 104, 170401 (2010).
[104]
F. Andreoli, G. Carvacho, L. Santodonato, R. Chaves, and F. Sciarrino, “Maximal violation of
n
-locality
inequalities in a star-shaped quantum network,” New J. Phys. 19, 113020 (2017).
[105]
A. Tavakoli, P. Skrzypczyk, D. Cavalcanti, and A. Ac´ın, “Nonlocal correlations in the star-network configuration,”
Phys. Rev. A 90, 062109 (2014).
[106]
D. Rosset, C. Branciard, T. J. Barnea, G. P¨utz, N. Brunner, and N. Gisin, “Nonlinear Bell inequalities tailored
for quantum networks,” Phys. Rev. Lett. 116, 010403 (2016).
[107] A. Tavakoli, “Bell-type inequalities for arbitrary noncyclic networks,” Phys. Rev. A 93, 030101(R) (2016).
[108]
J. Pearl, “On the Testability of Causal Models with Latent and Instrumental Variables,” in Proc. 11th Conf.
Uncertainty in Artificial Intelligence (1995) pp. 435–443.
[109]
B. Bonet, “Instrumentality Tests Revisited,” in Proc. 17th Conf. Uncertainty in Artificial Intelligence (2001)
pp. 48–55.
[110]
R. J. Evans, “Graphical methods for inequality constraints in marginalized DAGs,” in IEEE International
Workshop on Machine Learning for Signal Processing (2012).
[111]
R. Chaves, G. Carvacho, I. Agresti, V. D. Giulio, L. Aolita, S. Giacomini, and F. Sciarrino, “Quantum violation
of an instrumental test,” Nat. Phy. 14, 291 (2017).
[112]
T. Van Himbeeck, J. Bohr Brask, S. Pironio, R. Ramanathan, A. Bel´en Sainz, and E. Wolfe, “Quantum violations
in the Instrumental scenario and their relations to the Bell scenario,” Quantum 3, 186 (2019).
53