Beyond the Cabello-Severini-Winter framework: Making
sense of contextuality without sharpness of measurements
Ravi Kunjwal
Perimeter Institute for Theoretical Physics,
31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5.
September 4, 2019
We develop a hypergraph-theoretic frame-
work for Spekkens contextuality applied to
Kochen-Specker (KS) type scenarios that goes
beyond the Cabello-Severini-Winter (CSW)
framework. To do this, we add new
hypergraph-theoretic ingredients to the CSW
framework. We then obtain noise-robust non-
contextuality inequalities in this generalized
framework by applying the assumption of
(Spekkens) noncontextuality to both prepara-
tions and measurements. The resulting frame-
work goes beyond the CSW framework in both
senses, conceptual and technical. On the con-
ceptual level: 1) as in any treatment based on
the generalized notion of noncontextuality
`
a la
Spekkens, we relax the assumption of outcome
determinism inherent to the Kochen-Specker
theorem but retain measurement noncontex-
tuality, besides introducing preparation non-
contextuality, 2) we do not require the exclu-
sivity principle that pairwise exclusive mea-
surement events must all be mutually exclu-
sive as a fundamental constraint on mea-
surement events of interest in an experimen-
tal test of contextuality, given that this prop-
erty is not true of general quantum measure-
ments, and 3) as a result, we do not need to
presume that measurement events of interest
are “sharp” (for any definition of sharpness),
where this notion of sharpness is meant to im-
ply the exclusivity principle. On the techni-
cal level, we go beyond the CSW framework
in the following senses: 1) we introduce a
source events hypergraph besides the mea-
surement events hypergraph usually consid-
ered and define a new operational quantity
Corr that appears in our inequalities, 2) we de-
fine a new hypergraph invariant – the weighted
max-predictability that is necessary for our
analysis and appears in our inequalities, and 3)
our noise-robust noncontextuality inequalities
quantify tradeoff relations between three oper-
ational quantities Corr, R, and p
0
only one
of which (namely, R) corresponds to the Bell-
Ravi Kunjwal: rkunjwal@perimeterinstitute.ca
Kochen-Specker functionals appearing in the
CSW framework; when Corr = 1, the inequal-
ities formally reduce to CSW type bounds on
R. Along the way, we also consider in detail
the scope of our framework vis-
`
a-vis the CSW
framework, particularly the role of Specker’s
principle in the CSW framework, i.e., what the
principle means for an operational theory sat-
isfying it and why we don’t impose it in our
framework.
Contents
1 Introduction 2
2 Spekkens framework 5
2.1 Operational theory . . . . . . . . . . . 5
2.2 Ontological model . . . . . . . . . . . 6
2.3 Representation of coarse-graining . . . 7
2.3.1 Coarse-graining of measurements 7
2.3.2 Coarse-graining of preparations 8
2.4 Joint measurability (or compatibility) 9
2.5 Noncontextuality . . . . . . . . . . . . 9
2.6 An example of Spekkens contextuality:
the fair coin flip inequality . . . . . . . 10
2.7 Connection to Bell scenarios . . . . . . 12
3 Hypergraph approach to Kochen-
Specker scenarios in the Spekkens
framework 13
3.1 Measurements . . . . . . . . . . . . . . 13
3.1.1 Classification of probabilistic
models . . . . . . . . . . . . . . 14
3.1.2 Distinguishing two conse-
quences of Specker’s principle:
Structural Specker’s principle
vs. Statistical Specker’s principle 15
3.1.3 What does it mean for an oper-
ational theory to satisfy struc-
tural/statistical Specker’s prin-
ciple? . . . . . . . . . . . . . . 16
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arXiv:1709.01098v4 [quant-ph] 3 Sep 2019
3.1.4 Remark on the classification of
probabilistic models: why we
haven’t defined “quantum mod-
els” as those obtained from pro-
jective measurements . . . . . . 20
3.1.5 Scope of this framework . . . . 20
3.2 Sources . . . . . . . . . . . . . . . . . 21
4 A key hypergraph invariant: the
weighted max-predictability 23
5 Noise-robust noncontextuality inequali-
ties 24
5.1 Key notions from CSW . . . . . . . . 24
5.2 Key notion not from CSW:
source-measurement correlation, Corr 25
5.3 Obtaining the noise-robust noncontex-
tuality inequalities . . . . . . . . . . . 25
5.3.1 Expressing operational quanti-
ties in ontological terms . . . . 25
5.3.2 Derivation of the noncontextual
tradeoff for any graph G . . . . 26
5.3.3 When is the noncontextual
tradeoff violated? . . . . . . . . 27
5.4 Example: KCBS scenario . . . . . . . 27
6 Discussion 30
6.1 Measurement-measurement cor-
relations vs. source-measurement
correlations . . . . . . . . . . . . . . . 30
6.2 Can our noise-robust noncontextuality
inequalities be saturated by a noncon-
textual ontological model? . . . . . . . 30
6.2.1 The special case of facet-
defining Bell-KS inequalities:
Corr=1 . . . . . . . . . . . . . 30
6.2.2 The general case: Corr < 1 . . 30
6.3 Can trivial POVMs ever violate these
noncontextuality inequalities? . . . . . 31
6.3.1 The case p C
G
) . . . . . . 31
6.3.2 The case p
ConvHull(G
G
)|
ind
) . . . . . . 31
6.3.3 The general case p G
G
) . . 31
7 Conclusions 32
Acknowledgments 33
A Status of KS-contextuality as an experi-
mentally testable notion of nonclassical-
ity for POVMs in quantum theory 33
A.1 Limitations of KS-contextuality vis-`a-
vis POVMs . . . . . . . . . . . . . . . 34
A.1.1 KS-contextuality for POVMs in
the literature . . . . . . . . . . 34
A.1.2 Classifying probabilistic mod-
els: restriction of quantum
models to PVMs . . . . . . . . 35
A.2 Robustness of Bell nonlocality vis-`a-vis
POVMs . . . . . . . . . . . . . . . . . 36
B Ontological models without respecting
coarse-graining relations 37
B.1 How to construct a “KS-
noncontextual” ontological model
of the KCBS experiment [47] without
coarse-graining relations . . . . . . . . 37
B.2 How to construct a “preparation and
measurement noncontextual” ontologi-
cal model without coarse-graining rela-
tions . . . . . . . . . . . . . . . . . . . 37
C Trivial POVMs 38
C.1 Bell-CHSH scenario . . . . . . . . . . 38
C.2 CHSH-type contextuality scenario: 4-
cycle . . . . . . . . . . . . . . . . . . . 38
D The KS-uncolourable hypergraph Γ
18
40
References 41
1 Introduction
To say that quantum theory is counterintuitive, or
that it requires a revision of our classical intuitions,
requires us to be mathematically precise in our def-
inition of these classical intuitions. Once we have a
precise formulation of such classicality, we can begin
to investigate those features of quantum theory that
power its nonclassicality, i.e., its departure from our
classical intuitions, and thus prove theorems about
such nonclassicality. To the extent that a physical
theory is provisional, likely to be replaced by a better
theory in the future, it also makes sense to articu-
late such notions of classicality in as operational a
manner as possible. By ‘operational’, we refer to a
formulation of the theory that takes the operations
preparations, measurements, transformations that
can be carried out in an experiment as primitives and
which specifies the manner in which these operations
combine to produce the data in the experiment. Such
an operational formulation often suggests generaliza-
tions of the theory that can then be used to better
understand its axiomatics [13]. At the same time,
an operational formulation also lets us articulate our
notions of nonclassicality in a manner that is experi-
mentally testable and thus allows us to leverage this
nonclassicality in applications of the theory. Indeed,
a key area of research in quantum foundations and
quantum information is the development of methods
to assess nonclassicality in an experiment under min-
imal assumptions on the operational theory describ-
ing it. The paradigmatic example of this is the case of
Bell’s theorem and Bell experiments [411], where any
operational theory that is non-signalling between the
different spacelike separated wings of the experiment
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is allowed. The notion of classicality at play in Bell’s
theorem is the assumption of local causality: any non-
signalling theory that violates the assumption of local
causality is said to exhibit nonclassicality by the lights
of Bell’s theorem.
More recently, much work [1217] has been devoted
to obtaining constraints on operational statistics that
follow from a generalized notion of noncontextuality
proposed by Spekkens [18]. This notion of classicality
[18] has its roots in the Kochen-Specker (KS) theo-
rem [19], a no-go theorem that rules out the possibility
that a deterministic underlying ontological model [20]
could reproduce the operational statistics of (projec-
tive) quantum measurements in a manner that sat-
isfies the assumption of KS-noncontextuality. KS-
noncontextuality is the notion of classicality at play
in the Kochen-Specker theorem. The Spekkens frame-
work abandons the assumption of outcome determin-
ism [18] the idea that the ontic state of a system
fixes the outcome of any measurement deterministi-
cally that is intrinsic to KS-noncontextuality. It also
applies to general operational theories and extends
the notion of noncontextuality to general experimen-
tal procedures preparations, transformations, and
measurements rather than measurements alone.
Parallel to work along the lines of Spekkens [18],
work seeking to directly operationalize the Kochen-
Specker theorem (rather than revising the notion
of noncontextuality at play) culminated in two re-
cent approaches that classify theories by the de-
gree to which they violate the assumption of KS-
noncontextuality: the graph-theoretic framework of
Cabello, Severini, and Winter (CSW) [21, 22], where a
general approach to obtaining graph-theoretic bounds
on linear Bell-KS functionals was proposed, and the
related hypergraph framework of Ac´ın, Fritz, Lev-
errier, and Sainz (AFLS) [23], where an approach
to characterizing sets of correlations was proposed.
The CSW framework relates well-known graph in-
variants to: 1) upper bounds on Bell-KS inequali-
ties that follow from KS-noncontextuality, 2) upper
bounds on maximum quantum violations of these in-
equalities that can be obtained from projective mea-
surements, and 3) upper bounds on their violation
in general probabilistic theories [24] denoted E1
which satisfy the “exclusivity principle” [22]. Com-
plementary to this, the AFLS framework uses graph
invariants in the service of deciding whether a given
assignment of probabilities to measurement outcomes
in a KS-contextuality experiment belongs to a partic-
ular set of correlations; they showed that membership
in the quantum set of correlations (defined only for
projective measurements in quantum theory) cannot
be witnessed by a graph invariant, cf. Theorem 5.1.3
of Ref. [23]. Another recent approach due to Abram-
sky and Brandenburger [25] employs sheaf-theoretic
ideas to formulate KS-contextuality.
A key achievement of the frameworks of Refs. [22,
23, 25] is a formal unification of Bell scenarios
with KS-contextuality scenarios, treating them on
the same footing. Indeed, the perspective there is
to consider Bell scenarios as a special case of KS-
contextuality scenarios. What is lost in this math-
ematical unification, however, is the fact that Bell-
locality and KS-noncontextuality have physically dis-
tinct, if related, motivations. The physical situation
that Bell’s theorem refers to requires (at least) two
spacelike separated labs (where local measurements
are carried out) so that the assumption of local causal-
ity (or Bell-locality) can be applied.
1
On the other
hand, the physical situation that the Kochen-Specker
theorem refers to does not require spacelike separa-
tion as a necessary ingredient and one can there-
fore consider experiments in a single lab. However,
the assumption of KS-noncontextuality entails out-
come determinism [18], something not required by
local causality in Bell scenarios.
2
This difference
in the physical situation for the two kinds of ex-
periments is one of the reasons for generalizing KS-
noncontextuality to the notion of noncontextuality in
the Spekkens framework [18] (so that outcome deter-
minism is not assumed) while leaving Bell’s notion of
local causality untouched.
In the present paper we build a bridge from the
CSW approach, where KS-noncontextual correlations
are bounded by Bell-KS inequalities, to noise-robust
noncontextuality inequalities in the Spekkens frame-
work [18]. That is, we show how the constraints from
KS-noncontextuality in the framework of Ref. [22]
translate to constraints from generalized noncontex-
tuality in the framework of Ref. [18]. The resulting
operational criteria for contextuality `a la Spekkens
1
What do we mean by whether an assumption “can be ap-
plied”? Of course, mathematically, one can “apply” any as-
sumption one wants in the service of proving a theorem. But
insofar as the mathematics here is trying to model a real exper-
iment, the consistency of those assumptions with some essen-
tial facts of the experiment is the minimal requirement for any
no-go theorem derived from such assumptions to be physically
interesting. Hence, in the presence of signalling (implying the
absence of spacelike separation), it makes no sense to assume
local causality in a Bell experiment and derive the resulting
Bell inequalities: such an assumption on the ontological model
is already in conflict with the fact of signalling across the labs
and no Bell inequalities are needed to witness this fact. Bell in-
equalities only become physically interesting when the theories
being compared relative to them are all non-signalling: if the
experiment itself is signalling, any non-signalling description
locally causal, quantum, or in a general probabilistic theory
(GPT) is ipso facto ruled out.
2
Note that this assumption of outcome determinism doesn’t
affect the conclusions in a Bell scenario even if one adopted
it because of Fine’s theorem [26]: a locally deterministic on-
tological model entails the same set of (Bell-local) correlations
as a locally causal ontological model. Relaxing outcome deter-
minism, however, doesn’t mean the same thing for the kinds
of experiments envisaged by the Kochen-Specker theorem in
particular, it doesn’t mean that models satisfying factorizabil-
ity `a la Ref. [25] are the most general outcome-indeterministic
models – and thus considerations parallel to Fine’s theorem [26]
do not apply, cf. [27, 28].
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are noise-robust and therefore applicable to arbi-
trary positive operator-valued measures (POVMs)
and mixed states in quantum theory. Note that the
insights gleaned from frameworks such as those of
Refs. [22, 23, 25] regarding Bell nonlocality require
no revision in our approach. It is only in the appli-
cation of such frameworks (in particular, the CSW
framework) to the question of contextuality that we
seek to propose an alternative hypergraph framework
(formalizing Spekkens contextuality [18]) that is more
operationally motivated for experimental situations
where one cannot appeal to spacelike separation to
justify locality of the measurements.
3
For Kochen-
Specker type experimental scenarios, we will con-
sider the twin notions of preparation noncontextu-
ality and measurement noncontextuality taken to-
gether as a notion of classicality to obtain noise-
robust noncontextuality inequalities that generalize
the KS-noncontextuality inequalities of CSW. These
inequalities witness nonclassicality even when quan-
tum correlations arising from arbitrary (i.e., possibly
nonprojective) quantum measurements on any quan-
tum state are allowed. A key innovation of this ap-
proach is that it treats all measurements in an oper-
ational theory on an equal footing. No definition of
“sharpness” [2931] is needed to justify or derive non-
contextuality inequalities in this approach. Further-
more, if certain idealizations are presumed about the
operational statistics, then these inequalities formally
recover the usual Bell-KS inequalities `a la CSW. The
Bell-KS inequalities can be viewed as an instance of
the classical marginal problem [2527, 32, 33], i.e.,
as constraints on the (marginal) probability distri-
butions over subsets of a set of observables that fol-
low from requiring the existence of global joint prob-
ability distribution over the set of all observables.
Since the Bell-KS inequalities are only recovered un-
der certain idealizations, but not otherwise, the noise-
robust noncontextuality inequalities we obtain can-
not in general be viewed as arising from a classical
marginal problem. Hence, they cannot be understood
within existing frameworks that rely on this (reduc-
tion to the classical marginal problem) property to
formally unify the treatment of Bell-nonlocality and
KS-contextuality [22, 23, 25]. This is a crucial dis-
tinction relative to the usual Bell-KS inequality type
witnesses of KS-contextuality.
This paper is based on a previous contribution [16]
that laid the conceptual groundwork for the progress
we make here. Besides the noise-robust noncontex-
tuality inequalities that generalize constraints from
KS-noncontextuality in the CSW framework leverag-
ing the graph invariants of CSW [22] (cf. Section 5),
3
Nor the sharpness of the measurements to justify outcome
determinism. We discuss these issues in detail in particular,
the physical basis of KS-noncontextuality vis-`a-vis Bell-locality
and how that influences our framework – in Appendix A for the
interested reader.
the contributions of this paper also include:
An exposition of Specker’s principle and how dif-
ferent implications of it (e.g., consistent exclu-
sivity [23]) for a given operational theory arise in
the hypergraph framework (cf. Sections 3.1.2 and
3.1.3), in particular the results in Theorems 1, 2,
and Corollary 1.
Introduction of a hypergraph invariant the
weighted max-predictability that is key to
our noise-robust noncontextuality inequalities,
cf. Section 4. This invariant is also key to the
hypergraph framework of Ref. [34] which is com-
plementary to the present framework.
A detailed discussion of how KS-
noncontextuality for POVMs has been previously
treated in the literature and the limitations of
those treatments, cf. Appendices A and C.
Also, unlike for the case of KS-noncontextuality
inequalities, we show that trivial POVMs can
never violate our noise-robust noncontextuality
inequalities, cf. Section 6.3.
A discussion of coarse-graining relations in Sec-
tion 2.3 and their importance for contextual-
ity no-go theorems, in particular a discussion of
ontological models that do not respect coarse-
graining relations in Appendix B. We show,
in Appendix B, how relaxing the constraint
from coarse-graining relations on an ontological
model renders either notion of noncontextuality
whether Kochen-Specker [19] or Spekkens [18]
vacuous.
A discussion, by example, of why our generaliza-
tion of the CSW framework cannot accommodate
contextuality scenarios that are KS-uncolourable
in Appendix D and why one needs a distinct
framework, i.e., the framework of Ref. [34], to
treat KS-uncolourable scenarios.
The structure of this paper follows: Section 2 reviews
the Spekkens framework for generalized noncontextu-
ality [18]. Section 3 introduces a hypergraph frame-
work that shares features of traditional frameworks
for KS-contextuality [22, 23] but is also augmented
(relative to these traditional frameworks) with the in-
gredients necessary for obtaining noise-robust noncon-
textuality inequalities. In particular, its subsections
3.1.2 and 3.1.3 discuss Specker’s principle [35] and
define its different implications for contextuality sce-
narios `a la Ref. [23]. Section 4 defines a new hyper-
graph invariant the weighted max-predictability
that we need later on as a crucial new ingredient in
our inequalities. Section 5 obtains noise-robust non-
contextuality inequalities within the framework de-
fined in Section 3 and using the hypergraph invariant
of Section 4 in addition to two graph invariants from
the CSW framework [22]. These inequalities can be
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seen as special cases of the general approach outlined
in Ref. [16]. In Section 6, we include discussions on
various features of our noise-robust noncontextuality
inequalities, in particular the fact that trivial POVMs
can never violate them. Section 7 concludes with some
open questions and directions for future research.
2 Spekkens framework
We concern ourselves with prepare-and-measure ex-
periments. A schematic of such an experiment is
shown in Figure 1 where, for the sake of simplicity, we
imagine a single source device that can perform any
preparation procedure of interest (rather than a col-
lection of source devices, each implementing a partic-
ular preparation procedure) and a single measurement
device that can perform any measurement procedure
of interest (rather than a collection of measurement
devices, each implementing a particular measurement
procedure). Note that this is just a conceptual ab-
straction: in particular, the various possible measure-
ment settings on the measurement device may, for ex-
ample, correspond to incompatible measurement pro-
cedures in quantum theory. The fact that we repre-
sent the different measurement settings by choices of
knob settings M M on a single measurement de-
vice does not mean that it’s physically possible to im-
plement all the measurement procedures represented
by M jointly; it only means that the experimenter
can choose to implement any of the measurements in
the set M in a particular prepare-and-measure exper-
iment. The same is true for our abstraction of prepa-
ration procedures to knob settings (S S) and out-
comes (s V
S
) of a single source device: it’s not that
the same device can physically implement all possible
preparation procedures; it’s just that an experimenter
can choose to implement any procedure in the set S
in a particular prepare-and-measure experiment.
We will consider two levels of description of
prepare-and-measure experiments represented by
Fig. 1: operational and ontological. The operational
description will be specified by an operational theory
that takes source and measurement devices as primi-
tives and describes the experiment solely in terms of
the probabilities associated to their input/output be-
haviour. The ontological description will be specified
by an ontological model that takes the system that
passes between the source and measurement devices
as primitive and describes the experiment in terms of
probabilities associated to properties of this system,
deriving the operational description as a consequence
of coarse-graining over these properties. Let us look
at each description in turn.
2.1 Operational theory
We now describe the various components of Fig. 1 in
more detail. The source device has a source setting
Measurement
Source
Figure 1: A prepare-and-measure experiment.
labelled by S that can be chosen from a set S. The
set S represents, in general, some subset of the set of
all source settings, S , that are admissible in the op-
erational theory, i.e., S S . In a particular prepare-
and-measure experiment, S will typically be a finite
set of source settings. Choosing the setting S pre-
pares a system according to an ensemble of prepara-
tion procedures, denoted {(p(s|S), P
[s|S]
)}
sV
S
, where
{p(s|S)}
sV
S
is a probability distribution over the
preparation procedures {P
[s|S]
}
sV
S
in the ensemble.
This means that the source device has one classical
input S and two outputs: one output is a classical
label s V
S
identifying the preparation procedure
(in the ensemble {(p(s|S), P
[s|S]
)}
sV
S
) that is car-
ried out when source outcome s is observed for source
setting S (this source event is denoted [s|S]), and the
other output is a system prepared according to the
source event [s|S], i.e., preparation procedure P
[s|S]
,
with probability p(s|S). Thus, the assemblage of pos-
sible ensembles that the source device can prepare can
be denoted by {{(p(s|S), P
[s|S]
)}
sV
S
}
SS
.
On the other hand, the measurement device has
two inputs, one a classical input M M specifying
the choice of measurement setting to be implemented,
and the other input receives the system prepared ac-
cording to prepartion procedure P
[s|S]
and on which
this measurement M is carried out. The measurement
device has one classical output m V
M
denoting the
outcome of the measurement M implemented on a
system prepared according to P
[s|S]
, and which occurs
with probability p(m|M, S, s). The set M represents,
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in general, some subset of the set of all measurement
settings, M , that are admissible in the operational
theory, i.e., M M . In a particular prepare-and-
measure experiment, M will typically be a finite set
of measurement settings.
We will be interested in the operational joint prob-
ability p(m, s|M, S) p(m|M, S, s)p(s|S) for this
prepare-and-measure experiment for various choices
of M M, S S. Note how this operational de-
scription takes as primitive the operations carried out
in the lab and restricts itself to specifying the prob-
abilities of classical outcomes (i.e., m, s) given some
interventions (i.e., classical inputs, M, S). So far, we
haven’t assumed any structure on the operational the-
ory describing the schematic of Fig. 1 beyond the fact
that it is a catalogue of input/output probabilities
{{p(m, s|M, S) [0, 1]}
mV
M
,sV
S
}
MM,SS
for various interventions S S and M M that we
will consider in a prepare-and-measure experiment.
We now require more structure in the operational the-
ory underlying this experiment, beyond a mere spec-
ification of these probabilities.
We require that the operational theory admits
equivalence relations that partition experimental pro-
cedures of any type, whether preparations or measure-
ments, into equivalence classes of that type. These
equivalence relations are defined relative to the op-
erational probabilities (not necessarily restricted to a
particular prepare-and-measure experiment) that are
admissible in the theory. We will call these equiv-
alence relations “operational equivalences”, in keep-
ing with standard terminology [18]. This means that
any distinctions of labels between procedures in an
equivalence class of procedures do not affect the oper-
ational probabilities associated with the procedures.
We specify these equivalence relations for measure-
ment and preparation procedures below.
Two measurement events [m|M] and [m
0
|M
0
] are
said to be operationally equivalent, denoted [m|M ] '
[m
0
|M
0
], if there exists no source event in the opera-
tional theory that can distinguish them, i.e.,
p(m, s|M, S) = p(m
0
, s|M
0
, S) [s|S], s V
S
, S S .
(1)
Note that the statistical indistinguishability of [m|M]
and [m
0
|M
0
] must hold for all possible source settings
S in the operational theory, not merely the source
settings S that are of direct interest in a particular
prepare-and-measure experiment. Hence, the “dis-
tinction of labels”, [m|M] or [m
0
|M
0
], is empirically
inconsequential since the two procedures are, in prin-
ciple, indistinguishable by the lights of the operational
theory.
Similarly, two source events [s|S] and [s
0
|S
0
] are said
to be operationally equivalent, denoted [s|S] ' [s
0
|S
0
],
if there exists no measurement event in the opera-
tional theory that can distinguish them, i.e.,
p(m, s|M, S) = p(m, s
0
|M, S
0
),
[m|M], m V
M
, M M . (2)
Again, the statistical indistinguishability of [s|S] and
[s
0
|S
0
] must hold for all possible measurement settings
M , not merely those (i.e., M) that are of direct inter-
est in a particular prepare-and-measure experiment.
Similar to measurement events, the “distinction of la-
bels”, [s|S] or [s
0
|S
0
], is empirically inconsequential
since the two procedures are, in principle, indistin-
guishable by the lights of the operational theory.
Given this equivalence structure for preparation
and measurement procedures in the operational the-
ory, we can now formalize the notion of a context:
Definition 1. A context is any distinction of labels
between operationally equivalent procedures in the op-
erational theory.
To see concrete examples of the kinds of contexts
that will be of interest to us in this paper, con-
sider quantum theory. Any mixed quantum state ad-
mits multiple convex decompositions in terms of other
quantum states, i.e., it can be prepared by coarse-
graining over distinct ensembles of quantum states,
each ensemble denoted by a different label. In this
case, the “distinction of labels” between different de-
compositions denotes a distinction of preparation en-
sembles, which instantiates our notion of a prepara-
tion context. Similarly, a given positive operator can
be implemented by different positive operator-valued
measures (POVMs), and the distinction of labels de-
noting these different POVMs instantiates our notion
of a measurement context.
2.2 Ontological model
Given the operational description of the experiment
in terms of probabilities p(m, s|M, S), we want to
explore the properties of any underlying ontological
model for this operational description. Any such on-
tological model, defined within the ontological mod-
els framework [20], takes as primitive the physical
system (rather than operations on it) that passes
between the source and measurement devices, i.e.,
its basic objects are ontic states of the system, de-
noted λ Λ, that represent intrinsic properties of
the physical system. When a preparation proce-
dure [s|S] is carried out, the source device samples
from the space of ontic states Λ according to a prob-
ability distribution {µ(λ|S, s) [0, 1]}
λΛ
, where
P
λΛ
µ(λ|S, s) = 1, and the joint distribution over
s and λ given S, i.e., {µ(λ, s|S)}
λΛ
, is given by
µ(λ, s|S) µ(λ|S, s)p(s|S). On the other hand, when
a system in ontic state λ is input to the measure-
ment device with measurement setting M M, the
probability distribution over the measurement out-
comes is given by {ξ(m|M, λ) [0, 1]}
mV
M
, where
Accepted in Quantum 2019-09-01, click title to verify. Published under CC-BY 4.0. 6
P
mV
M
ξ(m|M, λ) = 1. The operational statistics
{{p(m, s|M, S) [0, 1]}
mV
M
,sV
S
}
MM,SS
results from a coarse-graining over λ, i.e.,
p(m, s|M, S) =
X
λΛ
ξ(m|M, λ)µ(λ, s|S), (3)
for all m V
M
, s V
S
, M M, S S.
Note that the definition of an ontological model
above extends to the definition of an ontological model
of the operational theory (as opposed to a particular
fragment of the theory representing the experiment)
when we take M = M and S = S .
2.3 Representation of coarse-graining
We will now specify how coarse-graining of procedures
in a prepare-and-measure experiment is represented
in its description, whether operational or ontological.
Namely, if a procedure is defined as a coarse-graining
of other procedures, then we require that the repre-
sentation of such a procedure is defined by the same
coarse-graining of the representation of the other pro-
cedures.
4
Implicit in this discussion is the assump-
tion that the operational theory allows one to define
new procedures in the set M or S by coarse-graining
other procedures in these sets, i.e., both M and S
are closed under coarse-grainings. In particular, one
can consider coarse-graining measurement and source
settings (belonging to sets M and S, respectively) ac-
tually implemented in the lab to define new measure-
ment and source settings that belong to M \M and
S \S, respectively.
5
2.3.1 Coarse-graining of measurements
Let us see how this works for the case of measurement
procedures: if a measurement procedure M with mea-
surement events {[m|M ]}
mV
M
is defined as a coarse-
graining of another measurement procedure
˜
M with
measurement events {[ ˜m|
˜
M]}
˜mV
˜
M
, symbolically de-
noted by
[m|M]
X
˜m
p(m|˜m)[ ˜m|
˜
M],
where m, ˜m : p(m|˜m) {0, 1},
X
m
p(m|˜m) = 1,
(4)
4
Quantum theory is an example of an operational theory
that satisfies this requirement because of the linearity of the
Born rule with respect to both preparations and measurements.
The same is true, more generally, of general probabilistic theo-
ries (GPTs) [1, 24]. We require this feature in any ontological
model as well, regardless of its (non)contextuality.
5
Similarly, we also allow probabilistic mixtures of (prepa-
ration or measurement) procedures in the operational theory
to define new procedures, i.e., the theory is convex. See the
last paragraph of Section 2.5 for the role of this convexity in
experimental tests of contextuality and Section 2.6 for an ex-
ample where a probabilistic mixture of measurement settings
is required in a proof of contextuality.
then its representation in the operational description
as well as in the ontological description satisfies this
coarse-graining relation.
6
More explicitly, the coarse-
graining relation of Eq. (4) denotes the following post-
processing of
˜
M: for each m V
M
, relabel each
outcome ˜m V
˜
M
to outcome m with probability
p(m|˜m) {0, 1}; the logical disjunction of those ˜m
which are relabelled to m with probability 1 then de-
fines the measurement event [m|M ]. Now, in the op-
erational theory, this post-processing is represented
by
[s|S], where s V
S
, S S :
p(m, s|M, S)
X
˜m
p(m|˜m)p( ˜m, s|
˜
M, S), (5)
and in the ontological model it is represented by
λ Λ : ξ(m|M, λ)
X
˜m
p(m|˜m)ξ( ˜m|
˜
M, λ). (6)
As an example, consider a three-outcome measure-
ment
˜
M with outcomes ˜m {1, 2, 3}, which can
be classically post-processed to obtain a two-outcome
measurement M with outcomes m {0, 1}, such that
p(m = 0|˜m = 1) = p(m = 0|˜m = 2) = 1 and
p(m = 1|˜m = 3) = 1. The measurement events of
M are then just
[m = 0|M ] [ ˜m = 1|
˜
M] + [ ˜m = 2|
˜
M], (7)
[m = 1|M ] [ ˜m = 3|
˜
M], (8)
where the + sign denotes (just as the summation
sign in the definition of [m|M] in Eq. (4) did) logical
disjunction, i.e., measurement event [m = 0|M] is said
to occur when [ ˜m = 1|
˜
M] or [ ˜m = 2|
˜
M] occurs. The
operational and ontological representations of these
measurement events are then given by
[s|S], where s V
S
, S S :
p(m = 0, s|M, S)
2
X
˜m=1
p( ˜m, s|
˜
M, S), (9)
p(m = 1, s|M, S) p( ˜m = 3, s|
˜
M, S), (10)
λ Λ :
ξ(m = 0|M, λ)
2
X
˜m=1
ξ( ˜m|
˜
M, λ), (11)
ξ(m = 1|M, λ) ξ( ˜m = 3|
˜
M, λ). (12)
This requirement on the representation of coarse-
graining of measurements is particularly important
(and often implicit) when the notion of a mea-
surement context is instantiated by compatibility
6
Note that Eq. (4) is not an operational equivalence between
independent procedures. It is a definition of a new procedure
obtained by coarse-graining another procedure.
Accepted in Quantum 2019-09-01, click title to verify. Published under CC-BY 4.0. 7
(or joint measurability), as in the case of KS-
contextuality, where one needs to consider coarse-
grainings of distinct measurements. For example,
consider a measurement setting M
12
with outcomes
(m
1
, m
2
) V
1
× V
2
that is coarse-grained over
m
2
to define an effective measurement setting M
(2)
1
with measurement events {[m
1
|M
(2)
1
]}
m
1
V
1
. Sym-
bolically, [m
1
|M
(2)
1
]
P
m
2
[(m
1
, m
2
)|M
12
], which
is represented in the operational theory as [s|S] :
p(m
1
, s|M
(2)
1
, S)
P
m
2
p((m
1
, m
2
), s|M
12
, S) and
in the ontological model as λ : ξ(m
1
|M
(2)
1
, λ)
P
m
2
ξ((m
1
, m
2
)|M
12
, λ). Similarly, consider another
measurement setting M
13
with outcomes (m
1
, m
3
)
V
1
× V
3
that is coarse-grained over m
3
to de-
fine an effective measurement setting M
(3)
1
with
measurement events {[m
1
|M
(3)
1
]}
m
1
V
1
. Symboli-
cally, [m
1
|M
(3)
1
]
P
m
3
[(m
1
, m
3
)|M
13
], which is
represented in the operational theory as [s|S] :
p(m
1
, s|M
(3)
1
, S)
P
m
3
p((m
1
, m
3
), s|M
13
, S) and
in the ontological model as λ : ξ(m
1
|M
(3)
1
, λ)
P
m
3
ξ((m
1
, m
3
)|M
13
, λ).
Now, imagine that the following oper-
ational equivalence holds at the opera-
tional level: [m
1
|M
(2)
1
] ' [m
1
|M
(3)
1
]. KS-
noncontextuality is then the assumption that
P
m
2
ξ((m
1
, m
2
)|M
12
, λ) =
P
m
3
ξ((m
1
, m
3
)|M
13
, λ)
(i.e., ξ(m
1
|M
(2)
1
, λ) = ξ(m
1
|M
(3)
1
, λ)) for all λ and
that ξ((m
1
, m
2
)|M
12
, λ), ξ((m
1
, m
3
)|M
13
, λ) {0, 1}
for all λ. This assumption applied to multiple
(compatible) subsets of a set of carefully chosen
measurements can then provide a proof of the KS
theorem, i.e., there exist sets of measurements in
quantum theory such that their operational statis-
tics cannot be emulated by a KS-noncontextual
ontological model.
The key point here is this: the requirement that
coarse-graining relations between measurements be
respected by their representations in the ontological
model is independent of the KS-(non)contextuality of
the ontological model.
7
However, this requirement is
necessary for the assumption of KS-noncontextuality
to produce a contradiction with quantum theory; on
the other hand, a KS-contextual ontological model
(while respecting the coarse-graining relations) can
always emulate quantum theory. In this sense, the
representation of coarse-grainings is baked into an on-
tological model from the beginning (just as it is baked
into an operational description), before any claims
about its (non)contextuality.
8
7
In our example, this requirement has to do with the defini-
tions of ξ(m
1
|M
(2)
1
, λ) and ξ(m
1
|M
(3)
1
, λ), not their ontological
equivalence. The ontological equivalence only comes into play
when invoking KS-noncontextuality.
8
One could, of course, choose to not respect the coarse-
graining relations and define a notion of an ontological model
without them. In such a model, one could treat every mea-
2.3.2 Coarse-graining of preparations
Let us now consider the representation of coarse-
grainings for preparation procedures. This works
in a way similar to the case of measurement proce-
dures which we have already outlined. If an ensemble
of source events {[s|S]}
sV
S
is defined as a coarse-
graining of another ensemble, {[˜s|
˜
S]}
˜sV
˜
S
, symboli-
cally denoted as
[s|S]
X
˜s
p(s|˜s)[˜s|
˜
S], where
s, ˜s : p(s|˜s) {0, 1},
X
s
p(s|˜s) = 1, (13)
then its representation should satisfy the same coarse-
graining relation in any description, operational or
ontological. More explicitly, this coarse-graining de-
notes the following post-processing: for any s V
S
,
relabel each outcome ˜s V
˜
S
to outcome s with prob-
ability p(s|˜s) {0, 1}; the logical disjunction of those
˜s which are relabelled to s with probability 1 then de-
fines the source event [s|S]. Now, in the operational
theory, this coarse-graining is represented by
[m|M], where m V
M
, M M :
p(m, s|M, S)
X
˜s
p(s|˜s)p(m, ˜s|M,
˜
S), (14)
and in the ontological model it is represented by
λ Λ : µ(λ, s|S)
X
˜s
p(s|˜s)µ(λ, ˜s|
˜
S). (15)
In this paper, we will focus on a specific type of coarse-
graining: namely, completely coarse-graining over the
outcomes of a source setting, say {[˜s|
˜
S]}
˜sV
˜
S
, to yield
an effective one-outcome source-setting, denoted
˜
S
>
,
associated with a single source event {[>|
˜
S
>
]}, where
[>|
˜
S
>
]
P
˜s
[˜s|
˜
S]. In the operational theory, this
coarse-graining is represented by
[m|M], where m V
M
, M M :
p(m, >|M,
˜
S
>
)
X
˜s
p(m, ˜s|M,
˜
S), (16)
and in the ontological model it is represented by
λ Λ : µ(λ, >|
˜
S
>
)
X
˜s
µ(λ, ˜s|
˜
S). (17)
surement obtained by coarse-graining another (parent) mea-
surement as a fundamentally new measurement with response
functions not respecting the coarse-graining relations with the
parent measurement’s response functions, even if such coarse-
graining relations are respected in the operational description.
Such an ontological model, however, will not be able to ar-
ticulate the ingredients needed for a proof of the KS theorem
and we will not consider it here. Indeed, in the absence of
the requirement that coarse-graining relations be respected in
an ontological model, one can easily construct an ontological
model that is “KS-noncontextual” for any operational theory.
The interested reader may look at Appendix B for more details,
perhaps after looking at Section 2.5 for the relevant definitions
of noncontextuality.
Accepted in Quantum 2019-09-01, click title to verify. Published under CC-BY 4.0. 8
Hence, we use the notation [>|
˜
S
>
] to denote the
source event that “at least one of the source outcomes
in the set V
˜
S
occurs for source setting
˜
S (i.e., the
logical disjunction of ˜s V
˜
S
), formally denoting the
choice of
˜
S and the subsequent coarse-graining over ˜s
by the “source setting”
˜
S
>
and the definite outcome
of this source setting by >”. This source event al-
ways occurs, i.e., p(>|
˜
S
>
) = 1, so p(m, >|M,
˜
S
>
) =
p(m|M,
˜
S
>
, >) and µ(λ, >|
˜
S
>
) = µ(λ|
˜
S
>
, >).
This notion of coarse-graining over all the outcomes
of a source setting allows us to define a notion of
operational equivalence between the source settings
themselves. More precisely, two source settings S and
S
0
are said to be operationally equivalent, denoted
[>|S
>
] ' [>|S
0
>
], if no measurement event can distin-
guish them once all their outcomes are coarse-grained
over, i.e.,
X
sV
S
p(m, s|M, S) =
X
s
0
V
S
0
p(m, s
0
|M, S
0
)
[m|M], m V
M
, M M . (18)
In quantum theory, this would correspond
to the operational equivalence
P
s
p(s|S)ρ
[s|S]
=
P
s
0
p(s
0
|S
0
)ρ
[s
0
|S
0
]
for the density operator obtained
by completely coarse-graining over two distinct en-
sembles of quantum states, {(p(s|S), ρ
[s|S]
)}
sV
S
and
{(p(s
0
|S
0
), ρ
[s
0
|S
0
]
)}
s
0
V
S
0
on some Hilbert space H.
2.4 Joint measurability (or compatibility)
A given measurement procedure, {[m|M]}
mV
M
for
some M M , in the operational description can
be coarse-grained in many different ways to define
new effective measurement procedures. The coarse-
grained measurement procedures thus obtained from
{[m|M]}
mV
M
are then said to be jointly measurable
(or compatible), i.e., they can be jointly implemented
by the same measurement procedure {[m|M ]}
mV
M
which we refer to as their parent or joint measure-
ment. Formally, a set C of measurement procedures
{{[m
i
|M
i
]}
m
i
V
M
i
i {1, 2, 3, . . . , |C|}}
is said to be jointly measurable (or compatible) if it
arises from coarse-grainings of a single measurement
procedure M M , i.e., for all {[m
i
|M
i
]}
m
i
V
M
i
C
[m
i
|M
i
]
X
mV
M
p(m
i
|m)[m|M], (19)
where for all i, m, m
i
: p(m
i
|m) ∈ {0, 1} and
P
m
i
V
M
i
p(m
i
|m) = 1. In terms of the operational
probabilities, this means that
[s|S], s V
S
, S S and ∀{[m
i
|M
i
]}
m
i
V
M
i
C :
p(m
i
, s|M
i
, S)
X
mV
M
p(m
i
|m)p(m, s|M, S). (20)
If, on the other hand, a set of measurement proce-
dures cannot arise from coarse-grainings of any single
measurement procedure, then the measurement pro-
cedures in the set are said to be incompatible, i.e.,
they cannot be jointly implemented.
Note that we will also often refer to a measurement
procedure {[m
i
|M
i
]}
m
i
V
M
i
by just its measurement
setting, M
i
, and thus speak of the (in)compatibility
of measurement settings. Another notion that we will
need to refer to is the joint measurability of measure-
ment events: a set of measurement events that arise
as outcomes of a single measurement setting are said
to be jointly measurable, e.g., all the measurement
events in {[m|M ]}
mV
M
are jointly measurable since
they arise as outcomes of a single measurement set-
ting M.
As a quantum example, consider a commuting pair
of projective measurements, say {Π
1
, I Π
1
} and
{Π
2
, I Π
2
}, where Π
1
and Π
2
are projectors on
some Hilbert space H such that Π
1
Π
2
= Π
2
Π
1
and
I is the identity operator on H. This pair is jointly
implementable since they can be obtained by coarse-
graining the outcomes of the joint projective measure-
ment given by {Π
1
Π
2
, Π
1
(I Π
2
), (I Π
1
2
, (I
Π
1
)(I Π
2
)}.
2.5 Noncontextuality
It is always possible to build an ontological model
reproducing the predictions of any operational the-
ory, while respecting the coarse-graining relations.
9
A trivial example of such an ontological model is one
where ontic states λ are identified with the prepara-
tion procedures P
[s|S]
(where s V
S
and S S )
and we have µ(λ, s|S) δ
λ,λ
[s|S ]
p(s|S), where ontic
state λ
[s|S]
is the one deterministically sampled by the
preparation procedure P
[s|S]
. Further, the response
functions are identified with operational probabili-
ties as ξ(m|M, λ
[s|S]
) p(m|M, S, s). Then we have
P
λΛ
ξ(m|M, λ)µ(λ, s|S) = ξ(m|M, λ
[s|S]
)p(s|S) =
p(m, s|M, S). Also, coarse-graining relations of the
type [ ˜m|
˜